Book cover animation

Refer to the book as: Dul, J. (2026). Necessary Condition Analysis (NCA): Principles and Application. Chapman & Hall/CRC Press.

Preface

“Is coffee a necessary condition for you?” Someone asked me this question after hearing that I had taken time away in Italy to focus on writing my book, and that I like Italy’s coffee culture. “No, it is not; it only helps.” Focus, time, perseverance, and several other things are true necessary conditions. Without any of them, the book would not exist.

And here it is! In the book I present the principles of Necessary Condition Analysis (NCA) and their application in empirical studies in research and practice. NCA is an approach that uses necessity causal logic to identify necessary conditions from data. The core idea is that a necessary condition must be present to make the outcome possible, and that its absence guarantees the absence of the outcome. Compensation is not possible.

My intention with this book is to help researchers, data analysts, practitioners, and others develop a deeper understanding of the NCA approach and apply it effectively and with high-quality standards. The book brings together the fundamentals of NCA and integrates the most recent methodological advancements and tools. It synthesizes ideas from my earlier publications that describe NCA at introductory levels (e.g., Dul, 2016b, 2020) as well as more advanced treatments (Dul, Van der Laan, et al., 2020; Dul, 2021, 2024a), while also introducing more background, new topics and recent developments. From 2021 to 2026, a regularly updated online predecessor of this book was available (Dul, 2021), which is now archived and integrated into the present book.

The first chapter of this book provides a general introduction to the possibilities and current use of NCA. Next, Part I contains five chapters on the principles of NCA, covering NCA’s causality, theory, mathematics, statistics, and its credibility for identifying necessity from data. Part II has six chapters about the application of these principles in empirical studies discussing NCA’s hypothesis formulation, data collection, data analysis, reporting, and how to apply NCA in multimethod research and in practice. I finish the book with a summary and a personal reflection.

The book’s content expresses my current understanding and interpretation of NCA. NCA develops over time, just like other methodological approaches do. As NCA is applied more widely, new questions, topics, insights, and ideas will emerge. An online version of the book with updates and additional materials is available via the book-page: https://jandul.github.io/NCA/. For the physical version of the book see Dul (2026).

I hope this book serves those who wish to gain a comprehensive understanding of the principles of NCA and their application, both in academic research and practical contexts.

Enjoy discovering and applying NCA!

Leiden, March 2026

Acknowledgements

I created NCA during a long process that spans more than a decade. NCA and the content of this book are not only the result of my own thinking and reading but also of many discussions with fellow researchers during in person and online meetings, at conferences, workshops, seminars, webinars, and in email exchanges. These interactions have played an important role in shaping NCA.

I wish to express my gratitude to the universities and other research organizations that invited me to deliver NCA seminars and workshops, both in its early years and more recently. These engagements enabled valuable face-to-face discussions that contributed to the development of NCA. I would like to mention the following institutions in alphabetical order: AGH University of Krakow (Poland), Bocconi University (Italy), Chalmers University (Sweden), Deakin University (Australia), Eindhoven University of Technology (Netherlands), Erasmus University (Netherlands), Federal University of Pernambuco (Brazil), George Washington University (USA), Helmut Schmidt University (Germany), INSEAD Business School (France), Jagiellonian University (Poland), LUISS (Italy), Mines Paris Tech (France), Nottingham University (UK), Oslo University (Norway), Oxford University (UK), Penn State University (USA), Radboud University of Nijmegen (Netherlands), RIVM (Netherlands), Tilburg University (Netherlands), University of Amsterdam (Netherlands), TNO (Netherlands), University of Bayreuth (Germany), University of Beira Interior (Portugal), University of Bologna (Italy), University of Coimbra (Portugal), University of Côte d’Azur (France), University of Durham (UK), University of Groningen (Netherlands), University of Kassel (Germany), University of Lincoln (UK), University of Maastricht (Netherlands), University of Paris (France), University of Porto (Portugal), University of Sevilla (Spain), University of Trento (Italy), University of Valencia (Spain), and Wroclaw University of Economics and Business (Poland).

I would also like to express my special appreciation to the Rotterdam School of Management at Erasmus University (Netherlands). Since the incubation of NCA, the school has supported me in developing and disseminating NCA. In particular, I wish to thank Erik van Raaij and Finn Wynstra for their enduring support.

I extend my acknowledgements to the dedicated team of NCA ambassadors and supporters who have contributed in different roles to the dissemination and advancement of this approach, ranging from student assistants to co-authors and co-developers: Jorick Alberga, Florence Allard-Poesie, Tatiana Andreeva, Joost Baart, Gijs van Biezen, Ruben Blom, Stefan Breet, Jon Bokrantz, Govert Buijs, Silvia Dello Russo, Monique van Donzel, Ger Haan, Tony Hak$^\dagger$, Sven Hauff, Patrycja Klimas, Wilfred Knol, Roelof Kuik, Erwin van der Laan$^\dagger$, Wangoo Lee, Igor Marchetti, Magdalena Marczewska, Justine Massu, Indy Oostrom, Erik van Raaij, Henk van Rhee, Nicole Richter, Krista Schellevis, Chloé Schwitzgebel, Daniele Spinelli, Zsófia Tóth, and Caroline Witte. Their support greatly accelerated the adoption of NCA in research.

Throughout the development of NCA, I engaged in extensive and ongoing discussions with numerous scholars and practitioners regarding the positioning of NCA in relation to mainstream approaches. In particular, I learned a lot from my conversations with Gary Goertz, regarding the positioning of NCA vis-à-vis regression analysis and statistics; Barbara Vis and Claude Rubinson on using NCA within the context of QCA; Oskar Kosch on the relationship between NCA and machine learning; Roelof Kuik on the detailed mathematical and statistical foundations of NCA, and Ger Haan on strategies for making NCA appealing to practitioners.

Finally, I would like to thank Bob Bastian, Gijs van Biezen, Jon Bokrantz, Patrycja Klimas, Roelof Kuik, Igor Marchetti, and José Luis Roldán for commenting on a draft version of this book.

Thank you all. Without your help, NCA would not be what it is today.

Chapter 1 Introduction

NCA is a methodological approach that uses a necessity causal perspective for identifying necessary conditions from data. Necessity causal logic in NCA implies that if a certain level of the condition is not present, a certain level of the outcome will not occur: If not $X$, then not $Y$. No other factor can compensate for the absence of the necessary condition (or a required level of it). The necessary condition allows the outcome to exist, but does not produce it.¹ A necessary condition operates independently of the rest of the causal structure and serves as a ‘critical success factor’ or ‘must-have’. When absent, it creates a ‘bottleneck’ or ‘constraint’ that prevents the target outcome, which cannot be compensated by other factors. For example, a student must have a high school diploma for admission to a university. Without such a diploma, the student will not be admitted. Other factors like a student’s high motivation cannot compensate for the lack of a diploma.

Necessity causal logic and NCA differ from sufficiency causal logic and related methods. In sufficiency causal logic the condition produces the outcome: if $X$, then $Y$. A single factor may be sufficient for the outcome, for example when students graduate, they receive a diploma. Often, not a single factor but a combination of factors (a ‘configuration’) may be sufficient for the outcome: if [combination of $X$’s], then $Y$. A university diploma alone is not sufficient for admission to a PhD program; however, combining it with strong motivation, good grades, and a recommendation letter will secure it. Such a configurational sufficiency logic is used in methodologies and frameworks like Rothman’s Sufficient-Component Cause model in medicine (SCC, Rothman, 1976), Wright’s Necessary Element of a Sufficient Set in law (NESS, Wright, 1985), and Ragin’s Qualitative Comparative Analysis in the social sciences (QCA, Ragin, 1987).

Necessity causal logic also differs from probabilistic causal sufficiency logic where the single factor is a ‘contributing cause’ that helps to produce the outcome and contributes to the outcome ‘on average’: ‘if $X$, then probably $Y$’. Highly motivated people are more likely to be admitted to a PhD program than those with lower motivation, although there are highly motivated people who are not admitted and less motivated people who are admitted. Such a probabilistic causal sufficiency logic is applied in statistical methodologies for causal inference, such as in Pearl’s Directed Acyclic Graphs (DAGs) and do-operator approach (Pearl, 2009) and Angrist et al.’s Instrumental Variable approach (IV, Angrist et al., 1996).

The different causal lenses (single necessity causes, configurational sufficiency causes, and single probabilistic sufficiency causes) offer fundamentally distinct perspectives on causality. These lenses reflect different theoretical expectations (hypotheses) about how outcomes arise. This means that phenomena and data can be approached with different causal lenses (Chapter 2). The choice of lens may be guided by the specific nature of the phenomenon, the study question, or the analyst’s theoretical priorities². For instance, if the question is whether a certain resource (e.g., access to capital) is required for a firm to enter a market, a necessity lens is appropriate. This lens helps identify bottleneck factors: conditions that must be in place for success. In contrast, if the aim is to identify combinations of national conditions that are sufficient for strong climate policy commitment (e.g., public support + high income, or climate risk + high fossil fuel dependence), a configurational sufficiency perspective, as used in QCA, may be appropriate. This approach assumes that different pathways can each lead to the same outcome. If the goal is to understand whether self-efficacy increases the likelihood of engaging in a behavior, a probabilistic sufficiency lens is fitting. This approach helps estimate how much more likely the outcome is when a cause is present.

No lens is inherently superior; each has unique strengths and limitations. Thus, there are no better or worse lenses; there are no more or less relevant lenses. These lenses are different and shed different light on the same issues. What matters is that the chosen lens aligns with the goals of the study, the nature of the causal claim being made, and indeed the preference of the analyst. Once a causal lens is selected, it is essential to apply a methodological approach that matches the logic of that lens, ensuring theory–method fit, which is essential for the credibility and usefulness of the findings (Chapter 11).

This book’s emphasis on causality has both theoretical and practical reasons. Theoretically, scientific theories aim to explain why outcomes occur, not merely to describe patterns of co-occurrence (Chapters 3 and 7). Practically, changing or influencing an outcome requires intervention on causes of the outcome, not on co-occurring factors (Chapter 12). The assumption that $X$ is a necessary cause of $Y$ implies the assumption that $X$ temporally precedes $Y$ (Chapter 2).

NCA is an approach that is specifically designed to analyze necessity. It has two intertwined parts: a ‘soft’ part and a ‘hard’ part. The soft part is NCA’s soul: the causal perspective of necessity (NCA methodology). The hard part is NCA’s data analysis method (NCA method) that fits this perspective. Therefore, the term ‘NCA’ refers to a combination of the NCA methodology with its causal necessity perspective, and the corresponding NCA method for analyzing data.³ NCA’s necessity causal lens extends beyond common binary conditional necessity logic. Specifically, NCA adds causal reasoning to conditional logic, incorporates a continuous view on necessity, and permits exceptions (the ‘typicality’ perspective, see Chapter 2).

The book is organized as follows. Part I discusses the methodological principles of NCA, explaining its causality (e.g., compared to other causal perspectives), theory (e.g., main elements of a necessity theory, different types of necessity theories), mathematics (e.g., mathematical estimation of ceiling line, NCA parameters such as effect size, and measures of fit) and statistics (e.g., NCA’s null hypothesis tests, Monte Carlo simulations with NCA), and NCA’s credibility to identify necessity (e.g., using statistical and empirical criteria).

Part II of the book outlines how to conduct NCA. Specifically, this part discusses NCA’s hypothesis formulation, data collection, data analysis, reporting, how to apply NCA in a multimethod study, and how NCA can be used in practice. The book concludes with a summary and personal reflection on 10 years of NCA, and Part III provides additional materials.

With its innovative approach to causality and data analysis, NCA has rapidly entered the social, medical, and technical sciences.⁴ The publication in 2016 of NCA’s core paper (Dul, 2016b) marks the start of NCA, although proto-versions of NCA were introduced earlier (Dul et al., 2010; Dul & Hak, 2008).⁵ Subsequently, several extensions were developed, including a statistical significance test (Dul, Van der Laan, et al., 2020), and specialized software (Appendix B). Various publications, such as a textbook (Dul, 2020) and an online book (Dul, 2021) with practical guidelines on applying the method were developed to facilitate the understanding and dissemination of NCA. Furthermore, publications introducing and summarizing NCA have emerged across various fields of business and management, including Human Resource Management (Hauff et al., 2021), Marketing and Sales (Conde, 2025; Dul, Hauff, et al., 2021), Tourism Management (Tóth et al., 2019), Entrepreneurship (Linder et al., 2023), Supply Chain Management (Bokrantz & Dul, 2023), and International Business (Richter & Hauff, 2022). Also outside the business and management field, the method was introduced in fields such as Public Health (Greco et al., 2022), Clinical Psychology (Marchetti et al., 2026), Education (Tynan et al., 2020), and Creativity (Dul, Karwowski, et al., 2020).

Editors of academic journals have recognized the value of NCA by publishing editorial comments on the approach (e.g., Dul, 2025; Fainshmidt et al., 2020; Hernaus & Černe, 2022; S. Robinson et al., 2022; Sarstedt & Liu, 2023). Review articles (e.g., Aguinis et al., 2020; Boon et al., 2019; Dabić et al., 2021; Del Sordo & Zattoni, 2025; Frazier et al., 2017; Rozenkowska, 2023) and a plethora of journal articles suggest using NCA in future studies, for example, in the fields of education (e.g., Credé & Tynan, 2021), human resource management (e.g., Hauff, 2021; Parker & Knight, 2024), information systems (e.g., Sharma et al., 2024), international business/management (e.g., Richter et al., 2022; Zahoor et al., 2023), marketing (e.g., Guenther et al., 2023; Jovanovic & Morschett, 2022), operations and supply chain management (e.g., Acquah et al., 2023), strategy (e.g., Bouncken et al., 2020; Deist et al., 2023; Klimas et al., 2023; Ogundipe et al., 2022), and tourism and hospitality (e.g., Becker et al., 2023). At the time of writing this book, hundreds of articles that apply NCA have been published already. The number of publications in ‘Web of Science’ ranked journals⁶ that apply NCA has increased steadily. Appendix C shows these articles that have appeared through the end of 2025.

Published articles use NCA as the sole or primary method (Appendix C; Table ??), or combine NCA in a multimethod study (Chapter 11) with Qualitative Comparative Analysis (QCA, Table ??) or with regression-based methods. Examples of regression-based methods include multiple linear regression (MLR, Table ??) and structural equation modeling (SEM, Tables ?? and ??). NCA is also combined with a variety of other methods (Tables ?? and ??). Although the number of articles with NCA has grown rapidly, NCA is not always applied according to minimum standards (Chapter 10, Dul et al., 2023), and several misconceptions have been published.⁷ Therefore, the main goal of this book is to promote proper understanding and high-quality application of NCA.

Many articles that apply NCA conclude that necessity was identified, although the evidence is not always convincing. Other articles that apply NCA concluded that necessity was not found (e.g., Batey et al., 2021; Golini et al., 2016; Gu et al., 2022; Luo et al., 2022; Peng et al., 2022). Not finding a necessary condition might be a valuable result, because such a result shows that a supposed essential factor for an outcome does not need to be present and its absence can be compensated (e.g., Arenius et al., 2017). On the other hand, finding a large number of potential necessary conditions, for example in an exploration study (e.g., Gantert et al., 2022; Klimas et al., 2022; Stek & Schiele, 2021) does not mean that all are informative because several conditions may not have a theoretical meaning or may be trivial, which could have been easily identified before the analysis was done (Chapter 7).⁸

Conducting NCA consists of four stages:

Formulate the formal necessity hypothesis.
Collect data.
Analyze data.
Report results.

In the first stage (Chapter 7), necessity causal logic is employed to formulate a theoretical expectation about conditions that are necessary for the outcome⁹.
The second stage (Chapter 8) consists of collecting new or existing data for testing the hypothesis, which includes the consideration of study design, sampling or case selection, and measurement. NCA does not have a ‘measurement model’, as SEM and QCA have (in QCA it is called ‘calibration’). This means that any type of data can be used as input to NCA as long as the scores of $X$ and $Y$ are meaningful, valid, and reliable. NCA poses no new requirements on the data, although some aspects of data collection (case selection in qualitative studies; power analysis in quantitative studies, see for example, Dul (2024b) may differ. NCA can also be used with archival data to give a new perspective on previous findings (Dul et al., 2024).
In the third stage (Chapters 9, 11), NCA’s data analysis is conducted, which differs entirely from common types of data analysis.
In the fourth stage (Chapter 10), the results of NCA are reported.

Currently, most applications of NCA are in academic research, where researchers use the NCA approach to develop and test theories. NCA is also entering practical settings where NCA is applied by practitioners such as consultants and data analysts as an innovative approach for addressing challenges and exploring new opportunities. This is discussed in Chapter 12.

After discussing the fundamental backgrounds of the NCA methodology in Part I, the four stages of the NCA method are discussed in Part II.

Part I. Principles

Part I of the book about Principles presents the theoretical background and assumptions of NCA. Understanding the principles of NCA is important for recognizing the method’s possibilities and limitations. It is also essential for avoiding unrealistic expectations and misinterpretations.

The first chapter (Chapter 2) discusses NCA’s causality perspective and how it is different from conventional perspectives. The next chapter (Chapter 3) discusses what a necessity theory is. The subsequent two chapters present the mathematical (Chapter 4) and statistical (Chapter 5) backgrounds of NCA. The final chapter (Chapter 6) evaluates NCA’s credibility for identifying necessary conditions from data.

Chapter 2 Causality

2.1 Summary of this chapter

NCA uses a causal perspective that is (partly) different from common views on causality. This chapter presents NCA’s causal perspective by comparing it to other causal perspectives. The chapter begins by discussing what a cause is (Section 2.2), followed by a discussion of the commonly used sufficiency causality (Section 2.3). Next, necessity causality is discussed (Section 2.4). For both types of causality, three perspectives are distinguished: deterministic, probabilistic, and typicality (deterministic with exceptions). NCA uses the deterministic (or typicality) necessity causal perspective. In the next section (Section 2.5), NCA’s use of conditional logic (assuming a causal direction) is compared with the conventional view on conditional logic (in which no causal direction is assumed). Subsequently, another unique feature of NCA is discussed: the concept of necessary condition in degree (NiD), which extends the conventional binary approach (absence or presence of $X$ is necessary for absence or presence of $Y$) to a continuous approach: level $x$ of $X$ is necessary for level $y$ of $Y$ (Section 2.6). Finally, Section 2.7 discusses that, due to causal underdetermination, no causal perspective is superior to another.

2.2 What is a cause?

Causality is often loosely defined as the influence of a cause $X$ on an effect $Y$. General expressions like $X$ has an effect on $Y$, $X$ affects $Y$, or $X$ contributes to $Y$ without specifying the type of causality are commonplace. Causality is a complex subject that is extensively debated in the philosophy of science literature and elsewhere. The Scottish philosopher David Hume (1711-1776) laid the foundation for modern discussions about causation. Hume defined a cause as follows:

“… we may define a cause to be an object, follow’d by another, and where all the objects, similar to the first, are follow’d by objects, similar to the second: Or in other words, where, if the first object had not been, the second never had existed.” (Hume, 1756, p. 121).

The first part of this definition refers to sufficiency causality (if $X$, then $Y$).¹⁰ It includes a specification of the temporal order: $Y$ follows $X$, and a generalization: the statement applies also to “similar objects”. The second part of the definition refers to necessity causality (if not $X$, then not $Y$)¹¹ although without the generalization (for a further discussion see Dul, 2024a). The two types of causality are illustrated in Figure 2.1, showing observed $XY$ patterns where $X$ is the condition and $Y$ is the outcome that can have two values (absent/present).¹² When $X$ is a sufficient cause for $Y$ (Figure 2.1-left), observations (points, cases) can exist in all corners except corner 4: If $X$ is present, then $Y$ is present. When $X$ is a necessary cause for $Y$ (Figure 2.1-right), observations can exist in all corners except in corner 1: If $X$ is absent, then $Y$ is absent. The corner where observations are not possible is called the empty space and the corner where observations are possible the feasible area.

$XY$-tables illustrating sufficiency causality (Left) and necessity causality (Right) with binary concepts. Gray = observations (points, cases) are possible. White = no observations are possible.

Figure 2.1: $XY$-tables illustrating sufficiency causality (Left) and necessity causality (Right) with binary concepts. Gray = observations (points, cases) are possible. White = no observations are possible.

2.3 Sufficiency causality

Most phenomena in the social, biomedical and other sciences are studied from the perspective of sufficiency causality (if $X$, then $Y$). When people talk about a cause or causality they (almost always) implicitly mean sufficiency causality.

Three fundamentally different perspectives on sufficiency causality can be distinguished: deterministic, probabilistic, and typicality (Dul, 2024a). Table 2.1 summarizes these perspectives and lists some main methods that (implicitly) use this perspective when inferring causality.

Table 2.1: Perspectives on sufficiency causality: if $X$, then … $Y$.
Perspective	Logic	Main methods
Deterministic	if X, then Y	QCA, SCC, NESS
Probabilistic	if X, then Y	LR, MLR, SEM
Typicality	if X, then Y	Methods in physics
Note:
QCA = Qualitative Comparative Analysis
SCC = Sufficient Component Cause model
NESS = Necessary Element of a Sufficient Set
LR = Logistic (Logit) Regression
MLR = Multiple Linear Regression
SEM = Structural Equation Modeling

2.3.1 Deterministic sufficiency

Hume’s sufficiency definition has been interpreted as being about deterministic causality: if $X$, then always $Y$ (e.g., Baumgartner, 2009). Many phenomena in the physical world may be described by such a deterministic sufficiency causal perspective. For example, reaching the speed of sound produces a sonic boom. In the social sciences, single deterministic sufficiency causes are rare, as normally several factors together produce an outcome. Such a sufficiency perspective is used in ‘parts-whole’ methods and frameworks (Machamer et al., 2000; Varzi, 2019), for example, in the Sufficient-Component Cause model in medicine [SCC; Rothman (1976)], the Necessary Element of a Sufficient Set approach in law (NESS, Wright, 1985), and Qualitative Comparative Analysis in social sciences [QCA; Ragin (1987); Campbell & Fiss (2026)], use such a sufficiency perspective. The combination of causal factors (called configurations in QCA) is considered deterministically sufficient for the outcome: If [combination of $X$’s], then $Y$. Several combinations may be sufficient (‘equifinality’), such that none of these configurations is necessary for the outcome. The causal factors that are parts of the configurations are called INUS conditions: Insufficient but Necessary parts of Unnecessary but Sufficient configurations (Mackie, 1965). INUS conditions should not be confused with necessary conditions; INUS conditions are ‘locally’ necessary for the configuration to produce the outcome, and usually not overall necessary for the outcome itself (Dul, Vis, et al., 2021). NCA considers only overall necessary conditions for the outcome. Although the theoretical foundations of SCC, NESS and QCA are based on a deterministic ontology, in practice the determinism may be imperfect.¹³ Deterministic sufficiency means that corner 4 of Figure 2.1-left remains empty.

2.3.2 Probabilistic sufficiency

As single deterministic sufficiency causes are rare in the social sciences, a non-deterministic causal approach is often adopted: probabilistic sufficiency: ‘if $X$, then probably $Y$’, or ‘if $X$, then likely $Y$’. Pearl states that:

“According to one of the basic tenets of probabilistic causality, a cause should raise the probability of the effect.” (Pearl, 2009, p. 254)

The dominant framework of the probabilistic sufficiency causal perspective focuses on the average effects of ‘parts’ of the whole. The lower-left corner is not empty but is less populated than the other cells. It identifies the Average Causal Effect (ACE) or Average Treatment Effect (ATE) of a causal factor. The approach is currently mainstream in quantitative studies using statistical methods. Statistical models are based on the assumption of the probability distributions of the cause and the effect. The effect is produced by an unknown ‘data generation process’ that describes the underlying relationships between concepts and the stochastic processes involved in generating the observed data. When statistical models with regression-based methods are used to infer causality, the underlying causal perspective is probabilistic sufficiency. Examples of regression-based methods that are commonly used for inferring causality are multiple linear regression (MLR), logistic (logit) regression, and structural equation modeling (SEM). Probabilistic sufficiency may be inferred if the lower-right corner of $XY$-tables or $XY$-plots is less densely populated with observations than the other corners (Figures 2.1 and 2.3).

2.3.3 Typicality sufficiency

A recently developed non-deterministic and non-probabilistic perspective on causality is the ‘typicality’ perspective. The typicality sufficiency causal perspective was introduced in physics, including quantum mechanics and thermodynamics (e.g., Goldstein, 2012; Wilhelm, 2022). It challenges the deterministic ontology but does not adopt a probabilistic ontology. The typicality interpretation also gets attention in other scientific disciplines, including the social sciences (Wagner, 2020). This perspective can be described as deterministic with exceptions. It accepts the perspective if $X$, then $Y$ but fundamentally allows exceptions: if $X$, then typically $Y$, or if $X$, then almost always $Y$. The typicality perspective does not describe exceptions in terms of likelihood of occurrence, but in terms of cardinality (Wilhelm, 2022): the number of elements that deviates from the majority of the elements in the set. Typicality sufficiency means that corner 4 of Figure 2.1-left remains empty, but a few observations may be exceptions.

2.4 Necessity causality

The three types of perspectives on causality can also be applied to necessity causality. This is shown in Table 2.2.

2.4.1 Deterministic necessity

The second sentence of Hume’s definition of causality can be considered a deterministic view on necessity causality (Goertz & Mahoney, 2012). If $X$ does not occur, $Y$ will not happen. Whereas single sufficient causes are seldom on their own sufficient for an effect, single necessity causes alone can stop an effect from occurring if absent. For example, oxygen is necessary for human life but not sufficient on its own. There are multiple necessity causes for human life, and each will stop the outcome from occurring if it is absent. Deterministic necessity means that corner 1 of Figure 2.1-right remains empty.

2.4.2 Probabilistic necessity

The probabilistic necessity perspective can be phrased as: If not $X$, then probably not $Y$. Probabilistic necessity means that corner 1 of Figure 2.1-right may have observations but is less densely populated than the other corners.

2.4.3 Typicality necessity

The idea of typicality was developed for sufficiency causality but can also be applied to necessity causality (Dul, 2024a). It accepts a deterministic perspective with exceptions: if not $X$, then typically not $Y$, or if not $X$, then almost always not $Y$. Again, the typicality perspective does not describe exceptions in terms of probabilities but in terms of cardinality. Typicality necessity means that corner 1 of Figure 2.1-right remains empty, but a few observations may be exceptions. Exceptions are rare without quantifying what is “rare”. If an exception is not an error, it could be a unique counterexample (e.g., an innovative case), worth studying in detail. However, the phenomenon of interest is still described with deterministic necessity, excluding the exception.¹⁴ Note that NCA has a deterministic view on causality and that the ceiling line is a sharp border between an area with cases and an area without cases (Chapter 2). The presence of outliers (in the “empty” space) indicates that necessity does not exist, rather than that necessity exists with high probability.

With a deterministic view on necessity causality, a case with high $Y$ and low $X$ rejects necessity. The data point may be mistakenly in the empty area because of measurement error or sampling or case selection error. Such errors may be corrected, or the case may be removed from further consideration if the error cannot be repaired. However, if the case is not an error it is a special case: the rare exceptions may stay in the otherwise empty space, representing another phenomenon not captured by necessity. Only the presence of such exception may be the reason for adopting the typicality perspective. NCA keeps its deterministic approach, but allows rare exceptions. Having many outlier cases in the “empty” space that cannot be explained by errors is a reason to reject necessity (or to redefine the formal hypothesis, see Chapter 7 if these cases have a common characteristic), but is not a reason to accept necessity with a lenient view on (typicality) necessity.

Table 2.2: Perspectives on necessity causality: if not $X$, then … not $Y$.
Perspective	Logic	Main methods
Deterministic	if not X, then not Y	NCA, (QCA)
Probabilistic	if not X, then not Y	(PN)
Typicality	if not X, then not Y	NCA
Note:
NCA = Necessary Condition Analysis
QCA = Qualitative Comparative Analysis (only in kind; typicality implicit)
PN = Probability of Necessity (only binary variables; no generic necessity)

Table 2.2 maps different methods on the three causal perspectives. It shows that NCA can be used for the deterministic and typicality perspectives. The table includes also QCA for identifying necessity using a deterministic or typicality perspective. Although QCA’s main focus is on sufficiency, and QCA studies often ignore necessity, QCA can also identify necessity. A necessity analysis of QCA is less detailed than that of NCA, as it only identifies necessity-in-kind (NiK)¹⁵ and often misses necessary conditions (Torres & Godinho, 2022). The differences between necessity approaches of NCA and QCA are explained in Dul (2016a) and Vis & Dul (2018) and summarized in Section ??. The table also mentions Pearl’s Probability of Necessity (PN) approach for identifying probabilistic necessity using a counterfactual analysis. Pearl (2009) introduced the Probability of Necessity (PN) concept (where $X$ and $Y$ are binary) to address the question: “Was event $X$ necessary for event $Y$ to occur?” PN is the probability that $Y$ would not have occurred if $X$ had not occurred, given that both $X$ and $Y$ actually occurred.¹⁶ This is a counterfactual probability conditioned on the actual world where $X$ and $Y$ happened. PN evaluates a hypothetical world where $X$ did not happen. PN is used for identifying necessity causes at individual level, for example in legal reasoning, where the question is not whether in general $X$ is a probabilistic cause of $Y$, but rather whether $X$ was the necessity cause of $Y$ in a specific situation that has happened. PN assumes that $X$ and $Y$ are binary.

2.5 Causal versus conditional logic

NCA extends conditional logic with causality. In conditional logic, as used in philosophy and mathematics, relationships are described in terms of if-then statements. In the statement ‘if $A$, then $B$’, $A$ is the antecedent and $B$ is the consequent, and $A$ and $B$ usually have only two values: true or false. The truth or falsity of $B$ depends on the truth or falsity of $A$. In conditional logic, ‘if $A$, then $B$’ means that $A$ implies $B$, or if $A$ occurs, then $B$ occurs. This can be interpreted as $A$ being a sufficient condition for $B$. For example, being born in Italy ($A$) implies having Italian citizenship ($B$). This can be written as:

\[\begin{equation} \tag{2.1} A_s \Rightarrow B \end{equation}\]

Here the subscript ‘$s$’ emphasizes the interpretation of $A$ as a sufficient condition, and the double arrow $\Rightarrow$ means ‘implies’.

In conditional logic, the statement that $A$ is a necessary condition for $B$ implies that $B$ can only be true if $A$ is true: if $B$, then $A$. For example, oxygen is necessary for human life because being alive implies that the person has access to oxygen. Therefore, truth of $B$ (being alive) implies the truth of $A$ (oxygen). This can be written as:

\[\begin{equation} \tag{2.2} B \Rightarrow A_n \end{equation}\]

where the subscript ‘$n$’ emphasizes the necessary condition interpretation of $A$. This can be rewritten as:

\[\begin{equation} \tag{2.3} A_n \Leftarrow B \end{equation}\]

where the double arrow $\Leftarrow$ means “is implied by”.

Because $B$ can only be true when $A$ is true, the falsity of $A$ implies the falsity of $B$. This can be written as:

\[\begin{equation} \tag{2.4} \neg A_n \Rightarrow \neg B \end{equation}\]

\[\begin{equation} \tag{2.5} \neg B \Leftarrow \neg A_n \end{equation}\]

where the symbol $\neg$ indicates negation, or ‘no’ or ‘not’.

Equations (2.2) - (2.5) are four equivalent expressions for $A_n$ being necessary for $B$ from the perspective of conditional logic.

The phrase $B$ implies $A_n$ (2.2) can also be understood as $B$ is sufficient for $A_n$. Therefore, the expression $A_n$ is necessary for $B$ can be written in four equivalent ways:

$A_n$ is necessary for $B$ (oxygen is necessary for being alive).
$B$ is sufficient for $A_n$ (being alive is sufficient for oxygen).
$A_n$ is sufficient for $B$ (no oxygen is sufficient for not being alive).
$B$ is necessary for $A_n$ (not being alive is necessary for no oxygen).

A major difference between necessity in causal logic as used in NCA, and necessity logic as used in conditional logic is that NCA assumes a causal direction between $A_n$ and $B$ whereas conditional logic does not assume such a causal direction. This means that NCA assumes a temporal order: first $A_n$, followed by $B$. Conditional logic is ignorant about whether there is first oxygen and then life or first life and then oxygen. Therefore, NCA’s causal logic can be seen as an extension of conditional logic where a causal direction is assumed between $A_n$ and $B$. This temporal order excludes the possibility of ‘first $B$, then $A_n$’. The assumption of a temporal order may be justified when first the occurrence of $A_n$ is observed and afterwards the occurrence of $B$, or that it is theoretically implausible that $A_n$ occurs after $B$ (see Section 7.5.3 for a further discussion on temporality). It means that when NCA formulates that $A_n$ is a necessary condition for $B$, it is assumed that $A_n$ is a causal necessary condition for $B$. Assuming also no reverse or reciprocal causality (Section ??), it implies that only two of the four conditional expressions are causal statements:

Therefore, in NCA the antecedent $A_n$ is considered the necessity cause of the consequent $B$ (the effect). Statements 2 and 4 still apply as logical statements, but not as causal statements, assuming no reverse or reciprocal causality.

To stress that NCA makes a causal assumption about the relationship between antecedent and consequent, the symbols $A$ and $B$ or $P$ and $Q$ that are common in conventional conditional logic, are replaced by the symbols $X$ and $Y$ to express the cause-effect relationship, where $X$ is the cause and $Y$ is the effect. Then the two equivalent causal expressions for a necessity cause are:

\[\begin{equation} \tag{2.6} X \text{ is necessary for } Y \end{equation}\]

\[\begin{equation} \tag{2.7} \neg X \text{ is sufficient for } \neg Y \end{equation}\]

In the remainder of this book, the expression ‘$X$ is a necessary condition for $Y$’ implies that $X$ is a necessity cause of $Y$ and both expressions (2.6) and (2.7) apply. Dul (2016b) refers to expression (2.6) as the ‘necessity formulation’ of the necessary condition and expression (2.7) as the ‘sufficiency formulation’ of the necessary condition. In this book the statement $X$ is necessary for $Y$ is understood in a deterministic or typicality causal perspective: ‘$X$ is always necessary for $Y$’ or ‘$X$ is almost always/typically necessary for $Y$.’ Consequently, all or nearly all observations with the outcome $Y$ do have the necessity cause $X$, and that all or nearly all observations that do not have the necessity cause $X$ do not have the outcome $Y$.¹⁷

2.6 From binary necessity to continuous necessity

Another extension that NCA introduces to necessity logic involves the levels of condition $X$ and outcome $Y$. Until now, the book considered the levels of $X$ and $Y$ as binary. Only two values are possible: absent or present (like in Figure 2.1-right), 0 or 1, or false or true, etc. This binary approach is common in conditional logic as well as in the SCC, NESS, and QCA methods and frameworks.¹⁸

NCA extends the binary levels of condition and outcome to multiple possible levels by allowing $X$ and $Y$ to be discrete concepts (with a number of finite levels of more than 2) or continuous concepts (with an infinite number of levels) within minimum and maximum bounds of $X$ and $Y$. This has several consequences.

First, the $XY$-table is extended to an $XY$-plot as shown in Figure 2.2.

$XY$-plot with dichotomous (Left), discrete (Middle) and continuous (Right) values of condition $X$ and outcome $Y$.

Figure 2.2: $XY$-plot with dichotomous (Left), discrete (Middle) and continuous (Right) values of condition $X$ and outcome $Y$.

Instead of mapping observations on the cells of the $XY$-table, observations are now mapped on a 2D Euclidean coordinate system. The bounds of $X$ and $Y$ are again 0 and 1, but could be any other minimum and maximum values. The bounds on $X$ and $Y$ create a bounding box (Section 4.3). In a dichotomous necessity relationship where both the condition and the outcome are binary, the observations are mapped at corners of the $XY$-plot (Figure 2.2-left). When necessity applies, no observations are possible in the upper-left corner [0,1]. In a discrete necessity relationship observations can be on the intersections of a grid representing the possible discrete values of $X$ and $Y$ (Figure 2.2-middle), and in a continuous necessity relationship observations can exist anywhere in the unit square (Figure 2.2-right), but not in the upper-left corner. Combinations of dichotomous, discrete and continuous values are also possible in NCA. This is shown in Figure 2.3.

$XY$-plot with combinations of dichotomous, discrete, and continuous values of condition $X$ and outcome $Y$. The upper-left corner is empty if the presence/high value of $X$ is necessary for the presence/high value of $Y$.

Figure 2.3: $XY$-plot with combinations of dichotomous, discrete, and continuous values of condition $X$ and outcome $Y$. The upper-left corner is empty if the presence/high value of $X$ is necessary for the presence/high value of $Y$.

Second, whereas in the dichotomous situation the bounding box can be either full or empty, in the discrete and continuous situation it can be partly empty. The size of the empty space may vary, depending on the location of the observations. A border line (ceiling line) separates the empty space (ceiling zone) from the area where observations are possible (feasible area). Referring to the $XY$-plot in Figure 2.2 where both the condition and the outcome are dichotomous, the ceiling line between empty and feasible area is predefined: the step line connecting the observations [0,0], [0,1], and [1,1] is always the same. The full bounding box (unit square) is empty, such that points can be only on the ceiling line. No observations can be above or to the left of the ceiling line. In the discrete and continuous situations, the location of the ceiling line depends on the location of the observations in the unit square. This means that the location of the ceiling line and the size of the empty space vary. In any case, there can be no points above or to the left of the ceiling line. If necessity applies, points can only be on, below, or to the right of the ceiling: the feasible area. The ceiling line can be assumed (predefined) or estimated from data. In all situations, the assumption is that the ceiling line represents the sharp border between the empty space and the feasible area to represent deterministic necessity (see model fit measure Sharpness discussed in Section 4.5.6). When a relevant empty space exists that is caused by necessity, the necessary condition is formulated qualitatively as necessity-in-kind (NiK): $X$ is necessary for $Y$.

Third, the discrete and continuous situation adds the possibility of a quantitative formulation of a necessary condition statement: level of $X$ is necessary for level of $Y$. This is called necessity-in-degree (NiD) and is shown for the continuous situation in Figure 2.4. Any observation $C$ on the ceiling line defines a specific necessity-in-degree: level $X = x_c$ is necessary for level $Y = y_c$. Assuming an increasing ceiling line, the condition must have at least level $x_c$ for outcome level $y_c$. If the condition is less than $x_c$, the outcome will be less than $y_c$. If $X = x_c$, it cannot be concluded that $Y = y_c$, but only that $Y \leq y_c$. This situation can be described by level $x_c$ of $X$ is necessary but not sufficient for level $y_c$ of $Y$ (Section 3.8 for a further discussion on the meaning of necessary but not sufficient).

Figure 2.4: $XY$-plot illustrating necessity causality when $X$ and $Y$ are continuous (necessity-in-degree, NiD).

2.7 Causal underdetermination

A causal relationship cannot be directly observed. What can be observed are only sequences and patterns. For example, when a glass falls from the table, we see the motion of the hand, the contact with the glass, the glass’s acceleration, and its eventual impact on the ground. Nowhere in this stream of sensory data is there an observable entity called “causation”. The causal link must instead be inferred. Was the push responsible for the fall, was the table moved at the same moment, or did the glass fall by coincidence?

Because causality cannot be observed, a human interpretation of observed data must be made. The same observational data can support multiple causal explanations. This is the phenomenon known as causal underdetermination (e.g., Stanford, 2023). An observed relationship between $X$ and $Y$ may reflect that $X$ causes $Y$, $Y$ causes $X$, an unobserved cause, or mere chance. Underdetermination implies that a single dataset can be explained by different causal models, whether framed in terms of necessity, sufficiency, probabilistic tendencies, or typicality.

The idea that the same evidence can be “seen” differently is illustrated by the classic ambiguous picture below. Some viewers immediately see a young woman, while others see an old woman, and it can be hard to switch between the two. Likewise, the same data can support different causal interpretations, and once someone becomes familiar with one perspective, it may be difficult to adopt another.

$The same image can be seen in different ways. Likewise, the same data can be viewed through different causal lenses. From: My wife and my mother-in-law. They are both in this picture – find them. Prints \& Photographs Online Catalog, Library of Congress. https://www.loc.gov/pictures/item/2010652001/. Retrieved 1 October 2025.$

Figure 2.5: The same image can be seen in different ways. Likewise, the same data can be viewed through different causal lenses. From: My wife and my mother-in-law. They are both in this picture – find them. Prints & Photographs Online Catalog, Library of Congress. https://www.loc.gov/pictures/item/2010652001/. Retrieved 1 October 2025.

A related implication is the problem of non-identifiability (or observational equivalence): distinct models can generate the same observable data patterns and therefore fit the data equally well, even though they imply different underlying structures (e.g., Rothenberg, 1971). When this happens, the evidence cannot distinguish between these competing explanations. For example, when a correlation between $X$ and $Y$ is observed, this could be interpreted as a result of a probabilistic sufficiency relationship, or as a result of a necessity relationship (Appendix G). In principle, both interpretations are possible.

Given the inherent underdetermination of causal claims by data, theory becomes indispensable. Theory provides the assumptions and background knowledge needed to justify a causal model that is consistent with the observations. Without such guiding principles, data remain compatible with infinitely many interpretations. Theory specifies, for example, which mechanisms are plausible or which temporal order is even possible. The central role of theory in data analysis for causal inference is illustrated by Nancy Cartwright’s slogan:

“No causes in, no causes out” (Cartwright, 1989, pp. 39–90).

Some causal theories may be more plausible than others. For example, if we observe that plants exposed to more sunlight grow more, the explanation that sunlight causes plant growth is far more credible than the idea that plant growth causes sunlight. An observed relationship is considered spurious when an alternative explanation is clearly more credible because the theory is more plausible or because of better model fit.¹⁹

However, several causal theories may also be equally credible. This is the situation known as causal pluralism: different causal perspectives may each illuminate different aspects of the same phenomenon (Section ??). Causal pluralism can help to better understand the underlying causal mechanisms of the phenomenon of interest. If several causal theories are equally credible, it is also possible to select one theory for pragmatic reasons. Such reasons could be simplicity (Section 3.3), usefulness for prediction, computational elegance, compatibility with other accepted theories, interpretability, or practical actionability. A necessity theory could be a candidate.

Underdetermination and pluralism imply that no single causal perspective is inherently the “proper” one for every study question. In particular, commonly used probabilistic sufficiency perspectives and their associated regression tools are not inherently preferred. It is the analyst’s freedom to select an appropriate perspective and tools, and the analyst’s responsibility to justify the corresponding hypothesis and interpretation (Chapter 7).

This book is about the use of the necessity causal perspective and tools when studying phenomena. It therefore follows that a necessity-based theoretical framework is essential for describing such phenomena coherently. What constitutes such a necessity theory is discussed in the next chapter.

Chapter 3 Theory

3.1 Summary of this chapter

This chapter discusses causal theory in general and necessity theory in particular. A causal theory helps us understand phenomena, predict effects, and explain empirical findings. The chapter begins (Section 3.2) by discussing the four fundamental elements of a theory: (1) focal unit (e.g., person, organization, country), (2) concepts (varying characteristics of the focal unit), (3) propositions (causal relationships between the concepts), and (4) theoretical domain (generalization to where the theory holds). This is followed by a discussion of parsimonious necessity theories (Section 3.3) and types of necessity theories (Section 3.4), for example a pure necessity theory where all relationships between the concepts are described in terms of necessity, and an embedded necessity theory, where part of the theory has one or more necessity relationships. Next, the direction of a necessity proposition is discussed (Section 3.5), which depends on whether $X$ and $Y$ are present/have high value or are absent/have low value. The most common direction is that the presence or high value of $X$ is necessary for the presence or high value of $Y$ (high-high direction), which is the default when the direction is not specified. Other possible directions are high-low, low-high, and low-low. The following section (Section 3.6) discusses the expected data pattern when necessity holds. A consequence of a high-high necessary condition is that the upper-left corner in an $XY$-plot is expected to be empty. The next section (Section 3.7) discusses the possibility of having double necessity when multiple corners are expected to be empty. The chapter finishes with a discussion of the theoretical meaning of ‘necessary but not sufficient’ (Section 3.8).

3.2 Main elements of a theory

Because there are multiple ways to conceptualize theory, there is no consensus on what a theory is. This book adopts the ‘four elements’ approach to theory (Dul & Hak, 2008), which enables straightforward empirical testing. It uses a broad and inclusive definition of what constitutes a theory, ranging from established theories that are broadly accepted to theoretical assertions by an analyst made in a specific study. Not only academic theories but also ‘theories-in-use’ qualify as a theory. At a minimum, a theory must have four fundamental elements: focal unit, concepts, propositions, and theoretical domain. The focal unit is the object to which the theory applies: country, person, firm, project, etc. The concepts are the variant characteristics of the focal unit representing properties of a phenomenon that can increase or decrease. In NCA, a concept that is the cause is called the ‘condition’ and a concept that is the effect is called the ‘outcome’. The proposition describes the relationship between cause and effect, including the causal explanation. In NCA, the causal relationship is a necessity relationship: $X$ is necessary for $Y$, meaning that $X$ is (almost always) necessary for $Y$ (Dul, 2024a). The words ‘almost always’ may be omitted from the necessity proposition, allowing both a deterministic and a typicality perspective. As theories in the social sciences are usually semantic (expressed in words), specific levels of the condition (cause) and the outcome (effect) are not specified. The theoretical domain is the set of cases of the focal unit where the theory is supposed to hold, and is defined by the boundary conditions of the theory. A case is one particular example of the focal unit: a specific country, a specific person, etc. For example, a theory that proposes a relationship between physical exercise and stress might be applicable to working adults, but not to children. In that case, the theoretical domain includes only working adults.

For example, the established Theory of Planned Behavior (TPB) explains how an individual’s intention for a specific non-habitual behavior relates to the performance of that behavior (Ajzen, 1991). The focal unit is ‘individual’ or ‘person’, two of its concepts are ‘Intention for the behavior’ (the cause) and ‘Performing the behavior’ (the effect), the proposition describes the causal relationship between these concepts, and one possible theoretical domain is ‘all people in the world’.

The relationship between the two TPB concepts is usually specified as a probabilistic sufficiency relationship, for example ‘Intention to perform a behavior likely to result in Performing the behavior’, or, as the founder of TPB puts it: “As a general rule, the stronger the intention to engage in a behavior, the more likely should be its performance” (Ajzen, 1991, p. 181). Recent interpretations of TPB suggest formulating the relationship between the two concepts in terms of necessity (Eccarius & Chen, 2024; e.g., Frommeyer et al., 2022; Rozenkowska, 2023): for example ‘Intention to perform a behavior is necessary for Performing the behavior’. The key difference between a conventional (probabilistic sufficiency) theory and a necessity theory is the difference in specification of the causal relationship, in other words the causal perspective (Dul, 2024a) that is used to describe the relationship between the concepts (i.e., the proposition).

A theory is only complete if it is accompanied by a plausible substantive explanation of why the cause affects the outcome. For a necessity proposition, the theory should explain (1) why all cases from the theoretical domain with the outcome have the condition, (2) why without the condition, the outcome does not exist, (3) why the absence of the condition is not compensable by other factors (no possibility of substitution), and (4) that the condition precedes the outcome (fulfilling the temporality requirement of causality). Details about how to formulate a formal necessity hypothesis embedded in a necessity theory are explained in Chapter 7. When applying NCA, a necessity theory that causally explains the necessity relationship is essential.

3.3 Parsimonious necessity theories

Necessity theories are inherently parsimonious: single concepts make a strong, actionable prediction: the absence of a necessary condition perfectly predicts the absence of the outcome. Starting with William of Ockham’s parsimony principle²⁰ (also known as Occam’s Razor), many famous scholars including Newton²¹, Einstein²², Popper²³, and Simon²⁴ emphasized the importance of keeping theories simple, such that they are understandable, practically useful, and falsifiable.

Because necessary conditions operate in isolation from the rest of the causal structure (see Section 4.6), even the simplest necessity theory with only one or a few necessary conditions can be actionable. Figure 3.1 shows a necessity theory with two concepts: one condition $X$ and one outcome $Y$. To emphasize the necessity perspective used to describe the relationship, nc (necessary cause or necessary condition) is placed above the arrow.

Figure 3.1: Conceptual model of a necessity theory with one condition and one outcome.

It is also possible that multiple concepts (e.g., $X_1$, $X_2$, $X_3$) are necessary for the same outcome ($Y$) as shown in Figure 3.2-left, or that one or more conditions are necessary for several outcomes (e.g., $Y_1$, $Y_2$, $Y_3$), which is shown in Figure 3.2-right. Note that conditions $X_1$, $X_2$, and $X_3$ are all individual necessary conditions. If any one condition is missing (below its required level) the outcome cannot occur (at its target level), while the presence of all conditions is normally not sufficient for the outcome because other contributing factors must be in place as well.

Conceptual models of a necessity theory with three conditions and one outcome (Left), and of a theory with one condition and three outcomes (Right).

Figure 3.2: Conceptual models of a necessity theory with three conditions and one outcome (Left), and of a theory with one condition and three outcomes (Right).

Furthermore, several necessary conditions can combine into a necessity causal chain where $X_1$ is necessary for $X_2$, and $X_2$ is necessary for $Y$. This situation is shown in Figure 3.3 and is called ‘chain necessity’ of $X_1$ for $Y$. Chain necessity differs from ‘direct necessity’ as shown in Figure 3.1. In the dichotomous necessity-in-kind (NiK) situation, the necessity of $X_1$ for $X_2$ and of $X_2$ for $Y$ implies the necessity of $X_1$ for $Y$. However, in the necessity-in-degree (NiD) situation it is possible that the chain necessity of $X_1$ for $Y$ is absent despite the necessity between the elements of the chain ($X_1$ for $X_2$ and $X_2$ for $Y$), as discussed in Section 4.6.4.

Figure 3.3: Conceptual model of a chain of necessary conditions. $X_1$ is necessary for $X_2$ and $X_2$ is necessary for $Y$.

Necessity models are inherently parsimonious. A single necessary condition can stop the outcome, independently of other conditions and other contributing factors. However, in sufficiency-based perspectives, a single factor is seldom enough to produce the outcome. For a probabilistic or configurational sufficiency model, a series of factors are included to model causal complexity. This applies to both probabilistic sufficiency (e.g., structural causal models) and configurational sufficiency (e.g., as in QCA). Parts of the relationships of sufficiency models could (also) be necessity relationships. For the necessity analysis these parts can be isolated (as in Figure 3.1) from the rest of the causal structure and independently be analyzed for necessity (Chapter 11).

3.4 Types of necessity theories

In addition to the established/emerging distinction, necessity theories can also be classified according to the extent to which necessity propositions are part of the theory. Bokrantz & Dul (2023) distinguish between pure necessity theories, which are theories that consist only of necessity relationships between the concepts and embedded necessity theories, which are theories that consist of a mix of necessity relationships and other relationships (e.g., probabilistic sufficiency or configurational sufficiency). In embedded necessity theories, some of the relationships may be necessity relationships (e.g., $X_1 \xrightarrow{nc\ } Y$), whereas others may be probabilistically sufficient relationships ($X_2 \xrightarrow{+\ } Y$). It is also possible that one condition has both causal roles in a model such that the presence of $X$ is necessary for the presence of $Y$, but also that $X$ likely increases $Y$ ($X \xrightarrow{nc\ ,\ +} Y$).²⁵

Typology of necessity theories with examples. AMO = Ability-Motivation-Opportunity. TPB = Theory of Planned Behavior. TAM = Technology Acceptance Model. SDT = Self-Determination Theory.

Figure 3.4: Typology of necessity theories with examples. AMO = Ability-Motivation-Opportunity. TPB = Theory of Planned Behavior. TAM = Technology Acceptance Model. SDT = Self-Determination Theory.

Figure 3.4 shows the four possible types of necessity theories by also considering if the necessity theory is established or new.
Established pure necessity theories, where all relationships in the theory are necessity relationships as in Figures 3.1, 3.2, and 3.3 in Section 3.2 are relatively scarce. One example is the early theory of Guilford (1967) that intelligence is necessary for creativity. Shortly after NCA was introduced (Dul, 2016b), this theory was empirically tested with NCA and supported (Karwowski et al., 2016). Another established pure necessity theory is the original Ability, Motivation, Opportunity model (AMO) or Motivation, Opportunity, Ability (MOA) model of human behavior. Necessity lies at the core of this theory assuming that A, M, and O are single necessary conditions for behavior.²⁶

Established embedded necessity theories are broadly accepted theories consisting of both necessity and probabilistic sufficiency relationships. Often, these theories are originally formulated as probabilistic sufficiency theories for all their relationships, and revisited to suggest a necessity perspective for one or more relationships. An example is Frommeyer et al. (2022)’s interpretation of one relationship of the Theory of Planned Behavior (TPB): Intention of the behavior is necessary for Performance of the behavior, while keeping the other relationships as probabilistic sufficiency.²⁷ Another example is the Technology Acceptance Model (TAM) that is usually interpreted and studied as a probabilistic sufficiency theory and that was recently revisited with a necessity causal perspective for all relationships while maintaining also the probabilistic sufficiency perspective (Erdmann & Toro-Dupouy, 2025; Hassan et al., 2025; Kopplin, 2023; Low & Ramayah, 2023; e.g., Richter et al., 2020; Su et al., 2023).²⁸ Also the Self-Determination Theory (SDT) was recently revisited from the perspective of necessity only (Ding & Kuvaas, 2023, 2025) or in combination with the probabilistic sufficiency perspective (Cassia & Magno, 2024). SDT suggests that humans have three basic psychological needs (motivation, competence, relatedness) that must be satisfied for optimal human development and well-being.

Many established theories exist (not shown in Figure 3.4) that could be revisited from a necessity perspective to develop pure or embedded necessity theories because their founders clearly employed necessity causal reasoning to formulate these theories. Examples include Porter’s theory on competitive advantage of nations (Porter, 1985), Wernerfelt’s resource-based view of the firm (Wernerfelt, 1984), Teece’s theory on dynamic capabilities (Teece et al., 1997; Teece, 2007), Rogers’ theory on person-centered psychotherapy (Rogers, 1957), Turner’s framework for project management success (Turner, 2009), and Meehl’s concept of ‘specific etiology’ in medicine (Meehl, 1962). Until recently, such theories were mostly tested using regression-based methods that implicitly assume probabilistic sufficiency, suggesting theory-method misfit (Table ??). With NCA available, testing necessity with a dedicated methodology becomes possible.

Since the introduction of NCA, the number of emerging pure necessity theories has risen. Appendix ?? includes examples of NCA studies in which only necessity relationships are theoretically formulated and tested with NCA. After replication and further theorizing, these emerging necessity theories may become established necessity theories. An example of an empirical study with an emerging pure necessity theory is a study by Yan et al. (2023) that proposes and finds four necessary conditions for a country’s high level of early COVID-19 mortality rate: high levels of a delayed first response, political decentralization, elderly populations, and urbanization. An example of a theoretical study with an emerging pure necessity theory is the study by Andrevski & Miller (2022) about defining the concept ‘strategic forbearance’. When firms are attacked by competitors, forbearance is defined by three necessary conditions: the presence of awareness of the attack, the presence of capabilities to react, and the absence of a motivation to react.

Many studies formulate emerging embedded necessity theories. Such studies often start with a probabilistic sufficiency theory based on earlier empirical regression-based studies and then suggest that one or more of the relationships could (also) be formulated as necessity relationships. In an empirical study Lee & Jeong (2021) propose an emerging embedded necessity theory in which two dimensions of tourist happiness experience (Hedonic enjoyment and Personal expressiveness) are related to Satisfaction (customer’s judgment about their fulfillment with a product or service) and Place attachment (bonding between individuals and places). This is shown in the conceptual model of Figure 3.5. By conducting both NCA and structural equation modeling (SEM), they conclude that Hedonic enjoyment (pleasure, comfort) has both a necessity relationship and a probabilistic sufficiency relationship with Satisfaction, but only a necessity relationship with Place attachment. Furthermore, they conclude that Personal expressiveness (a subjective experience of self-realization that one is acting in such a way that one is truly being oneself) has no probabilistic sufficiency relationship with Satisfaction, and both a necessity relationship and a probabilistic sufficiency relationship with Place attachment. Other examples of emerging embedded necessity theories may be found in empirical studies that apply NCA in combination with regression-based methods (Tables ??, ??, and ?? in Appendix C). Liehr & Hauff (2022) propose in a theoretical study an emerging embedded necessity theory about leadership competencies for employee innovative behavior. They review a large number of competencies that contribute to innovative behavior and conclude that three competencies (according to the literature) have a probabilistic sufficiency relationship with innovative behavior but are not necessary (rewards, sharing expertise, resource management) and four competencies are (also) necessary: providing support, communicating a vision, granting autonomy and discretion, and providing feedback.

Example of an emerging embedded necessity theory for the effect of two dimensions of happiness experiences (Hedonic enjoyment and Personal expressiveness) on two outcomes (Satisfaction and Place attachment). The symbols + represents a positive probabilistic relationship and the symbol nc represents a necessity relationship with high-high direction (After Lee & Jeong, 2021).

Figure 3.5: Example of an emerging embedded necessity theory for the effect of two dimensions of happiness experiences (Hedonic enjoyment and Personal expressiveness) on two outcomes (Satisfaction and Place attachment). The symbols + represents a positive probabilistic relationship and the symbol nc represents a necessity relationship with high-high direction (After Lee & Jeong, 2021).

3.5 Direction of a necessity relationship

In Figure 3.5 the direction of the probabilistic sufficiency relationship is given by a + symbol above the arrow. This convention indicates a positive effect: $X$ increases $Y$ or $X$ has a positive effect on $Y$ or more precisely, that a higher value of $X$ increases the probability of a higher value of $Y$. Similarly, a – symbol indicates a negative effect: $X$ decreases or has a negative effect on $Y$: a higher value of $X$ decreases the probability of a higher value of $Y$.

Such single symbols do not convey meaning for a necessity relationship. In a necessity proposition the direction is not described by a verb as in $X$ increases $Y$ or $X$ has a positive effect on $Y$. For a necessity proposition a noun or adverb (or a combination) is used to describe relationships between levels of $X$ and $Y$. For example, in the dichotomous situation absence of $X$ is necessary for presence of $Y$ or in the continuous situation low level of $X$ is necessary for high level of $Y$. For a necessity relationship four possible directions exist (Figure 3.6).

First, for a necessary condition with a high-high direction, the presence/high value of $X$ is necessary for presence/high value of $Y$. This is symbolized as (see Figure 3.6- top-left). For example, high level of Urbanization is necessary for a high level of Early COVID-19 mortality (Yan et al., 2023).

Second, for a necessary condition with a low-high direction, the absence/low value of $X$ is necessary for the presence/high value of $Y$ (). For example, low Intensity of production is necessary for high Environmental sustainability in agriculture (Lankoski & Lankoski, 2023).

Third, for a necessary condition with a high-low direction, the presence/high value of $X$ is necessary for the absence/low value of $Y$ (). For example, the presence/high value of social support is necessary for the absence/low value of stress.

Fourth, for a necessary condition with a low-low direction, the absence/low value of $X$ is necessary for the absence/low value of $Y$ (). For example, the absence of rain is necessary for the absence of a wet surface.

The third and fourth formulations of a necessary condition are less common than the first and second. Often the absence/low value of the outcome is redefined as the presence/high level of the opposite of the outcome. For example, the presence/high value of social support is necessary for the presence/high value of relaxation or the absence of rain is necessary for the presence of a dry surface. Thus, the direction of the necessary condition depends on the definition of the concepts that are part of the proposition.

In the absence of a specified direction, the implied direction is , as shown in Figure 3.5. Throughout this book, the default assumption is the direction, unless stated otherwise.

Figure 3.6 shows the four possible conceptual models depending on the direction of the necessary condition.

Conceptual models for necessary conditions. Top-Left: high-high = presence/high value of $X$ is necessary for presence/high value of $Y$. Top-Right: low-high = absence/low value of $X$ is necessary for presence/high value of $Y$. Bottom-Left: high-low = presence/high value of $X$ is necessary for absence/low value of $Y$. Bottom-Right: low-low = absence/low value of $X$ is necessary for absence/low value of $Y$.

Figure 3.6: Conceptual models for necessary conditions. Top-Left: high-high = presence/high value of $X$ is necessary for presence/high value of $Y$. Top-Right: low-high = absence/low value of $X$ is necessary for presence/high value of $Y$. Bottom-Left: high-low = presence/high value of $X$ is necessary for absence/low value of $Y$. Bottom-Right: low-low = absence/low value of $X$ is necessary for absence/low value of $Y$.

$XY$-tables showing empty corners without observations depending on the direction of the necessity relationship. Top-Left: high-high = presence of $X$ is necessary for presence of $Y$. Top-Right: low-high = absence of $X$ is necessary for presence of $Y$. Bottom-Left: high-low = presence of $X$ is necessary for absence of $Y$. Bottom-Right: low-low = absence of $X$ is necessary for absence of $Y$.

Figure 3.7: $XY$-tables showing empty corners without observations depending on the direction of the necessity relationship. Top-Left: high-high = presence of $X$ is necessary for presence of $Y$. Top-Right: low-high = absence of $X$ is necessary for presence of $Y$. Bottom-Left: high-low = presence of $X$ is necessary for absence of $Y$. Bottom-Right: low-low = absence of $X$ is necessary for absence of $Y$.

3.6 Expected empty corner

Depending on the direction of necessity, different corners in the $XY$-table or $XY$-plot are expected to be empty. For the dichotomous situation, the $XY$-table of Figure 2.1-right illustrates necessity causality when the presence of $X$ is necessary for the presence of $Y$. Corner 1 cannot have observations (points, cases) whereas the other corners may have observations. This is also shown in Figure 3.7-top-left. In Figure 3.7 rows represent the two values of the outcome $Y$ (Absent/Present), and columns the two values of the condition $X$. The condition $X$ increases to the right, and the outcome $Y$ increases upward. Each corner in the table represents a certain combination of $X$ and $Y$. For example, the upper-left corner represents absence of $X$ and presence of $Y$. A gray corner with a black dot indicates that the corner may contain observations. A white corner without dots indicates that no observations are possible.

When the presence of $X$ is necessary for the presence of $Y$, the upper-left corner is empty. This corresponds to the necessity statement ‘if not $X$, then not $Y$’ (see the necessity statement of expression (2.6) in Chapter 2). The outcome is absent when the condition is absent. This means that the upper-left corner does not have observations, and that the lower-left corner has observations. Similarly, when the outcome is present, the condition must also be present: the upper-right corner has observations. The content of the lower-right corner is irrelevant for necessity: this corner may or may not have observations when the necessary condition applies. For necessity, the emptiness of the upper-left corner is essential.

For other directions of the necessary condition, other corners in the $XY$-table remain empty when necessity applies. When necessity is valid, the upper-right corner remains empty. Similarly, for necessity, the lower-left corner is empty, and for necessity, the lower-right corner stays unoccupied. Therefore, the empty corner depends on how the necessary condition is formulated in the theory (and how the direction of the axes and the concepts are defined). Each formulation of the necessity proposition results in a specific corner of the $XY$-table being empty, referred to as the expected empty corner.

For the continuous situation, Figure 3.8 shows the four directions and corresponding empty corners. Here an $XY$-plot instead of a $XY$-table is used as a visualization method for representing that a low/high value of the condition is necessary for a low/high value of the outcome. The condition $X$ increases to the right, and the outcome $Y$ increases upward. Each corner in the plot represents a certain combination of low/high $X$- and $Y$-values. For example, the upper-left corner represents low level of $X$ and high level of $Y$. A gray area indicates that the area may contain observations.

$XY$-plots showing the direction of a necessity relationship. Top-Left: a high value of $X$ is necessary for a high value of $Y$ (+ nc +, high-high). Top-Right: a low value of $X$ is necessary for a high value of $Y$ (- nc +, low-high). Bottom-Left: a high value of $X$ is necessary for a low value of $Y$ (+ nc -, high-low). Bottom-Right: a low value of $X$ is necessary for a low value of $Y$ (- nc -, low-low).

Figure 3.8: $XY$-plots showing the direction of a necessity relationship. Top-Left: a high value of $X$ is necessary for a high value of $Y$ (+ nc +, high-high). Top-Right: a low value of $X$ is necessary for a high value of $Y$ (- nc +, low-high). Bottom-Left: a high value of $X$ is necessary for a low value of $Y$ (+ nc -, high-low). Bottom-Right: a low value of $X$ is necessary for a low value of $Y$ (- nc -, low-low).

3.7 Double necessity

It is possible that a single factor has two necessity relationships with the outcome at the same time, implying that more than one corner is expected to be empty. When the concepts are continuous, it is possible that two adjacent corners are empty. This allows theorizing that an optimum (not low, not high) level of $X$ is necessary for a high level of $Y$, implying that the upper-left and the upper-right corners are expected to be empty (Figure 3.9-left). For example, in team performance, moderate levels of conflict ($X$) may be necessary for high team innovation performance ($Y$). Low conflict leads to group think, which may limit performance; high conflict leads to dysfunction, which also limits team innovation performance. Therefore, a moderate level of conflict enables teams to challenge each other productively, providing the possibility for high innovation performance.
Similarly, it may be theorized that an optimum (not low, not high) level of $X$ is necessary for a low level of $Y$, implying that the lower-left and lower-right corners are expected to be empty (Figure 3.9-right). For example, in stress management, moderate levels of workload ($X$) may be necessary for low employee burnout ($Y$). Low workload can result in boredom and lack of purpose, which may limit the possibility of low burnout; high workload can cause stress and exhaustion, which also limits the possibility of low burnout. Therefore, a moderate workload is necessary for low burnout because both too low and too high workload create psychological strain that prevents burnout from remaining low.

$$XY$-plot showing an optimum level of the necessary condition. Left: An optimum level of $x_{c1} <x < x_{c2}$ is necessary for a high level of $y = y_c$. Right: An optimum level of $x_{c1} < x < x_{c2}$ is necessary for a low level of $y = y_c$.$ $$XY$-plot showing an optimum level of the necessary condition. Left: An optimum level of $x_{c1} <x < x_{c2}$ is necessary for a high level of $y = y_c$. Right: An optimum level of $x_{c1} < x < x_{c2}$ is necessary for a low level of $y = y_c$.$

Figure 3.9: $XY$-plot showing an optimum level of the necessary condition. Left: An optimum level of $x_{c1} <x < x_{c2}$ is necessary for a high level of $y = y_c$. Right: An optimum level of $x_{c1} < x < x_{c2}$ is necessary for a low level of $y = y_c$.

It is also possible to theorize that a high level of $X$ is necessary for an extreme (low or high) level of $Y$. Then the upper-left and the lower-left corners are expected to be empty (Figure 3.10-left). Similarly, it may be theorized that a low level of $X$ is necessary for an extreme (low or high) level of $Y$. Then the upper-right and the lower-right corners are expected to be empty (Figure 3.10-right).

$$XY$-plot showing necessity for an extreme level of the outcome. Left: A high level of $x > x_c$ is necessary for an extreme (low or high) level of $y = y_{c1}$ or $y = y_{c2}$. Right: A low level of $x < x_c$ is necessary for an extreme (low or high) level of $y = y_{c1}$ or $y = y_{c2}$.$ $$XY$-plot showing necessity for an extreme level of the outcome. Left: A high level of $x > x_c$ is necessary for an extreme (low or high) level of $y = y_{c1}$ or $y = y_{c2}$. Right: A low level of $x < x_c$ is necessary for an extreme (low or high) level of $y = y_{c1}$ or $y = y_{c2}$.$

Figure 3.10: $XY$-plot showing necessity for an extreme level of the outcome. Left: A high level of $x > x_c$ is necessary for an extreme (low or high) level of $y = y_{c1}$ or $y = y_{c2}$. Right: A low level of $x < x_c$ is necessary for an extreme (low or high) level of $y = y_{c1}$ or $y = y_{c2}$.

The situations shown in Figures 3.9 and 3.10 do not have dichotomous equivalents, as this would imply that the outcome or the condition is always absent (Figure 3.11-top-left) or always present (Figure 3.11- top-right), or that the condition is always present (Figure 3.11-bottom-left), or always absent (Figure 3.11-bottom-right).

$XY$-tables showing two adjacent empty corners without observations. Top-Left: presence of $Y$ is not possible. Top-Right: absence of $Y$ is not possible. Bottom-Left: absence of $X$ is not possible. Bottom-Right: presence of $X$ is not possible.

Figure 3.11: $XY$-tables showing two adjacent empty corners without observations. Top-Left: presence of $Y$ is not possible. Top-Right: absence of $Y$ is not possible. Bottom-Left: absence of $X$ is not possible. Bottom-Right: presence of $X$ is not possible.

When two opposite corners are empty, as shown in Figure 3.12-left, two claims apply: the presence/high level of $X$ is necessary for the presence/high level of $Y$ (empty upper-left corner) and that the absence/low level of $X$ is necessary for the absence/low level of $Y$ (empty lower-right corner). Similarly, it may be theorized that the absence/low level of $X$ is necessary for the presence/high level of $Y$ (empty upper-right corner) and that the presence/high level of $X$ is necessary for the absence/low level of $Y$ (empty lower-left corner). This is shown in Figure 3.12-right. This situation has dichotomous equivalents, as shown in Figure 3.13.

$Double necessity. Left: A high level of $x > x_{c1}$ is necessary for a high level of $y = y_{c1}$, and a low level of $x < x_{c2}$ is necessary for a low level of $y = y_{c2}$. Right: A low level of $x < x_{c1}$ is necessary for a high level of $y = y_{c1}$, and a high level of $x > x_{c2}$ is necessary for a low level of $y = y_{c2}$.$ $Double necessity. Left: A high level of $x > x_{c1}$ is necessary for a high level of $y = y_{c1}$, and a low level of $x < x_{c2}$ is necessary for a low level of $y = y_{c2}$. Right: A low level of $x < x_{c1}$ is necessary for a high level of $y = y_{c1}$, and a high level of $x > x_{c2}$ is necessary for a low level of $y = y_{c2}$.$

Figure 3.12: Double necessity. Left: A high level of $x > x_{c1}$ is necessary for a high level of $y = y_{c1}$, and a low level of $x < x_{c2}$ is necessary for a low level of $y = y_{c2}$. Right: A low level of $x < x_{c1}$ is necessary for a high level of $y = y_{c1}$, and a high level of $x > x_{c2}$ is necessary for a low level of $y = y_{c2}$.

$XY$-tables showing two opposite empty corners without observations. Left: the presence of $X$ is necessary for the presence of $Y$ and the absence of $X$ is necessary for the absence of $Y$. Right: the absence of $X$ is necessary for the presence of $Y$ and the presence of $X$ is necessary for the absence of $Y$.

Figure 3.13: $XY$-tables showing two opposite empty corners without observations. Left: the presence of $X$ is necessary for the presence of $Y$ and the absence of $X$ is necessary for the absence of $Y$. Right: the absence of $X$ is necessary for the presence of $Y$ and the presence of $X$ is necessary for the absence of $Y$.

3.8 The meaning of ‘necessary but not sufficient’

To understand what is necessary but not sufficient, this section first explains what is meant by necessary and sufficient. In the dichotomous situation, ‘necessary and sufficient’ means that the upper-left corner and the lower-right corner of the $XY$-plot are both empty. This is shown in Figure 3.13, which is interpreted as a double necessity relationship of $X$ for $Y$: a necessity relationship where the presence of $X$ is necessary for the presence of $Y$ (upper-left corner empty) and a necessity relationship where the absence of $X$ is necessary for the absence of $Y$ (lower-right corner empty). By logic, in the dichotomous situation the second relationship can be reformulated as a sufficiency relationship where the presence of $X$ is sufficient for the presence of $Y$, as shown in Figure 2.1-left. The first relationship (necessity) and the second relationship (sufficiency) now have the same $X$-value (presence) and the same $Y$-value (presence). This allows a single statement that the presence of $X$ is necessary and sufficient for the presence of $Y$. If the sufficiency relationship does not hold, the formulation of the combined relationship would be: the presence of $X$ is necessary but not sufficient for the presence of $Y$. This means that the lower-right corner is not empty.

In discrete and continuous situations, necessity and sufficiency are more complex. Like in the dichotomous situation, the upper-left corner and the lower-right corner are both empty. This is shown in Figure 3.12-left, which is interpreted as a double necessity relationship of $X$ for $Y$: a necessity relationship where a high level of $X$ is necessary for a high level of $Y$ (upper-left corner empty) and a necessity relationship where a low level of $X$ is necessary for a low level of $Y$ (lower-right corner empty). By logic, the second relationship can again be reformulated as a sufficiency relationship: a not-low level of $X$ is sufficient for a not-low level of $Y$. However, a ‘not-low’ level is usually not the same as a ‘high’ level. For example, in Figure 3.12-left, level $X = x_{c1}$ is necessary for level $Y = y_{c1}$, whereas level $X = x_{c2}$ is sufficient for level $Y = y_{c2}$. There are no common $X$-levels and common $Y$-levels that would allow a single statement that level $X = x_{c}$ is necessary and sufficient for level $Y = y_{c}$, unless the upper ceiling line and the lower ceiling line (floor line) overlap or intersect at the point (1,1). If the lines overlap there are only observations on the ceiling line and the necessity and sufficiency statement holds for all observations. If the lines intersect at point [1,1], only at that point the necessity and sufficiency statement holds. Both situations are, however, exceptional.

In all other situations the statement that level $X = x_{c}$ is necessary but not sufficient for level $Y = y_{c}$ has a broader meaning than that the lower-right corner is not empty. If the lower-right corner can have observations, observations are normally possible in the entire area below the ceiling line, not just in the lower-right corner. This allows a single statement that $X = x_{c}$ is necessary but not sufficient for $Y = y_{c}$. When $X = x_{c}$, no observations are above point $C$ on the ceiling line, whereas observations below point $C_1$ are possible. In other words, for a given level $X = x_{c}$, it is not possible to have values of $Y > y_{c}$ (necessity applies) but it is possible to have values $Y \leq y_{c}$ (“non-sufficiency” applies). Consequently, where in the dichotomous situation the statement $X$ is necessary but not sufficient for $Y$ means that observations are possible in the lower-right corner, in the discrete and continuous situations it means that there are observations below the ceiling line. NCA focuses on the identification of necessity (rather than on the identification of non-sufficiency) in any corner of the $XY$-table or the $XY$-plot. If the focus is on the lower-right corner, the emptiness of this corner is interpreted as indicating that the absence or a low level of $X$ is necessary for the absence or a low level of $Y$.

Chapter 4 Mathematics

4.1 Summary of this chapter

This chapter describes the mathematics of a necessity relationship between a pair of variables: $X$ is necessary for $Y$. The chapter focuses on necessity-in-degree (NiD) that describes the general situation of two continuous variables with an empty space in the upper-left corner of the $XY$-plot. Using a conventional Euclidean coordinate system where the $X$-axis runs horizontally with increasing values from left to right, and the $Y$-axis runs vertically with increasing values from bottom to top, this indicates that a high level of $X$ is necessary for a high level of $Y$. The mathematical approach described here can be extended to situations with dichotomous or discrete variables, or to analyzing other corners, though this is not addressed in this chapter. The chapter first presents the branches of mathematics applied in NCA (Section 4.2). In the subsequent section NCA’s mathematical model specification is presented (Section 4.3). It includes the specification of the bounding box, the ceiling line, and the model’s degrees of freedom. This is followed by a discussion on how to estimate a necessity model by estimating parameters related to the necessity effect size and model fit. Next, the situation of multiple necessary conditions is discussed, explaining why NCA considers multiple $X_jY$-planes (multiple projections), rather than one multidimensional space, and explaining how a chain of necessity can be analyzed (Section 4.6).

4.2 Branches of mathematics

Rather than relying on statistics and probability theory, NCA employs two branches of mathematics: geometry and algebra. Geometry describes relationships between objects and spaces including points, lines, angles, and planes. NCA focuses on points in the two-dimensional space ($XY$-plane), describes relationships between points (e.g., ceiling line) and considers the size of an area (e.g., effect size). Algebra uses symbols, letters, and numbers and describes their relationships and operations including variables, constants, functions, equations, and inequalities. NCA uses functions to describe the ceiling line with constants and variables, uses equations to calculate effect size, and uses inequalities to specify (in)feasible areas. When describing NCA, geometry and algebra are intertwined. NCA is not a statistical data analysis method per se, although it uses tools from statistics (Chapter 5).

4.3 Model specification

In NCA, variables $X$ and $Y$ can be dichotomous (two possible levels), discrete (more than two possible levels) or continuous variables (infinite number of possible levels). With numeric variables (including ratio- and interval-scale variables), effect size and other necessity parameters can be calculated. With categorical variables, these calculations are not possible unless equal distances between ordered categories are assumed (e.g., “low”, “medium”, “high”). NCA can handle both non-random and random variables. When NCA uses random variables for $X$ and $Y$, NCA only assumes that the univariate distributions are bounded, but no assumption is needed about the shape of the distribution.²⁹

NCA’s model specification consists of specifying the bounding box and the ceiling line.

4.3.1 Bounding box

NCA assumes that the condition and the outcome are bounded. There are two reasons for this. First, variables represent properties of phenomena. Many properties are finite, represented by minimum and maximum values of the variables, rather than being infinite (e.g., height of people). Second, the assumption that $X$ and $Y$ are bounded allows the definition and calculation of the necessity effect size (Section 4.4.1) and other NCA parameters (Sections 4.4.2 and 4.5). Assuming bounds on $(X,Y)$ means that there are four constants that define the ranges of the condition $X$ and the outcome $Y$:

\[\begin{equation} \tag{4.1} x_{\min} \leq x \leq x_{\max}\quad \text{ and } \quad y_{\min} \leq y \leq y_{\max} \end{equation}\]

These bounds define the bounding box in an $XY$-plot. The bounding box is defined by the minimum and maximum values of $X$ and $Y$, that is $[x_{\min}, x_{\max}] \times [y_{\min}, y_{\max}]$. The feasible and empty areas, as well as the ceiling line and its extensions, are defined relative to this bounding box.

The scope ($S$) is the area of the bounding box and is mathematically expressed as:

\[\begin{equation} \tag{4.2} S = {(x_{\max} - x_{\min})}\cdot{(y_{\max} - y_{\min})} \end{equation}\]

The bounding box can be constructed in two ways: theoretically and empirically. The theoretical scope is defined by theoretical bounds of $X$ and $Y$ when the theoretical bounds are known or presumed.³⁰ The empirical scope is defined by the observed minimum and maximum values of $X$ and $Y$. This option is often selected when theoretical bounds are unknown.

Without loss of generality (for the purpose of NCA), the variables can be linearly transformed to the unit square $[0,1]\times[0,1]$ to obtain a scope value of 1 (Section 8.5.2).³¹ Then a point $(x,y)$ gets transformed by min-max normalization into $(x',y')$ as follows:

\[\begin{equation} \tag{4.3} x' = \frac{x - x_{\min}}{x_{\max} - x_{\min}} \quad \text{ and } \quad y' = \frac{y - y_{\min}}{y_{\max} - y_{\min}} \end{equation}\]

4.3.2 Ceiling line

When necessity exists, the bounding box has an area where points are possible (feasible area) and an area where points are impossible (empty space or ceiling zone). The ceiling line³² is a line within the bounding box that separates these two areas. Specifying the ceiling line involves choosing its location (in which corner) and its form.

The location depends on the direction of the hypothesis (Section 3.5), which defines the expected empty corner. For a high-high hypothesis (presence/high value of $X$ is necessary for presence/high value of $Y$), the expected empty corner is the upper-left corner of the $XY$-plot (corner 1).³³

To specify the form of the ceiling line, several options exist. Figure 4.1 shows three classes of non-decreasing ceiling lines in a bounding box. The first class is the Ceiling Envelopment Free Disposal Hull (CE-FDH), which is a stepwise linear ceiling line consisting of several horizontal and vertical line segments (Figure 4.1-top-left), the second class is Ceiling Envelopment Variable Returns to Scale (CE-VRS), which is a concave piecewise linear ceiling consisting of several oblique line segments (Figure 4.1-top-right), and the final one is a linear ceiling line consisting of one segment (Figure 4.1-bottom). The ceiling line represents an upper bound on the outcome in the feasible area (the gray area). The ceiling line is considered a part of the feasible area.

Three classes of ceiling lines within a bounding box for high-high necessity. Top-Left: stepwise linear. Top-Right: concave piecewise linear. Bottom: linear. White: ceiling zone. Gray: feasible area. Numbered points are peers (dominant point with no cases above and to the left of it) connected by solid line segments constructing the ceiling line. The dashed lines are extensions such that the ceiling line envelopes the feasible area.

Figure 4.1: Three classes of ceiling lines within a bounding box for high-high necessity. Top-Left: stepwise linear. Top-Right: concave piecewise linear. Bottom: linear. White: ceiling zone. Gray: feasible area. Numbered points are peers (dominant point with no cases above and to the left of it) connected by solid line segments constructing the ceiling line. The dashed lines are extensions such that the ceiling line envelopes the feasible area.

A ceiling line is assumed to be non-decreasing for two reasons.³⁴ First, it allows the expression that at least a certain level of $X$ is necessary for a certain level of $Y$. Second, a non-decreasing ceiling line has the ‘no-reversal’ (no change of direction) property. This means that if a point $(x,y)$ is in the ceiling zone, then any point $(x',y')$ to the left or above it (a point with $x'\leq x$ and $y'\geq y$) and within the bounding box is also in the ceiling zone. Similarly, if a point $(x,y)$ is in the feasible area, then any point $(x',y')$ in the bounding box with $x' \geq x$ and $y' \leq y$ is also in the feasible area. This property is used for the mathematical expression of the ceiling line (see below) and for two measures of imperfect necessity, namely purity and solidity (Sections 4.5.4 and 4.5.5, respectively).

4.3.2.1 Stepwise linear ceiling line

NCA uses the stepwise linear ceiling line (CE-FDH) as the base ceiling model. It refers to the Free Disposal Hull (FDH) technique (Deprins et al., 1984) that originates in production economics. This application of FDH identifies the efficiency of decision-making units (DMUs) with a certain production input ($X$) and output ($Y$). The technique assumes the free disposability property meaning that if a DMU can produce a certain output, it can also reduce the amount of output without using additional input, and use more input without increasing output. FDH identifies the most efficient DMUs (points with maximum output for a given input and minimum input for a given output). The outer boundary of the hull (‘envelope’ or ‘frontier’) connects the most efficient points (called ‘peers’). Points below the frontier are considered inefficient.

NCA uses the FDH technique in 2D. In NCA’s context, the hull is called the feasible area, and the disposability property is called the no-reversal property. The feasible area is the smallest area defined by peers: a strictly increasing set of points, $(x_i,y_i)_{i=1,\ldots,P}$ for which

\[\begin{equation} \tag{4.4} x_i<x_{i+1}\quad\text{and}\quad y_i<y_{i+1}\quad\text{for}\quad i=1,\ldots,P-1 \end{equation}\]

Where $P$ is the number of peers. Peers are points that dominate other points. When the upper-left corner is the expected empty corner (as in Figure 4.1), no points exist to the left and above the peers.

A class of ceiling lines is defined by the way that peers are defined and connected. The CE-FDH-class of ceiling lines consists of all models with a feasible area generated by a set of $P$ peers connected by horizontal and vertical line segments (Figure 4.1-top-left).

Let the peers be $(x_i,y_i)_{i=1,\ldots,P}$. The feasible space generated by these peers is the hypograph³⁵ of the ceiling function $f(x)$, defined for $x\in[x_1,x_{\max}]$ by

\[\begin{equation} \tag{4.5} f_{\mathrm{CE-FDH}}(x)= \begin{cases} y_i & \text{if } x_i\le x<x_{i+1}\quad (i=1,\ldots,P-1),\\ y_P & \text{if } x_P\le x\le x_{\max} \end{cases} \end{equation}\]

and $f_{\mathrm{CE-FDH}}(x)$ is not defined for $x<x_1$.

The generalized inverse is:

\[\begin{equation} \tag{4.6} f_{\mathrm{CE-FDH}}^{-}(y)=\inf\{x\mid f_{\mathrm{CE-FDH}}(x)\ge y\}= \begin{cases} x_1 & \text{if } y\le y_1,\\ x_{i+1} & \text{if } y_i<y\le y_{i+1}\quad (i=1,\ldots,P-1) \end{cases} \end{equation}\]

and $f_{\mathrm{CE-FDH}}^{-}(y)$ is undefined (infeasible) if $y>y_P$.

The associated feasible space is generated by all points on the line segments from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$ for $i=1,\ldots,P-1$.

The ceiling line is given by the sequence of line segments from $(x_1,y_{\min})$ to $(x_1,y_1)$, to $(x_2,y_1)$, to $(x_2,y_2)$, and so on, ending at $(x_P,y_P)$ and $(x_{\max},y_P)$. Possibly, the first segment from $(x_1,y_{\min})$ to $(x_1,y_1)$ and/or the last segment from $(x_P,y_P)$ to $(x_{\max},y_P)$ has length equal to zero (starts and ends at the bounding box). When the first peer starts at $x = x_{min}$ and the last peer ends at $y = y_{max}$ the bounding box is called a tight bounding box.

4.3.2.2 Concave piecewise linear (CE-VRS) ceiling line

The CE-VRS-class of ceiling line consists of all models with a set of $P$ peers such that the sequence of peers forms a concave sequence, i.e. the slopes between consecutive peers strictly decrease:

\[\begin{equation} \tag{4.7} \frac{y_{i+1}-y_i}{x_{i+1}-x_i} > \frac{y_{i+2}-y_{i+1}}{x_{i+2}-x_{i+1}} \quad\text{for}\quad i=1,\ldots,P-2 \end{equation}\]

The ceiling function is given by linear interpolation between consecutive peers:

\[\begin{equation} \tag{4.8} f_{\mathrm{CE-VRS}}(x)=y_{i-1}+\frac{y_i-y_{i-1}}{x_i-x_{i-1}}(x-x_{i-1}) \quad\text{if}\quad x_{i-1}\le x\le x_i\quad (i=2,\ldots,P) \end{equation}\]

and $f_{\mathrm{CE-VRS}}(x)=y_P$ if $x\ge x_P$, while $f_{\mathrm{CE-VRS}}(x)$ is not defined for $x<x_1$.

The generalized inverse is:

\[\begin{equation} \tag{4.9} f_{\mathrm{CE-VRS}}^{-}(y)=\inf\{x\mid f_{\mathrm{CE-VRS}}(x)\ge y\}= \begin{cases} x_1 & \text{if } y\le y_1,\\[2mm] x_i+\dfrac{x_{i+1}-x_i}{y_{i+1}-y_i}(y-y_i) & \text{if } y_i<y\le y_{i+1}\quad (i=1,\ldots,P-1) \end{cases} \end{equation}\]

and $f_{\mathrm{CE-VRS}}^{-}(y)$ is undefined (infeasible) if $y>y_P$.

The associated feasible space is generated by all points on the line segments from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$ for $i=1,\ldots,P-1$.

The ceiling line follows the linear segments from $(x_1,y_{\min})$ to $(x_1,y_1)$, then to $(x_2,y_2)$, and so on, ending at $(x_P,y_P)$ and $(x_{\max},y_P)$. Possibly, the first and/or last segment has length equal to zero.

4.3.2.3 Linear ceiling line

The class of linear ceiling lines consists of models with a ceiling that is a (positive-length) segment of a line. The ceiling function is

\[\begin{equation} \tag{4.10} f_{\mathrm{LIN}}(x)=\max\{y_{\min},\min\{y_{\max},a+bx\}\} \end{equation}\]

with $b>0$ to ensure the line is strictly increasing and with $(a+bx_{\min},\,a+bx_{\max}]\cap[y_{\min},y_{\max}]$ to ensure the line intersects the bounding box with a segment of positive length. This is equivalent to

\[\begin{equation} \tag{4.11} a+bx_{\max}>y_{\min}\quad\text{and}\quad a+bx_{\min}<y_{\max} \end{equation}\]

which in turn is equivalent to

\[\begin{equation} \tag{4.12} y_{\min}-bx_{\max}<a<y_{\max}-bx_{\min}. \end{equation}\]

The line $y=a+bx$ intersects the boundary of the bounding box in two points $(x_1,y_1)$ and $(x_2,y_2)$ with

\[\begin{equation} \tag{4.13} (x_1,y_1)=\big(\max\{x_{\min},(y_{\min}-a)/b\},\ \max\{y_{\min},a+bx_{\min}\}\big) \end{equation}\]

and

\[\begin{equation} \tag{4.14} (x_2,y_2)=\big(\min\{x_{\max},(y_{\max}-a)/b\},\ \min\{y_{\max},a+bx_{\max}\}\big). \end{equation}\]

The set of two points $(x_1,y_1)$ and $(x_2,y_2)$ forms a set of peers. In terms of these peers, the ceiling can be written as

\[\begin{equation} \tag{4.15} f_{\mathrm{LIN}}(x)= \begin{cases} y_1+\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)=a+bx & \text{if } x_1\le x\le x_2,\\ y_2 & \text{if } x\ge x_2 \end{cases} \end{equation}\]

with $f_{\mathrm{LIN}}(x)$ not defined for $x<x_1$.

The generalized inverse is:

\[\begin{equation} \tag{4.16} f_{\mathrm{LIN}}^{-}(y)=\inf\{x\mid f_{\mathrm{LIN}}(x)\ge y\}= \begin{cases} x_1 & \text{if } y\le y_1,\\[2mm] x_1+\dfrac{x_2-x_1}{y_2-y_1}(y-y_1)=\dfrac{y-a}{b} & \text{if } y_1<y\le y_2 \end{cases} \end{equation}\]

and $f_{\mathrm{LIN}}^{-}(y)$ is undefined (infeasible) if $y>y_2$.

The ceiling line consists of the line segments from $(x_1,y_{\min})$ to $(x_1,y_1)$, then to $(x_2,y_2)$, and finally to $(x_{\max},y_2)$. Possibly, the first and/or last segment has length equal to zero.

4.3.3 Model degrees of freedom

A necessity model specifies the bounding box with the ceiling line that is inside it. The ceiling line is assumed not to be outside the bounding box. Model degrees of freedom refers to the minimum number of parameters that are needed to describe the necessity model. The bounding box has $4$ degrees of freedom. It can be described by, for example, the coordinates of the lower-left and the upper right corner points: $x_{\min}, y_{\min}, x_{\max}, y_{\max}$.

Since a sequence of peers in the two classes of segmented ceiling lines (CE-FDH and CE-VRS) is defined in terms of strict inequalities, each peer can change position slightly without violating the conditions. Hence, the degrees of freedom of each of the CE-FDH and CE-VRS classes is $2P$.

Since the class of linear ceiling lines only imposes restrictions on $a$ and $b$ through strict inequalities, the degrees of freedom equals $2$.

Therefore the general expression for the degrees of freedom of a necessity model is:

\[\begin{equation} \tag{4.17} df_{\text{model}} = \begin{cases} 6, & \text{linear ceiling line}. \\ 2P + 4, & \text{segmented ceiling line}. \end{cases} \end{equation}\]

where $P$ is the number of peers of the segmented ceiling lines (CE-FDH and CE-VRS).

4.4 Mathematical estimation

Mathematical estimation techniques using different ceiling lines. CE-FDH: stepwise linear ceiling line. CR-FDH: linear ceiling line (shown as the lower solid line). C-LP: linear ceiling line (shown as dashed line). CE-VRS: concave piecewise linear ceiling line (shown as segments passing through A, B, C, D, and E). CR-VRS: linear ceiling line (shown as the upper solid line). QR (Quantile Regression): linear ceiling line (shown as the dashed-dotted line in the middle). Population ceiling line: Y = 0.3 + X. Population effect size = 0.25. 50 points randomly selected from the population.

Figure 4.2: Mathematical estimation techniques using different ceiling lines. CE-FDH: stepwise linear ceiling line. CR-FDH: linear ceiling line (shown as the lower solid line). C-LP: linear ceiling line (shown as dashed line). CE-VRS: concave piecewise linear ceiling line (shown as segments passing through A, B, C, D, and E). CR-VRS: linear ceiling line (shown as the upper solid line). QR (Quantile Regression): linear ceiling line (shown as the dashed-dotted line in the middle). Population ceiling line: Y = 0.3 + X. Population effect size = 0.25. 50 points randomly selected from the population.

The mathematical estimation of the necessity model consists of estimating the parameters of the bounding box and the ceiling line from data. This can be done with the NCA software (Appendix B).

The estimation of the bounding box requires that the analyst selects the type of bounding box (defined by the theoretical or empirical scope). For the theoretical scope, the analyst specifies $x_{min}$, $y_{min}$, $x_{max}$ and $y_{max}$. For the empirical scope the software estimates these values by taking the observed extremes.

The estimation of the ceiling line requires that the analyst selects the form of the ceiling line. If a segmented ceiling line is selected (CE-FDH or CE-VRS), the software provides the estimated $x$- and $y$-coordinates of the peers. To select a linear ceiling line there are several options for this line. The Ceiling Regression Free Disposal Hull ceiling line (CR-FDH) is a trend line through the CE-FDH peers using least squares optimization. The Ceiling Regression Variable Returns to Scale (CR-VRS) ceiling line is the trend line through the CE-VRS peers using least squares optimization. The Ceiling - Linear Programming (C-LP) ceiling line is the line that is tight to the CE-FDH peers such that the sum of heights of the CE-FDH peers is minimized (“least sum”). The different lines are shown in Figure 4.2,³⁶

After selecting form of the line, the software provides the values of the parameters to describe the ceiling line (e.g., intercept and slope for the linear ceiling line). Criteria for selecting the type of bounding box and form of the ceiling line are discussed in Section ??.

NCA’s mathematical estimations with the above ceiling techniques are based on the location of the peers. CE-FDH and CE-VRS use peers directly, while the linear ceiling lines CR-FDH, CR-VRS and C-LP ceiling line use them indirectly. The use of only points near the border is motivated by several fundamental considerations.³⁷ First, NCA’s primary object of interest is the empty space above the ceiling: the region where observations are impossible. This follows necessity logic: if not $X$, then not $Y$. The ceiling zone is the complement of the feasible area in the bounding box. NCA quantifies the size of this zone and relates it to the size of the bounding box, which requires knowing the maximum feasible area but does not require modeling how points are distributed within that feasible area. Consequently, only points close to the ceiling are informative for locating the boundary. Points far below it do not contribute to identifying the boundary.

Second, conceptually, in NCA’s deterministic logic, the ceiling is an abrupt transition from a region that can contain points to a region that cannot, more like a cliff edge than a gradual slope. The characteristic feature of a ceiling is thus a sharp boundary between feasibility and infeasibility. Methods that estimate the ceiling indirectly via a gradual (non-abrupt) distributional model of all data risk misrepresenting this discontinuity. A gradual model can blur the boundary and thereby estimate a “soft” transition that is not characteristic of a true ceiling.

Third, ceiling estimates should be insensitive to data far below the ceiling. For example, increasing the density of points in the lower-right corner should not have an effect on how the ceiling is constructed. Such points would pull the estimated ceiling estimates downward. This inflates the estimated empty space and thus overstates the true effect size. For this reason, ceiling estimation techniques that rely on fitting the full distribution under the ceiling are generally risky and should be applied with caution.³⁸ This is illustrated with the distribution-based Quantile Regression (QR) ceiling line. In Figure 4.2, this line lies among the other ceilings (that all only use points near the ceiling) and approximates the true ceiling line well. However, after altering the distribution of points by adding observations in the lower-right corner, the QR-based ceiling becomes biased, whereas ceilings estimated from on upper points remain stable (4.3). Broader evidence from Monte Carlo simulations likewise suggests that a distribution-based method like QR can exhibit bias, even at larger sample sizes, warranting prudence in their use (Section ??).

Comparison of an estimation techniques based on the distribution of all data below the ceiling: QR = Quantile Regression (shown as the near-horizontal dash-dotted line) and based on upper peers (the other lines). Population ceiling line = $Y = 0.3 + X$. Population effect size is 0.25. First 50 points randomly selected from the population, next 50 points added in the lower-right corner.

Figure 4.3: Comparison of an estimation techniques based on the distribution of all data below the ceiling: QR = Quantile Regression (shown as the near-horizontal dash-dotted line) and based on upper peers (the other lines). Population ceiling line = $Y = 0.3 + X$. Population effect size is 0.25. First 50 points randomly selected from the population, next 50 points added in the lower-right corner.

4.4.1 Effect size

When a high value of $X$ is necessary for a high value of $Y$, the ceiling line determines the maximum possible $y$-value ($y$ = $y_{c}$) for a given $x$-value, while the bounding box determines the absolute maximum possible $y$-value ($y=y_{\max}$). The difference between the two can be considered the constraint that $X$ puts on $Y$. The increase of the ceiling line expresses the change in constraint under varying values of $X$. The maximum constraint occurs for $x = x_{\min}$. The effect builds up with increasing $x$ at a rate determined by the increase in the ceiling line. The total effect is then the area between the ceiling line and the maximum possible $y = y_{\max}$.

Referring to Figure 4.1, the general expression for the necessity effect size is as follows. Let the bounding box be $B=[x_{\min},x_{\max}]\times[y_{\min},y_{\max}]$ with $x_{\max}>x_{\min}$ and $y_{\max}>y_{\min}$ and let $c(x)$ be the ceiling value of defined on $[x_1,x_{\max}]$ (with $x_1$ the leftmost $x$ where the ceiling is defined).

The scope $S$ is the area of the bounding box:

\[\begin{equation} \tag{4.18} S=(x_{\max}-x_{\min})(y_{\max}-y_{\min}) \end{equation}\]

The feasible area is the area under the ceiling line $f(x)$ (only for $x\in[x_1,x_{\max}]$):

\[\begin{equation} \tag{4.19} F=\int_{x_1}^{x_{\max}}\big(f(x)-y_{\min}\big)\,dx \end{equation}\]

The ceiling zone $C$ is the complement of the feasible area within the bounding box:

\[\begin{equation} \tag{4.20} C=S-F \end{equation}\]

Equivalently, $C$ can be written as the sum of the left strip (where the ceiling is undefined) and the area above the ceiling for $x\in[x_1,x_{\max}]$:

\[\begin{equation} \tag{4.21} C=(x_1-x_{\min})(y_{\max}-y_{\min})+\int_{x_1}^{x_{\max}}\big(y_{\max}-f(x)\big)\,dx \end{equation}\]

Consequently the effect size $d$ is:

\[\begin{equation} \tag{4.22} d=\frac{C}{S} =1-\frac{F}{S} \end{equation}\]

The effect size for the different ceiling classes can be derived from the ceiling line equations.

4.4.1.1 Class of stepwise linear ceiling lines

For the CE-FDH class of ceiling line Equation (4.5), the feasible area is:

\[\begin{equation} \tag{4.23} F_{\mathrm{CE-FDH}} =\sum_{i=1}^{P-1}(x_{i+1}-x_i)(y_i-y_{\min}) +(x_{\max}-x_P)(y_P-y_{\min}) \end{equation}\]

Therefore, the effect size is:

\[\begin{equation} \tag{4.24} d_{\mathrm{CE-FDH}} =1-\frac{F_{\mathrm{CE-FDH}}}{S} =1-\frac{ \sum_{i=1}^{P-1}(x_{i+1}-x_i)(y_i-y_{\min}) +(x_{\max}-x_P)(y_P-y_{\min}) }{ (x_{\max}-x_{\min})(y_{\max}-y_{\min}) } \end{equation}\]

4.4.1.2 Class of concave piecewise linear ceiling lines

For the CE-VRS class of ceiling line Equation (4.8), the feasible area is:

\[\begin{equation} \tag{4.25} F_{\mathrm{CE-VRS}} =\sum_{i=1}^{P-1}(x_{i+1}-x_i)\left(\frac{y_i+y_{i+1}}{2}-y_{\min}\right) +(x_{\max}-x_P)(y_P-y_{\min}). \end{equation}\]

Therefore, the effect size is: \[\begin{equation} \tag{4.26} d_{\mathrm{CE-VRS}} =1-\frac{F_{\mathrm{CE-VRS}}}{S} =1-\frac{ \sum_{i=1}^{P-1}(x_{i+1}-x_i)\left(\frac{y_i+y_{i+1}}{2}-y_{\min}\right) +(x_{\max}-x_P)(y_P-y_{\min}) }{ (x_{\max}-x_{\min})(y_{\max}-y_{\min}) }. \end{equation}\]

4.4.1.3 Class of linear ceiling lines

For the class of linear ceiling line, let the ceiling line intersect the bounding box at two points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_{\min}\le x_1 < x_2 \le x_{\max}$. The feasible area under the ceiling within the bounding box is:

\tag{4.27} \[\begin{equation} F_{\mathrm{LIN}} =(x_2-x_1)\left(\frac{y_1+y_2}{2}-y_{\min}\right) +(x_{\max}-x_2)(y_{\max}-y_{\min}) \end{equation}\]

Therefore, the effect size is:

\tag{4.30} \[\begin{equation} d_{\mathrm{LIN}} =1-\frac{F_{\mathrm{LIN}}}{S} =1-\frac{ (x_2-x_1)\left(\frac{y_1+y_2}{2}-y_{\min}\right) +(x_{\max}-x_2)(y_{\max}-y_{\min}) }{ (x_{\max}-x_{\min})(y_{\max}-y_{\min}) } \end{equation}\]

4.4.2 Necessity inefficiency

As illustrated in Figure 4.4, for the standard case of a linear ceiling line above the diagonal, $X$ may not constrain $Y$ over the entire ranges of $X$ and $Y$ within the bounding box. The ceiling line intersects with the bounding box: $[x_{\min}, y_{\operatorname{cmin}}]$ and $[x_{\operatorname{cmax}}, y_{\max}]$. The vertical extension is the distance $y_{\operatorname{1}} - y_{\min}$, and the horizontal extension $x_{\max} - x_{\operatorname{2}}$.

$Necessity inefficiency refers to the part of the feasible area where $Y$ is not constrained by $X$ in the area $[x_{min}, x_{max}]\times[y_{min}, y_{1}]$ (outcome inefficiency), and to the part of the feasible area where $X$ does not constrain $Y$ in the area $[x_{2}, x_{max}] \times [y_{min}, y_{max}]$ (condition inefficiency).$

Figure 4.4: Necessity inefficiency refers to the part of the feasible area where $Y$ is not constrained by $X$ in the area $[x_{min}, x_{max}]\times[y_{min}, y_{1}]$ (outcome inefficiency), and to the part of the feasible area where $X$ does not constrain $Y$ in the area $[x_{2}, x_{max}] \times [y_{min}, y_{max}]$ (condition inefficiency).

When $(x_c,y_c)$ is a point on the ceiling line and $x < x_{c}$, then $x$ is a bottleneck for $y = y_{c}$. Only an increased value of $x$ such that $x\geq x_{c}$ enables a value $y = y_{c}$. For reaching the value of $y = y_{\max}$ it is necessary to have a value of $x\geq x_{\operatorname{2}}$, where $x_{\operatorname{2}}$ is the value of $x$ of the point where the ceiling line crosses the $y = y_{\max}$ line. Thus, for enabling the maximum possible value $y_{\max}$, $x$ should be at least $x = x_{\operatorname{2}}$. Increasing $x$ beyond $x = x_{\operatorname{2}}$ for enabling $y = y_{\max}$ is not needed as $y$ is no longer constrained. This is called ‘inefficiency’ regarding enabling the maximum outcome.

Condition inefficiency specifies the extent to which $x$ does not constrain $y$ for levels of $x_{\operatorname{2}}\leq x\leq x_{\max}$. Only for $x_{\min}\leq x\leq\ x_{\operatorname{2}}$, $x$ constrains $y$ (ceiling line exists for $x_{\min} \leq x\leq x_{\operatorname{2}}$). Condition inefficiency is expressed as a percentage:

\[\begin{equation} \tag{4.31} \text{condition inefficiency} = \frac{x_{\max} - x_{\operatorname{2}}}{x_{\max} - x_{\min}} * 100\% \end{equation}\]

where $x_{\max}$ and $x_{\min}$ are the minimum and maximum values of $x$ (bounds) and $x_{\operatorname{2}}$ is the value of $x$ of the point where the ceiling line crosses the $y = y_{\max}$ line.

Similarly, outcome inefficiency specifies the extent to which $Y$ is not constrained by $X$ for lower levels of $y_{\min}\leq y\leq y_{\operatorname{1}}$, where $y_{\operatorname{1}}$ is the value of $y$ of the point where the ceiling line crosses the $x = x_{\min}$ line. Only for $y_{\operatorname{1}}\leq y\leq y_{\max}$, $y$ is constrained by $X$ (ceiling line exists for $y_{\operatorname{1}}\leq y\leq y_{\max}$). Outcome inefficiency is expressed as a percentage:

\[\begin{equation} \tag{4.32} \text{outcome inefficiency} = \frac{y_{\operatorname{1}} - y_{\min}}{y_{\max} - y_{\min}} * 100\% \end{equation}\]

where $y_{\min}$ and $y_{\max}$ are the minimum and maximum values of $Y$ (bounds), and $y_{\operatorname{1}}$ is the value of $y$ of the point where the ceiling line crosses the $x = x_{\min}$ line. Outcome inefficiency is 0% if $X$ constrains $Y$ for all values of $Y$; outcome inefficiency is 100% if $Y$ is not constrained for any value of $X$.

4.5 Model fit

NCA’s mathematical model specification differs from a statistical model specification. The latter includes a random term called the error term or disturbance ($\varepsilon$) representing the variation in the data. NCA does not specify the distribution of data under the ceiling and therefore the model does not include the random term. Thus, NCA’s model consists of a bounding box with a ceiling line and is deterministic (possibly with noise and exceptions, see Section below, not probabilistic.

Since NCA cannot rely on statistical (distribution-based) approaches for evaluating model fit, NCA-specific metrics of model fit are developed to quantify how well an NCA model captures the underlying necessity pattern in the data, and how closely the model’s predicted values match observed values. In this section eight metrics of model fit are presented: complexity, fit, ceiling accuracy, exceptions, noise, support, spread, and sharpness.

4.5.1 Complexity

The complexity metric ($cp$) can be used to balance parsimony against accuracy. A complex model fits data well (accuracy) but needs more parameters, whereas the model with low complexity is attractive for generalization and practical usefulness, but may lack accuracy.

The proposed complexity measure relates to the model degrees of freedom (Section 4.3.3). When the degrees of freedom for the bounding box are fixed at four, the model degrees of freedom depend only on the ceiling line. The ceiling line degrees of freedom equal the model degrees of freedom minus 4. That is:

\[\begin{equation} \tag{4.33} df_{\text{ceiling}} = \begin{cases} 2, & \text{linear ceiling line} \\ 2P, & \text{segmented ceiling line} \end{cases} \end{equation}\]

where $P$ is the number of peers on the piecewise linear ceiling as defined in Section 4.3.3.

The more degrees of freedom, the more complex the ceiling is. The linear ceiling line has the lowest complexity. This base level is set at 1 (degrees of freedom divided by 2) and represents 1 pair of parameters needed for describing a linear ceiling line (intercept, slope). The complexity of a segmented ceiling line can be described by the number of peers. Each peer can be described by 1 pair of parameters ($x$-coordinate and $y$-coordinate). Therefore, complexity is defined as:

\[\begin{equation} \tag{4.34} complextity_{\text{}} = \begin{cases} 1, & \text{linear ceiling line} \\ P, & \text{segmented ceiling line} \end{cases} \end{equation}\]

where $P$ is the number of peers.

4.5.2 Fit

The model fit metric fit ($ft$) compares a selected ceiling line with the CE-FDH ceiling line. CE-FDH follows the border closely and gives the maximum possible effect size under strict necessity (no cases above the ceiling). By definition, fit of CE-FDH equals 100%. For other selected ceiling lines, fit is usually lower. The fit metric indicates how closely the selected ceiling line follows the pattern of the boundary like the CE-FDH does.³⁹

Fit can be expressed as follows:

\[\begin{equation} \tag{4.35} fit = \begin{cases} \frac{d_{\text{ceiling}}}{d_{\text{CE-FDH}}} \times 100\% & \text{if } d_{\text{ceiling}} \leq d_{\text{CE-FDH}} \\ \text{undefined} & \text{if } d_{\text{ceiling}} > d_{\text{CE-FDH}} \end{cases} \end{equation}\]

where $d_{ceiling}$ is the effect size of the selected ceiling line and $d_{CE-FDH}$ is the effect size of the CE-FDH ceiling line. Fit is undefined when the effect size of the selected ceiling line exceeds that of the CE-FDH ceiling line, to avoid interpretational issues.

4.5.3 Ceiling accuracy

Model fit metric ceiling accuracy ($ca$) refers to the emptiness of the ceiling zone. It quantifies the degree to which the expected empty corner is indeed empty. A selected ceiling line defines the ceiling zone. With a strict interpretation of necessity, the ceiling zone should not have points, and all points should be in the feasible area. Figure 4.5-left shows an example of two points being in the ceiling zone. The presence of points in the ceiling zone may have several explanations. First, the point can be an observation (case) from an empirical study with measurement error or sampling error (Section ??). Second, the selected ceiling line may be too low, resulting in points above it.⁴⁰ Third, from the typicality perspective of necessity (Section 2.4.3), the point might represent a rare permissible exception, transforming the expression ‘$X$ is always necessary for $Y$’ into ‘$X$ is typically necessary for $Y$’.

Overshoot and undershoot in estimation of the ceiling line. Left: points in the ceiling zone. Right: no points in the ceiling zone and no points just below the ceiling line.

Figure 4.5: Overshoot and undershoot in estimation of the ceiling line. Left: points in the ceiling zone. Right: no points in the ceiling zone and no points just below the ceiling line.

For good model fit nearly all points are in the feasible area. Ceiling accuracy is the percentage of points in the feasible area (on or below the ceiling line):

\[\begin{equation} \tag{4.36} Ceiling\,accuracy = \left( 1 - \frac{n_{\text{ceiling zone}}}{n}\right) \times 100\% \end{equation}\]

where $n_{ceiling\,zone}$ is the number of points in the ceiling zone, and $n$ is the total number of points.⁴¹

4.5.4 Purity - noise and exceptions

A disadvantage of the model fit metric ceiling accuracy is that it does not account for the distance between the point and the ceiling line. Points in the ceiling zone that are farther from the ceiling line present a greater challenge to necessity than points that are closer. The purity measure of a point captures the proximity of a point to the ceiling line (Figure 4.6).

Purity: capturing the proximity of a point in the ceiling zone to the ceiling line. $P$ = general point in the ceiling zone; $P_e$ = extreme point in the low-purity part of the ceiling zone. $P_p$ = point in the medium-purity part of the ceiling zone. See the text for an explanation of A and B.

Figure 4.6: Purity: capturing the proximity of a point in the ceiling zone to the ceiling line. $P$ = general point in the ceiling zone; $P_e$ = extreme point in the low-purity part of the ceiling zone. $P_p$ = point in the medium-purity part of the ceiling zone. See the text for an explanation of A and B.

Purity is a measure of closeness of a point P in the ceiling zone to the ceiling. Referring to Figure 4.6-left, it is defined as:

\[\begin{equation} \tag{4.37} purity\,(P) = \frac{\text{size of area of F} \,{\cap}\, \text{LR(P)}} {\text{size of area of LR(P)}} \end{equation}\]

where F is the feasible area, and LR(P) is the area at the lower-right side of point P within the bounding box. Referring to the linear ceiling line in Figure 4.6-left, the numerator of Equation (4.37) is A and the denominator A + B. Therefore, the purity of a point P for a linear ceiling line is:

\[\begin{equation} \tag{4.38} purity\,(P) = \frac{A}{A+B} \end{equation}\]

where the dark-gray area A is the intersection between the feasible area and the LR(P) area, and the dotted area B is the intersection between the ceiling zone and the LR(P) area.

The highest purity value of 1 refers to a point on or below the ceiling line (B = 0, point is not in the ceiling zone). A point above the ceiling line has a purity value < 1. A point in the upper-left corner of the ceiling zone is furthest from the ceiling line, and has the lowest purity value, which equals $1 - d$ where $d$ is the effect size.

Figure 4.6-right shows a dotted line representing points with the same purity value, here 0.9. This line is called the iso-purity line. For a linear ceiling line the iso-purity line is part of an ellipse⁴² and for a stepwise linear function with horizontal and vertical parts it consists of a combination of elliptic and linear segments. The iso-purity line and the ceiling line divide the bounding box into three zones. The low-purity zone is the area above the iso-purity line and within the box where the points are far from the ceiling line. Here the purity is low (purity < 0.9 in Figure 4.6-right). The medium-purity zone is the area between the iso-purity line and the ceiling line, where the points are close to the ceiling (0.9 $\leq$ purity < 1 in Figure 4.6-right). The saturated-purity zone consists of the feasible area including the ceiling line where purity = 1.

Two model fit metrics are derived from the purity measure: noise and exceptions.

The metric noise ($ns$) quantifies uncertainty around the ceiling line due to minor deviations. For good model fit there should be only a limited percentage of points in the medium-purity zone.⁴³

The model fit metric exceptions ($ex$) quantifies violations in the ceiling zone far from the ceiling line, indicating how strict (or tolerant) the necessity claim is. With a deterministic perspective on necessity no cases are allowed in the low-purity zone. If a rare point exists in the low-purity zone, this point may be considered an exception. For a typicality perspective, rare exceptions may be permitted in this zone. The proposed model fit metric exceptions captures the number of cases in the low-purity zone.⁴⁴

4.5.5 Solidity - support and spread

To ensure high levels of ceiling accuracy and low levels of noise, a high ceiling line well above the points may be selected as illustrated in Figure 4.5-right. However, when the selected ceiling line is far away from points under the ceiling, it may not be informative in the sense that no or only a few points are constrained by the condition, which can make the necessary condition trivial. When more points are close to the ceiling, more points are constrained by the condition and the necessity of the condition becomes more relevant. Solidity is a measure of closeness of a point P in the feasible area to the ceiling as illustrated in Figure 4.7.

Solidity: capturing the proximity of a point in the feasible area to the ceiling line. $P$ = general point in the feasible area. $P_s$ = point in the medium-solidity part of the ceiling zone. See the text for an explanation of A and B.

Figure 4.7: Solidity: capturing the proximity of a point in the feasible area to the ceiling line. $P$ = general point in the feasible area. $P_s$ = point in the medium-solidity part of the ceiling zone. See the text for an explanation of A and B.

Referring to Figure 4.7-left, it is defined as:

\[\begin{equation} \tag{4.39} solidity(P) = \frac{\text{size of area of C} \,{\cap}\, \text{UL(P)}} {\text{size of area of UL(P)}} \end{equation}\]

where C is the ceiling zone, and UL(P) is the area at the upper-left side of point P within the bounding box. Referring to the linear ceiling line in Figure 4.7-left, the numerator of Equation (4.39) is A and the denominator A + B. Therefore, the solidity equation for a linear ceiling line is:

\[\begin{equation} \tag{4.40} solidity\,(P) = \frac{A}{A+B} \end{equation}\]

where the dark-gray area A is the intersection between the ceiling zone and the UL(P) area, and the dotted area B is the intersection between the feasible area and the UL(P) area. The highest solidity value of 1 refers to a point on or above the ceiling line (B = 0, point is not in the feasible area). A point below the ceiling line has a solidity value < 1. A point in the lower-right corner of the feasible area is furthest from the ceiling line, and has the lowest solidity value.

Figure 4.7-right shows a dotted line representing points with the same solidity value, here 0.8. For a linear ceiling line the iso-solidity line is part of an ellipse. The iso-solidity line and the ceiling line divide the bounding box into three zones. The low-solidity zone is the area below the iso-solidity line and within the bounding box where the points are far from the ceiling line. Here the solidity is low (< 0.8 in Figure 4.7-right). The medium-solidity zone is the area between the iso-solidity line and the ceiling line, where the points are close to the ceiling. Here the solidity is high, though not perfect (0.8 $\leq$ solidity < 1 in Figure 4.7-right). The saturated-solidity zone consists of the ceiling zone and the ceiling line where the solidity = 1.

Details of the rationale for selecting the specific distance measure for purity and solidity can be found in (Kuik & Dul, 2026).⁴⁵

For good model fit there should be at least a certain percentage of points on the ceiling line or in the medium-solidity zone, which may represent support for necessity. Model fit metric support ($su$) quantifies how strongly the data in the feasible area support the ceiling.⁴⁶

While the model fit metric support evaluates the percentage of points on and just below the ceiling to support the ceiling, those points may be clustered and only support certain parts of the line. Preferably all parts of the ceiling are supported. The model fit metric spread ($sp$) indicates how evenly the $x$-positions of the supporting points cover the range ($x_{\min},x_{\max}$). To compute spread, the number of points on the ceiling line ($satS$) and in the medium-solidity zone ($medS$) are selected, and their $X$-values ($x_i$ for $i=1,\ldots,n$) are obtained. For this set of $n$ points the coefficient of variation ($CV$) is calculated by dividing the standard deviation ($sd$) by the mean ($\bar{x}$) as follows:

\[\begin{equation} \tag{4.41} \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i \end{equation}\]

\[\begin{equation} \tag{4.42} sd = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2} \end{equation}\]

\[\begin{equation} \tag{4.43} CV = \frac{sd}{\bar{x}} \end{equation}\]

To obtain a metric of evenness, the coefficient of variation is transformed as follows:
\[\begin{equation} \tag{4.44} spread = \frac{1}{1 + \frac{sd}{\bar{x}}} \end{equation}\]

The spread metric ranges from 0 to 1. High spread values (e.g., $\ge$ 0.5) are favored over low spread values. Spread is undefined if the value of $X$ equals zero.

4.5.6 Sharpness

The ceiling line serves as a clear boundary that distinguishes the ceiling zone from the feasible area. Sharpness ($sh$) refers to the abrupt change in point density when transitioning between the ceiling zone and the feasible area, and vice versa. Sharpness implies that the density of points in the medium-purity zone (medP; the area between the iso-purity line and the ceiling line) excluding the points on the ceiling line, is much lower than the density of points in the medium-solidity zone (medS; the area between the iso-solidity line and the ceiling line) excluding the points on the ceiling line, as illustrated in Figure 4.8.

Point $P_s$ is in the medium-solidity zone of the feasible area between ceiling line and iso-solidity line (dotted). Point $P_p$ is in the medium-purity zone of the ceiling zone between ceiling line and iso-purity line (dotted). Point $P_e$ is in the low-purity (empty) zone of the ceiling zone outside and may be considered an exception.

Figure 4.8: Point $P_s$ is in the medium-solidity zone of the feasible area between ceiling line and iso-solidity line (dotted). Point $P_p$ is in the medium-purity zone of the ceiling zone between ceiling line and iso-purity line (dotted). Point $P_e$ is in the low-purity (empty) zone of the ceiling zone outside and may be considered an exception.

sharpness is defined as:

\[\begin{equation} \tag{4.45} sharpness = \begin{cases} \frac{medS}{medS+medP} & \text{if } medS + medP \neq 0 \\ \text{NA} & \text{otherwise} \\ \end{cases} \end{equation}\]

where medS is the number of points in the medium-solidity zone excluding the points on the ceiling line, medP is the number of points in the medium-purity zone excluding the points on the ceiling line, and NA indicates that a value is not available. Sharpness defines among the cases that lie close to the ceiling (in the ribbon, see below), what fraction are on the “correct” side (below the ceiling) rather than violating it (above the ceiling). Sharpness has a value between 0 and 1. A sharpness value of 0.5 means that there is no distinction in density of points close above and close below the ceiling line. Positive sharpness indicates that there are more points close below the ceiling line than close above the ceiling line (as intended).

Table ?? shows sharpness values for different values of medP and medS.⁴⁷

4.5.7 NCA ribbon

The NCA ribbon is the combination of the medium-purity zone and the medium-solidity zone. It represents a type of uncertainty area around the ceiling line. It differs from a confidence interval in statistics but serves a similar goal to capture precision. The ribbon is the geometric area calculated from the ceiling line with specified medium-purity and medium-solidity zones (e.g., 0.9-iso and 0.8-iso, respectively), where a specified maximum (purity) or minimum (solidity) number of points (cases, observations) are located. Few points in the medium-purity zone and many points in the medium-solidity zone means high precision; more points in the medium-purity zone and fewer points in the medium-solidity zone means lower precision.

4.5.8 Summary model fit metrics

The eight metrics of NCA’s model fit with example guidelines are summarized in Table 4.1. The model fit metrics are part of the broader concept of credibility. NCA’s credibility for identifying necessary conditions is discussed in Chapter 6.

Table 4.1: Model fit measures for NCA
Metric	Description	Symbol	Guideline
Complexity	Half of ceiling degrees of freedom.	$cp$	Balance.
Fit	Alignment with boundary pattern (% effect size compared to CE-FDH effect size).	$ft$	High (e.g., $ft$ ≥ 80%).
Ceiling accuracy	Percentage of points in the feasible area.	$ca$	High (e.g., $ca$ ≥ 95%).
Noise	Percentage of points in the ceiling zone near the ceiling (medP-zone).	$ns$	Low (e.g., $ns$ ≤ 5%; 0.9 iso-purity).
Exceptions	Number of points in the ceiling zone far from the ceiling (lowP-zone).	$ex$	Very low (e.g., $ex$ ~0).
Support	Percentage of points in the feasible area on or near the ceiling (medS-zone).	$su$	High (e.g., $su$ ≥ 5%; 0.8 iso-solidity).
Spread	Evenness of $x$-positions of points on or near the ceiling (medS-zone).	$sp$	High (e.g., $sp$ ≥ 0.5).
Sharpness	Relative density of points in medium-purity (medP) and medium-solidity (medS) zones.	$sh$	High (e.g., $sh$ > 0.8).

4.6 Multiple necessary conditions

Until now the mathematics of necessity has been evaluated by analyzing a single pair of variables: one condition ($X$) and one outcome ($Y$). $X$ and $Y$ represent two (related) properties of the phenomenon of interest, and the relationship between these two properties is analyzed using geometry and algebra in the $XY$-plane. This is NCA’s basic approach, which analyzes necessary conditions one by one. This section first formalizes NCA’s basic approach of analyzing single necessary conditions. Subsequently, it illustrates this reasoning with an example and gives the arguments for it. Finally, the section shows how several linked necessary conditions can be analyzed as a chain of necessity.

4.6.1 Formalization of single condition approach

NCA intentionally reduces causal complexity by reducing a multidimensional constraint structure to single-condition necessity relations by projection, and assumes that these overall necessity constraints manifest as observable ceilings in the empirical (natural) distribution.

The formalization is as follows.⁴⁸ Let $Y \in \mathbb{R}$ denote an outcome variable and let
$\mathbf{X} = (X_1, \ldots, X_J)$ denote a vector of $J$ conditions. Assume that the data-generating process implies an upper bound on $Y$ given by a (possibly unknown and complex) function:

\[\begin{equation} Y \le f(\mathbf{X}) \end{equation}\]

where $\mathbf{X}$ is a vector of conditions and $f$ defines a multidimensional ceiling surface.

NCA does not attempt to model or estimate $f(\mathbf{X})$. Instead, for each condition $X_j$, NCA considers the projection of the feasible set onto the $(X_jY)$-plane:

\[\begin{equation} {(x_j, y) : \exists \mathbf{x}_{-j}\ \text{such that}\ y \le f(x_j, \mathbf{x}_{-j})}. \end{equation}\]

This projection induces a bivariate model with a ceiling function:

\[\begin{equation} Y \le f_j(X_j), \end{equation}\]

where \[\begin{equation} f_j(x_j) = sup_{\mathbf{x}_{-j}} f(x_j, \mathbf{x}_{-j}), \end{equation}\]

and $\mathbf{x}_{-j}$ denotes all conditions except $X_j$.

The function $f_j(x_j)$ defines the projected ceiling for condition $X_j$. The function $f(x_j, \mathbf{x}_{-j})$ represents the multidimensional ceiling, that is, the maximum attainable value of $Y$ for a given combination of all conditions. For a fixed value $x_j$, the supremum operator $\sup_{\mathbf{x}_{-j}}$ takes the least upper bound of the ceiling $f(x_j, \mathbf{x}_{-j})$ across all admissible values of the remaining conditions. Thus, $f_j(x_j)$ is the highest outcome level that can be achieved at condition level $x_j$ under the most permissive setting of the other conditions. Thus, even if all other conditions take their most lenient possible values, the outcome $Y$ cannot exceed the ceiling value $f_j(x_j)$.

In the context of necessity analysis, this construction ensures that the ceiling function $f_j$ captures the overall necessity constraint imposed by $X_j$: regardless of how the other conditions are configured, the outcome $Y$ cannot exceed $f_j(x_j)$. The empty space above the ceiling is NCA’s key interest, not the feasible area below the ceiling. Assuming a well-covered sample, meaning that for each $x_j$ the data include sufficiently permissive realizations of the unobserved conditions so that the upper boundary is observed, the estimated ceiling line is not biased by third variables that are omitted from the analysis. Conversely, insufficient boundary coverage (e.g., due to limited sampling or selection that removes near-boundary cases) may lead to underestimation of the ceiling (the true ceiling is higher) and thus overestimation of the effect size (the true effect size is smaller).

In other words, NCA assumes that the projected ceiling $f_j$ is recoverable from the empirical joint distribution of ($X_j,Y$), generated under real-world conditions (the natural distribution) and incorporating all interactions, dependencies, and selection mechanisms among the conditions. Under this coverage assumption, the observed data exhibit an empty region above $f_j(X_j)$, making the estimated ceiling a candidate for causal interpretation with additional theory.

4.6.2 Illustration

Section 3.3 discusses a necessity theory with multiple necessary conditions (Figure 3.2). For example, Figure 3.2 -left shows a theory that includes three properties of the phenomena (three conditions). In mathematical terms, each condition is a dimension of the multi-dimensional space that models the phenomena. Thus, a model with two conditions $X_1$ and $X_2$, and one outcome $Y$ results in a three-dimensional space. The $XY$-plot becomes a three-dimensional $X_1X_2Y$-space.

NCA’s overall (projected) necessity property is illustrated in Figure 4.9, which shows a volcano mountain where $X_1$ is the horizontal location in one direction, $X_2$ the horizontal location in another direction, and $Y$ the vertical elevation.⁴⁹

Figure 4.9: A nonlinear three-dimensional ceiling (surface) and its projections on the $X_1Y$ and $X_2Y$-planes (non-linear ceiling lines).

For example, a point within the mountain can be identified by specific values of $X_1$, $X_2$ and $Y$. The height of the mountain is a ceiling surface, which is the set of points representing different mountain heights depending on specific values of $X_1$, $X_2$. The ceiling surface is a very complex function. NCA is not interested in modeling or estimating the multidimensional ceiling (maximum value of the outcome for given combinations of values of the conditions). In contrast, NCA takes the projection of a multidimensional ceiling onto each $X_jY$-plane, resulting in separate ceiling lines for each condition. This projected overall ceiling line in an $X_jY$-plane represents the global necessity of the single condition $X_j$ for outcome $Y$ and is interpreted independently of other conditions and the rest of the causal structure. It is assumed that this global necessity (overall ceiling line) manifests in the real world and can be observed in the empirical (natural) distribution.

The projections of the three-dimensional ceiling surface of the volcano mountain on the $XY$-planes result in non-linear ceiling lines (Figure 4.9). When the three-dimensional ceiling surface is linear (e.g., $Y = X_1 + X_2$), the projections result in linear ceiling lines, as shown in Figure 4.10.

Figure 4.10: A linear multivariate three-dimensional ceiling ($Y = X_1 + X_2$) and its projections on the $X_1Y$- and $X_2Y$-planes (linear ceiling lines).

In a multidimensional space, NCA considers one projection at a time and therefore one condition at a time. With several conditions, NCA conducts successive analyses by considering the ceilings in the separate $X_jY$-planes. For example, with two conditions ($X_1$ and $X_2$), two separate analyses are performed: one in the $X_1Y$-plane and one in the $X_2Y$-plane. In geometric terms, NCA places a ceiling surface over the data and takes its orthogonal projections onto the $X_1Y$- and $X_2Y$-planes. Each condition thus has its own ceiling line, $f_1(X_1)$ and $f_2(X_2)$. Consequently, for a given value $Y = y_c$ two conditions must be satisfied: $X_1 \geq x_{1c}$ AND $X_2 \geq x_{2c}$. Therefore, the maximum possible $Y = y_c$ for given values $X_1 = x_{1c}$, and $X_2= x_{2c}$ is

\[\begin{equation} \tag{4.46} y_c = min \{f_1(x_{1c}), f_2(x_{2c})\} \end{equation}\]

where $y_c$ is a particular outcome value, $x_{jc}$ is the necessary value of the $j$-th condition, and $f_j$ is the ceiling line of condition $X_j$.

With more than two conditions, there is a multidimensional ceiling with projections $f_j(X_j)$.

The mathematical description of a necessity AND-combination of multiple necessary conditions is as follows: for achieving an outcome level $Y = y_c$, all conditions must satisfy $X_j \geq x_{jc}$. The maximum possible value of the outcome $Y = y_c$ for given values $X_j = x_{jc}$ is

\[\begin{equation} \tag{4.47} y_c = min_{j=1,\ldots,J} \{ f_j(x_{jc}) \} \end{equation}\]

Here, $y_c$ is a particular outcome value, $x_{jc}$ is the necessary value of the $j$-th condition, and $f_j$ is the ceiling line associated with condition $X_j$.

Although NCA performs separate analyses for each condition, these analyses are combined and considered jointly in NCA’s bottleneck table (Section ??). This allows answering questions such as: “What levels $x_{jc}$ of $X_j$ are necessary for a particular level $Y = y_c$?” and “For given levels $x_{jc}$ of $X_j$, what is the maximum possible level $y_c$ of $Y$?”

4.6.3 Arguments for single condition analyses

There are several reasons why NCA analyzes projections of the multidimensional ceiling.

The first reason is the fundamental choice to focus on single factors that are necessary for an outcome. This distinguishes NCA from conventional analyses that study multi-causal phenomena by considering combinations of factors and their joint effects. A multivariate analysis is the realistic option for predicting the presence of an outcome, because often only a combination of factors (and not a single factor) can produce the outcome. In the case of necessity, however, it is realistic and often theoretically sound to propose that a single factor (a necessary condition as a bottleneck) can prevent the outcome from occurring. Therefore, it is possible and useful to study the necessity of single factors for an outcome using projections.
The second reason is that NCA makes statements about single factors that are necessary independently of other factors in the natural distribution. Thus, the global necessity (with overall ceiling line) of a single factor does not depend on the level of other factors.⁵⁰ This allows for a “pure” and straightforward interpretation of necessity: “the factor is necessary” rather than “the factor is necessary depending on other factors.” Such pure necessity statements hold independently of other factors. However, the context in which a necessity statement holds is usually not unlimited. The domain in which the necessary condition is assumed to apply must be defined as the theoretical domain of the necessity theory (Chapter 3). Such a specification of the theoretical domain must be part of any necessity theory and related formal hypotheses (Chapter 7).
A third reason is that NCA aims to contribute to parsimonious (simple) theories by avoiding complexity that makes theories difficult to understand and less useful. This is a general goal of theory building in applied sciences. NCA provides an elegant way of reducing complexity, particularly in situations where it is hard or impossible to predict the outcome (e.g., when the explained variances of regression models are low or when accurate prediction require very complex models).
The fourth reason is that practical recommendations derived from single necessary conditions are immediately clear and useful: to achieve a target outcome, all necessary conditions must be satisfied; otherwise, failure of this outcome is guaranteed. The absence of a necessary condition cannot be compensated by other factors, including other necessary conditions.
The final reason is that identifying single necessary conditions is more efficient when using multiple analyses of planes (i.e., several ceiling lines) than when conducting a single analysis of a multidimensional space (i.e., one multidimensional ceiling) followed by projection. Modeling and describing a multidimensional ceiling can be complex (e.g., Figure 4.9). Such a modeling corresponds to frontier analysis (Section 4.4), which aims to predict the maximum outcome for a given combination of factors. In contrast, NCA describes the maximum possible outcome for a single condition and can therefore focus directly on the $XY$-planes and their ceiling lines.

4.6.4 Chain of necessity

Normally, a necessary condition is hypothesized (Chapter 7) and analyzed (Chapter 9) as a necessity relationship between $X$ and $Y$. When multiple necessary conditions are proposed, several overall necessity relationships between $X_j$ and $Y$ may be hypothesized. In addition to such parallel necessary conditions, a serial system—a chain of necessity conditions—can also be hypothesized. In that case, an overall relationship $X_1 \rightarrow Y$ (if present) still holds, but it may be (partly) explained by an indirect chain $X_1 \rightarrow \cdots \rightarrow Y$.

Section 3.3 discussed, in general terms, a necessity theory consisting of a chain of necessary conditions (Figure 3.3). For example, a certain amount of study time may be necessary for a certain level of knowledge retention (quiz score), which in turn may be necessary for a high exam grade. This section analyzes such a chain algebraically. Numerical examples are provided for a chain with two links.

4.6.4.1 Analysis of chain necessity

A general necessity chain consists of a sequence of necessary conditions: the first condition $X_1$, intermediate conditions $X_2, X_3, \ldots, X_J$, and the outcome $Y$:

\tag{4.48} \[\begin{equation} X_1 \rightarrow X_2 \rightarrow X_3 \rightarrow \cdots \rightarrow X_J \rightarrow Y \end{equation}\]

Here, $J$ is the number of necessary conditions (and thus the chain has $J$ links ending in $Y$). For example, an additional intermediate necessary condition between study hours and quiz score could be active class participation (engagement, asking questions).

For each link, a ceiling function bounds the subsequent variable:

\tag{4.49} \[\begin{equation} X_{j+1} \le f_j(X_j), \quad j = 1,\ldots,J-1, \qquad Y \le f_J(X_J). \end{equation}\]

The chain ceiling $f_{\mathrm{chain}}(X_1)$ expresses the implied necessity of $X_1$ for $Y$ through the chain. It is defined as the composition of the link-specific ceiling functions:

\tag{4.50} \[\begin{equation} Y \le f_J\!\big(f_{J-1}(\cdots f_1(X_1)\cdots)\big) = f_{\mathrm{chain}}(X_1). \end{equation}\]

Here, $f_J(X_J)$ is the ceiling function of the last link.

The phrase “a chain is as strong as its weakest link” also applies to a necessity chain. If any link fails to represent a necessity relationship, the chain collapses, and $X_1$ is not necessary for $Y$ through the chain. This situation can occur, for example, when a link is sufficient but not necessary, or when a link is described only by an average effect or correlation without necessity.

Throughout, all variables are assumed to be min–max normalized, and all ceilings are defined within the same bounding box.

In the linear case with the unit square bounding box ($[0,1] \times [0,1]$), each link is described by a truncated linear ceiling:

\tag{4.51} \[\begin{equation} f_j(X_j) = \min(a_j + b_j X_j,\,1), \qquad j=1,\ldots,J. \end{equation}\]

Here, $a_j$ is the intercept and $b_j$ is the slope. Assume that each link represents the standard case of necessity (a ceiling line on or above the diagonal in the unit square), i.e., $0<a_j<1$ and $1<a_j+b_j$.

The resulting chained ceiling is again linear:

\tag{4.52} \[\begin{equation} Y \le f_{\mathrm{chain}}(X_1) = a_{\mathrm{chain}} + b_{\mathrm{chain}}\,X_1. \end{equation}\]

The chain intercept and slope are:

\tag{4.53} \[\begin{equation} a_{\mathrm{chain}} = \sum_{j=1}^{J} \left( a_j \prod_{k=j+1}^{J} b_k \right), \end{equation}\]

\tag{4.54} \[\begin{equation} b_{\mathrm{chain}} = \prod_{j=1}^{J} b_j, \end{equation}\]

with the convention that an empty product equals $1$.

For ceilings on or above the diagonal, $a_{\mathrm{chain}} \ge 1-b_{\mathrm{chain}}$, which implies $d_{\mathrm{chain}} \le \tfrac{1}{2}b_{\mathrm{chain}}$. In the unit square bounding box $[0,1]\times[0,1]$, the effect size of the (linear) chain ceiling $Y \le a_{\mathrm{chain}} + b_{\mathrm{chain}}X_1$ is:

\tag{4.55} \[\begin{equation} d_{\mathrm{chain}} = \begin{cases} \dfrac{(1-a_{\mathrm{chain}})^2}{2\,b_{\mathrm{chain}}}, & \text{if } a_{\mathrm{chain}} < 1,\\[6pt] 0, & \text{otherwise.} \end{cases} \end{equation}\]

As shown in Equations (4.53) and (4.54), each intercept $a_j$ is weighted by the product of downstream slopes $b_{j+1},\ldots,b_J$, and the total intercept $a_{\mathrm{chain}}$ accumulates across the chain. As a result, composing multiple necessary conditions can shift the chain ceiling upward relative to a single link with a comparable slope. Consequently, the chain constraint can become less stringent and may even become non-binding. This explains why, even when each link has a positive effect size, the overall chain can yield a null effect of $X_1$ on $Y$ through the chain.

In other words, for ceilings on or above the diagonal within the bounding box, the “weakest link” principle extends to effect size: the chain effect size is less than or equal to the smallest link effect size.

Since ‘a chain is as strong as its weakest link’, the effect size $d_{chain}$ is less than or equal to the effect sizes of all links:

\[\begin{equation} \tag{4.56} d_{\text{chain}} \leq \min(d_1, d_2, \ldots, d_k) \end{equation}\]

where $d_i$ is the effect size of the $i$-th link in the chain.

4.6.4.2 Numeric examples

When the chain consists of two links and assuming ceiling lines above the diagonal, the intercept and slope of the chain necessity function of Equations (4.53) and (4.54), reduce to:

\[\begin{equation} \tag{4.57} a_{\text{chain}} = a_1b_2 + a_2 \end{equation}\]

\[\begin{equation} \tag{4.58} b_{\text{chain}} = b_1b_2 \end{equation}\]

If all variables are min-max normalized to $[0,1]$, the effect size of each link is, according to Equation (4.60):

\[\begin{equation} \tag{4.55} d_j = \frac{(1 - a_j)^2}{2b_j} \end{equation}\]

The equation for effect size $d_{chain}$ of the chain necessity of $X_1$ for $Y$ is:

\[\begin{equation} \tag{4.59} d_{\text{chain}} = \begin{cases} \displaystyle \frac{(1 - a_{\text{chain}})^2}{2b_{\text{chain}}}, & \text{if } a_{\text{chain}} < 1 \\ 0, & \text{otherwise} \end{cases} \end{equation}\]

Two numeric examples of a chain consisting of two links (with ceiling lines above the diagonal) illustrate the behavior that the chain is as strong as its weakest link. In the first example, the two ceiling functions are:

\[\begin{align*} f_1(X_1) = 0.4 + 2.0 \cdot X_1 \\ f_2(X_2) = 0.2 + 1.0 \cdot X_2 \end{align*}\]

According to Equations (4.58) and (4.57) the intercept and slope for the chain ceiling line are:

\[\begin{align*} a_{chain} = 0.6 \\ b_{chain} = 2.0 \end{align*}\]

Following Equation (4.55), the two effect sizes $d_1$ and $d_2$ are:

\[\begin{align*} d_1 = \frac{(1 - 0.4)^2}{2 \cdot 2.0} = \frac{0.36}{4.0} = 0.09 \\ d_2 = \frac{(1 - 0.2)^2}{2 \cdot 1.0} = \frac{0.64}{2.0} = 0.32 \end{align*}\]

According to Equation (4.59) the chain effect size is:

\[\begin{equation} \tag{4.60} d_{chain} = \frac{(1 - 0.6)^2}{2 \cdot 2.0} = \frac{0.16}{4} = 0.04 \end{equation}\]

The three $XY$-plots are shown in Figure 4.11.

$Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.09, $X_2 \rightarrow Y$ = 0.32, $X_1 \rightarrow Y$ = 0.04.$ $Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.09, $X_2 \rightarrow Y$ = 0.32, $X_1 \rightarrow Y$ = 0.04.$ $Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.09, $X_2 \rightarrow Y$ = 0.32, $X_1 \rightarrow Y$ = 0.04.$

Figure 4.11: Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.09, $X_2 \rightarrow Y$ = 0.32, $X_1 \rightarrow Y$ = 0.04.

The second example is an example where chain necessity vanishes. Two ceiling functions are: \[\begin{align*} f_1(X_1) = 0.4 + 0.7 \cdot X_1 \\ f_2(X_2) = 0.7 + 0.8 \cdot X_2 \end{align*}\]

According to Equations (4.58) and (4.57), the intercept and slope for the composite ceiling lines are: \[\begin{align*} a_{chain} = 1.02 \\ b_{chain} = 0.56 \end{align*}\]

According to Equation (4.55), the two effect sizes $d_1$ and $d_2$ are: \[\begin{align*} d_1 = \frac{(1 - 0.4)^2}{2 \cdot 0.7} = \frac{0.36}{1.4} \approx 0.26 \\ d_2 = \frac{(1 - 0.7)^2}{2 \cdot 0.8} = \frac{0.09}{1.6} \approx 0.06 \end{align*}\]

Referring to Equation (4.59), the chain effect size is: \[\begin{equation} d_{chain} = 0 \end{equation}\]

The three $XY$-plots are shown in Figure 4.12.

$Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.26, $X_2 \rightarrow Y$ = 0.06, $X_1 \rightarrow Y$ = 0.$ $Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.26, $X_2 \rightarrow Y$ = 0.06, $X_1 \rightarrow Y$ = 0.$ $Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.26, $X_2 \rightarrow Y$ = 0.06, $X_1 \rightarrow Y$ = 0.$

Figure 4.12: Example of a necessity chain. The effect sizes of the links are: $X_1 \rightarrow X_2$ = 0.26, $X_2 \rightarrow Y$ = 0.06, $X_1 \rightarrow Y$ = 0.

The examples illustrate that the chain effect size is smaller than the link effect sizes, and that the chain effect size can be zero while those of the links are not.

Chapter 5 Statistics

This chapter will be made available soon.

Chapter 6 Credibility

This chapter will be made available soon.

Part II. Application

Part II of the book about Application demonstrates how to apply the principles of NCA in an empirical study, whether in research or in a practice setting. The primary aim of an NCA study is to identify conditions $X$ (e.g., actionable factors) that enable or constrain an outcome $Y$, which may be either desirable (e.g., performance) or undesirable (e.g., disease). Identifying necessary conditions rather than average contributing factors provides novel insights in research and opens new possibilities for effective action in practice. Once the specific goal of the empirical study is defined, the NCA method involves formulating a formal necessity hypothesis ($X$ is necessary for $Y$) and testing this hypothesis using empirical data.

After establishing the goal, the application of NCA consists of four stages:

Formulate the necessary condition hypothesis.
Collect the data.
Analyze the data.
Report the results.

The first chapter (Chapter 7) discusses the steps that are needed for developing a formal necessity hypothesis. The next chapter (Chapter 8) discusses data collection in terms of the selection of study design (observational study, longitudinal study, case study, experimental study), sampling and selection of cases, measurement (getting scores for $X$ and $Y$), and the dataset. This is followed by Chapter 9, which discusses NCA’s data analysis. Chapter 10 provides guidance on reporting an NCA study. Chapter 11 illustrates the use of NCA in multimethod studies (NCA with regression and NCA with QCA) and Chapter 12 describes the application of NCA in practice. This part ends with a summary and personal reflections.

Chapter 7 Hypothesis

7.1 Summary of this chapter

This chapter presents the development of a formal necessary condition hypothesis. It is part of a necessity theory and should be testable, non-trivial, plausible, and accompanied by a causal explanation. First, it is explained why NCA adopts a hypothesis formulation-and-testing approach to answer research and practice questions (Section 7.2). The next section (Section 7.3) clarifies why the hypothesis development process differs from usual practice (e.g., lack of past research on necessary conditions). This is followed in Section 7.4 by an overview of knowledge sources that can be used to develop necessity hypotheses (explicit and tacit knowledge from academia and practice). Subsequently, the three steps for developing a formal hypothesis are presented, including decision tools to support this process (Section 7.5). Step 1 establishes a preliminary necessity theory that consists of selecting a “best guess” necessity hypothesis from available sources of information that is embedded in theory. Step 2 uses thought experiments to challenge the hypothesis and, if needed, refine it. Step 3 deepens the causal explanation, and formulates the final formal hypothesis to be tested with empirical data.

7.2 Hypothesis formulation and testing

After the theoretical or practical goal is set, and it is decided to apply NCA, often the question of interest is: “Is $X$ necessary for $Y$”? or “Which factors are necessary for $Y$?

NCA advocates starting the study by providing a preliminary answer to the question before collecting empirical data. Such an answer can be based on existing sources of knowledge. NCA is an approach in which the hypothesis is first formulated and subsequently tested with empirical data. In this deductive approach, the hypothesis can have three states: ‘preliminary’, ‘formal’ and ‘tested’. A preliminary necessity hypothesis is a ‘best guess’ answer to the above questions based on existing sources of knowledge and, for example, abductive reasoning. A formal necessity hypothesis is a deepened preliminary hypothesis that is embedded in theory and that was challenged by thought experiments regarding testability, triviality, plausibility, and completeness. A tested necessity hypothesis is a formal hypothesis that was subjected to tests using empirical data, where the outcome of the test is that the hypothesis is rejected or not-rejected (supported) by the observed evidence. In this book, a hypothesis is considered supported or plausible when it is not rejected after being challenged empirically. Rejection may arise from the absence of a plausible causal explanation, a very small necessity effect size, or a large $p$-value. However, support for, or plausibility of, a necessary condition does not automatically establish its truth. Greater confidence requires, for example, support from replication studies. Even then, alternative causal explanations for the observed empty corner are possible (Section 2.7). Ultimately, any conclusion about a hypothesis involves human judgment, as discussed in Sections 2.7 and ??. Such a judgment may persist only briefly or endure for a long time until rejection.

As discussed below, a formal hypothesis specifies the expected necessity relationship, provides a causal explanation, and specifies the domain where it is expected to hold, among others. This is also helpful in different subsequent stages of an empirical study:

Study design: It helps to focus the study based on existing knowledge (Section 8.2).
Sampling or case selection: It specifies from which theoretical domain cases must be sampled or selected (Section 8.3).
Results: It satisfies NCA’s requirement for theoretical support for a decision about (non-)rejection of the hypothesis (Sections ?? and ??).
Discussion: It helps to interpret why the hypothesis is rejected or not, and what the theoretical and practical consequences are (Chapters 10 and 12).

In principle, NCA can also be used in a hypothesis-building (inductive) approach. In situations where the phenomenon of interest is novel or poorly understood and existing knowledge is not available, empirical data may be collected to extract a preliminary necessity hypothesis from data (exploratory or theory-building study). Such a situation is further discussed in Section 8.3.2. However, this book focuses on using empirical data only for hypothesis-testing and not for exploration/hypothesis-building for two reasons:

It is normally possible to formulate a preliminary necessity hypothesis from existing knowledge (Section 7.3) without collecting new empirical data.
To obtain an empirically tested hypothesis, it is more efficient to formulate a hypothesis first, even if existing knowledge is scarce, followed by one round of data collection for testing, compared to first having a round of data collection for empirically observing a preliminary hypothesis, followed by a second round of data collection with new data for empirically testing this hypothesis. Without testing the hypothesis with new data, the hypothesis remains untested, even if it is formalized and refined afterwards. According to a basic principle of the scientific methodology, using the same data for building and testing does not qualify for a proper empirical test of the hypothesis. This ‘predesignation’ principle was articulated in the Neyman–Pearson framework of hypothesis testing (Neyman & Pearson, 1933) and has an analogue in modern machine learning, where one dataset is used for training and another for testing. In other words, “It is never OK to snoop at the data before formulating a hypothesis—at least not if the same data are to be used in testing that hypothesis” (Mayo, 1996).

As theories in the social sciences are usually semantic (expressed in words), a necessity hypothesis is formulated qualitatively as $X$ is necessary for $Y$ without specifying the exact levels of $X$ and $Y$, other than the possibility to specify presence/high level or absence/low level of $X$ and $Y$ (Chapter 3). Therefore, in this chapter only the in kind dichotomous version of the necessary condition (necessity-in-kind, NiK) is considered, where the presence/absence of $X$ is necessary for the presence/absence of $Y$ (Figure 3.6).

7.3 Why developing necessity hypotheses is different

Developing a necessity hypothesis differs from developing a conventional probabilistic sufficiency hypothesis. Conventional hypothesis development draws on a wealth of existing knowledge about average effects. Reviewing established theories, causal explanations, and prior empirical studies is usually a central part of this process. In such cases, a literature review provides the main justification for the hypothesis, demonstrating that it is plausible, theoretically grounded, and worth (re)testing with empirical data.

In contrast, the current body of knowledge about necessity is still limited. Theoretical frameworks and empirical studies specifically addressing necessity are not yet widespread, and the traditional literature on probabilistic sufficiency offers little direct guidance. That literature hints at important factors that potentially could be necessary conditions, but usually provides no evidence. It may also contain indirect hints or isolated statements about necessity logic and causality. Although this can serve as inspiration or secondary support for developing necessity hypotheses, more is needed for to develop well-grounded necessity hypotheses.

Because a cumulative body of theory and evidence on necessity causality is often lacking, developing necessity hypotheses requires an approach that relies more heavily on the analyst’s creativity, critical reasoning, and substantive expertise. By integrating solid knowledge of necessity logic and necessity causal reasoning with substantive domain knowledge, and by engaging in discussions with peers, analysts can formulate convincing and meaningful necessity hypotheses. It is often surprising how quickly a relevant preliminary necessity hypothesis can emerge once this reasoning process is clear.

Whereas developing an average-effect hypothesis generally involves dedicating most effort to the literature review, developing a necessity hypothesis centers on conceptual reasoning and logical argumentation. The goal is to construct a coherent narrative that justifies why a given condition must be present for the outcome to occur, and why the outcome will not occur if the condition is absent, and no compensation is possible.

Generally, such a narrative starts with identifying an important factor (known or presumed) that contributes to a desired outcome but is not the only factor that helps to produce the outcome. Conventional literature often establishes that such a factor is important on average.⁵¹ The next step is crucial for necessity reasoning: it formulates the expectation that the presence of a high level of the outcome will disappear if the condition is absent or is reduced, corresponding to necessity logic: if not $X$, then not $Y$. The narrative then explains why this disappearance occurs, for example, through what mechanisms the absence of $X$ undermines $Y$. This is followed by an argument that other factors cannot compensate for this absence: no substitute can replace the missing necessary condition; the outcome will not be possible without it. Finally, the narrative specifies the domain of applicability. This is the boundary within which the necessity claim holds. It clarifies whether it applies universally (e.g., oxygen is necessary for human life) or only within a specific group or context (e.g., being pregnant is necessary for having a child).

For example, it may be assumed that class attendance is necessary for school performance. A minimal narrative could be formulated as follows: Class attendance is widely recognized as an important determinant of school performance. Meta-analyses show that students who attend classes more regularly tend to achieve higher grades on average. Attendance is not the only factor: intelligence, motivation, prior preparation, and study habits also matter. However, class attendance is crucial, as it provides the behavioral foundation through which learning processes operate effectively. If class attendance is withdrawn, school performance deteriorates—a pattern consistent with necessity logic. This disappearance occurs for several reasons. Students who skip classes miss essential explanations, examples, and discussions on which exams and assignments depend. Absence removes opportunities for immediate correction and reinforcement, allowing misconceptions to persist. Reduced attendance may also weaken motivation and belonging, leading to further reductions in performance. Other factors cannot replace the absence of class attendance. Digital resources or peer notes can alleviate short-term gaps but do not replicate the real-time cognitive and social engagement that in-class participation provides. This necessity relationship applies primarily to students in traditional, face-to-face courses where class sessions deliver core content and interactive learning is integral to success. It may not apply universally, for example, in fully online, asynchronous courses where students can compensate for absence through other structured forms of engagement. Within the conventional classroom teaching format, however, attendance remains a necessary condition for school performance.⁵²

This narrative defines not only the hypothesis (Class attendance is necessary for high school performance), but gives also a causal explanation (student miss essential learning elements, lack of interactions and motivation, no compensation possible by digital resources) and defines the theoretical domain (conventional classroom teaching).

While creativity is central for formulating the narrative, existing information sources can help generate, sharpen and justifying hypotheses about necessary conditions. This chapter provides a detailed plan and practical tools for this process. While it may appear complex at first, it soon becomes almost automatic once one becomes familiar with the reasoning steps involved.

7.4 Knowledge sources for developing necessity hypotheses

The development of a formal necessity hypothesis begins with the selection of a preliminary hypothesis, based on existing knowledge. Various sources such as academic literature, practical insights, and expert judgment can inform this selection. To ensure the relevance of the necessity hypothesis for both theory and practice, knowledge from scholarly research, real-world experience, and the expertise of the analyst can be combined. Since formulating a necessity hypothesis is primarily a creative process it can rely both on explicit knowledge (documented knowledge from academia and practice) and on tacit knowledge (undocumented expert knowledge from scholars and practitioners). This process may be iterative and involve people with different knowledge backgrounds. The result is a defensible claim made by the analyst (and others), which requires a test with empirical data to be (non-)rejected. Or, according to Popper:

“[Great scientists] are men of bold ideas, but highly critical of their own ideas; they try to find whether their ideas are right by trying first to find whether they are not perhaps wrong.” (K. R. Popper, 1985, p. 119).

This section elaborates on the two sources of information for selecting a preliminary necessity hypothesis: explicit knowledge and tacit knowledge.

7.4.1 Explicit knowledge

Explicit knowledge is documented in the academic literature (e.g., books, journal articles) and practice literature (e.g., best practice documents, manuals). Since practice is more focused on ‘doing’ than on ‘documenting’, less explicit knowledge is available from practice than from academia. Practice documents could contain know-how and statements that refer to necessity using terms like ‘requirements’, ‘must have’, ‘critical success factors’, or ‘bottlenecks’, etc. This section focuses on academic literature since it contains much information that can be used to formulate a preliminary necessity hypothesis. Four approaches for identifying them can be distinguished:

Sometimes the academic literature includes explicit necessity theories, whose necessity hypotheses may need to be (re)tested;
The literature may also contain statements referring to necessity.
The literature contains causal models, which are formal models primarily developed for probabilistic sufficiency causality, but elements may be useful for necessity causality as well.
The literature contains causal narratives for describing probabilistic sufficiency relationships or more general causes that contribute to the outcome that can inspire necessity reasoning.

These sources of explicit scholarly knowledge can be used to derive a preliminary necessity hypothesis. Each source is discussed below in more detail.

7.4.1.1 Necessity theories

As discussed in Section 3.4, some theories explicitly state necessity relationships. Such theories can be established or emerging, and may be ‘pure’ necessity theories consisting only of necessity relationships, or ‘embedded’ necessity theories consisting of a combination of necessity relationship and other relationships. As the examples in Section 3.4 show, often, necessity theories have not been (extensively) tested empirically. (Re)testing selected necessity relationships from these theories can provide valuable insights into their empirical support and generalizability.

7.4.1.2 Necessity statements

Phrases using terms like ‘necessary’ or ‘necessary but not sufficient’ are very common in the academic literature (and in practice). Although several statements may use the word necessity loosely as a synonym for important, many refer explicitly to necessity logic and causality, in particular when it is stated ‘for what’ the condition is necessary. For example, in the field of supply chain management, Bokrantz & Dul (2023) systematically reviewed necessity statements in major journals and identified 157 statements that can be considered theoretical necessity claims. Most of these statements have not been tested at all, or not been tested properly for necessity (e.g., using regression analysis). A similar result was found by Richter & Hauff (2022) when analyzing necessity statements in the field of international business. The literature also uses terms that are directly related to necessity, without using the words ‘necessity’ or ‘necessary’. Examples of such words are listed in Table 7.1. Several terms reflect that the necessary condition must be present to have the outcome (enablers); others reflect that the absence of the necessary conditions guarantees the absence of the outcome (constraints). Existing necessity or equivalent statements can be reviewed, and selected statements can be candidates for a preliminary necessity hypothesis.

Table 7.1: Equivalent terms for $X$ is necessary for $Y$. Left: Enablers. Right: Constraints.
Enablers (The presence of X …)	Constraints (The absence of X …)
X is necessary for Y	X constrains Y
X is needed for Y	X limits Y
X is critical for Y	X blocks Y
X is crucial for Y	X bounds Y
X is essential for Y	X restricts Y
X is key for Y	X stops Y
X is fundamental for Y	X inhibits Y
X is pivotal for Y	X hinders Y
X is imperative for Y	X prevents Y
X is indispensable for Y	X impedes Y
X is a prerequisite for Y	X disables Y
X is a requirement for Y	X disallows Y
X is a conditio sine qua non for Y	X is a barrier for Y
X is a pre-condition for Y	X is a bottleneck for Y
X allows Y	X is a hurdle for Y
X enables Y	Without X there cannot be Y
X permits Y	No X then no Y
There must be X to have Y
X is a must-have for Y
Y requires X
Y needs X
Y demands X

7.4.1.3 Causal models

The academic literature contains numerous causal models based on probabilistic sufficiency relationships between variables. Such models are generally evaluated through regression-based methods. All or some of the relationships in the models could be revisited with a necessity perspective on the relationship. Revisiting the relationship with a new causal perspective is not a replication or robustness test of the existing model. It yields a new necessity model or theory that provides new insights.

An example of a probabilistic sufficiency model is shown in Figure 7.1 about nine factors that contribute to academic achievement. While these factors have only been modeled as having an average effects on the outcome and were tested with regression-based studies, some studies literature refer to these factor as critical (e.g., Ma & Wang, 2001), in the meaning of ‘highly influential’. This model could be revisited from the perspective of necessity causality thereby re-interpreting the word critical as necessary.

Example of a causal model (originally for describing probabilistic sufficiency) for academic achievement with 9 factors that contribute to academic achievement. After: @walberg1984improving.

Figure 7.1: Example of a causal model (originally for describing probabilistic sufficiency) for academic achievement with 9 factors that contribute to academic achievement. After: Walberg (1984).

It is also possible that two concepts that were not (directly) related in a probabilistic sufficiency-based model, are related in a necessity model. The original role of a concept (e.g., independent, dependent, mediator, moderator, control) is not relevant for formulating a new role for the concept in a different causal model (condition or outcome in a necessity model).

An additional possibility is to evaluate suggestions for future studies that are made in a study based on average effects. It is not uncommon that a publication about a model suggests to apply NCA in future studies to (also) investigate the phenomena from a necessity perspective. Likewise, literature reviews of studies with regression-based causal models and thus probabilistic sufficiency perspectives often suggest exploring the phenomenon with NCA and thus with a necessity perspective as well to further enrich the state of knowledge. Chapter 1 gives examples of publications that suggest using NCA in future research.

7.4.1.4 Causal narratives

When proposing or justifying hypotheses, academic publications often use causal narratives: stories with explanations that connect causes to effects. For example, in describing probabilistic sufficiency relations, the narrative outlines the mechanism through which a cause contributes to an effect. These causal mechanisms can be critically evaluated by assessing whether the absence of a particular factor disrupts the mechanism. By systematically analyzing causal terminology (Table 7.1), and identifying factors that could break the mechanism (potential necessary conditions) it may be possible to formulate a preliminary necessity hypothesis. For example, Hausknecht et al. (2008) argue that organizational commitment has a negative effect on absenteeism as follows:

[…] [O]rganizational commitment is defined as a collective sense of affective or emotional attachment to an organization […]. Employees are thought to develop organizational attachments via experiences in the unit where they work […]. For example, employees of high-commitment units may engage in more community maintenance behaviors, including regular attendance at work. Thus, high-commitment work units are likely to be associated with stricter attendance norms […]. At the individual level, meta-analytic evidence has supported the negative relationship between commitment and absenteeism […]. In view of these findings, and the premise that high commitment reflects a strong collective attachment to organizational values and goals, including attendance at work, we expect that at the unit level of analysis, organizational commitment is negatively related to absenteeism. (Hausknecht et al., 2008, p. 1255).

Although in this quote the relationship between organizational commitment and absenteeism is clearly described as a probabilistic sufficiency relationship, such a causal narrative could be approached from the perspective of necessity as well. It could be wondered if the absence of organizational commitment would stop low absenteeism. In this example, the authors themselves explicitly suggest this by stating:

When the level of analysis is the work unit, organizational commitment may be a necessary but insufficient condition for low absenteeism. (p. 1225)

and after conducting a regression analysis they conclude (although this cannot be supported by a regression-based analysis):

Taken together, the findings […] show that high organizational commitment is necessary to avoid high levels of absenteeism. (Hausknecht et al., 2008, p. 1239)

Explicit statements about necessity on causal reasoning in the literature may support the necessity argument, even though necessity was not tested in the original study (Section 7.4.1.2).

7.4.2 Tacit knowledge

Undocumented expert knowledge can be used to formulate a preliminary necessity hypothesis. Experts often possess theoretical or practical insights that can be interpreted in terms of necessity, in particular when these insights are described with words from Table 7.1. This approach is especially useful in fields where explicit knowledge is scarce, while experts believe necessity relationships exist.

Two types of experts can be distinguished: academic experts (scholars) and practitioner experts (practitioners). Scholars are persons with deep theoretical or empirical knowledge developed through research. Drawing upon their expertise, comprehensive understanding of the academic literature, and possibly practical experience, they may identify potential necessary conditions either directly by recognizing explicit hypotheses, or indirectly, by recognizing important factors for an outcome that could also be interpreted as necessary conditions. Practitioners are professionals with hands-on experience in a field outside academia (e.g., consultants, managers, employees, engineers, policy-makers, educators, designers). While they may not articulate their knowledge as hypotheses, they often operate based on an implicit “theory in use” (Dul & Hak, 2008), a solid belief grounded in practice and experience that explains why a certain factor causes a certain effect. These informal theories may include necessity logic. This is not unlikely as practitioners commonly focus on factors that are required for success (‘critical success factors’), or on fail factors that prevent success by identifying and addressing ‘bottlenecks’. Their practical insights can be systematically translated into preliminary necessity hypotheses.

An array of qualitative methods exists to capture experts’ tacit knowledge, ranging from informal conversations, in-depth interviews, and creative group sessions to more structured approaches such as think-aloud protocols, and Delphi studies. However, tacit knowledge has a complex status in academia. It is often valued implicitly (many scholars rely on it), but devalued explicitly when used as evidence or justification, because it does not fit the norms of transparency, replicability, and formal theorizing that define “scientific” knowledge. If a hypothesis is (partly) developed based on tacit practitioner knowledge, the approach used for obtaining this knowledge can be reported and it can be justified that such knowledge is not anecdotal, but accumulated through extensive experience with the practice, system, or process, thus empirically grounded.

7.5 Developing a formal necessity hypothesis

A formal necessity hypothesis has five characteristics:

Theory-grounded: the hypothesis is embedded in a necessity theory. According to Chapter 3, a theory is defined when four elements are defined: focal unit, concepts (condition and outcome), (direction of) the hypothesis⁵³, and theoretical domain.

Testable. A hypothesis is empirically testable if the concepts $X$ and $Y$ can be operationalized into measurable variables.

Non-trivial. A hypothesis is non-trivial when both the absence of the condition and presence of the outcome are possible, otherwise it is trivial.

Plausible. A hypothesis is plausible when virtually no cases are reasonably expected in the empty corner.

Complete. A hypothesis is complete when a plausible explanation for why condition $X$ is necessary for outcome $Y$ is available.

The development process of a formal necessity hypothesis consists of three steps that are shown in Figures 7.2, 7.4, and 7.6, respectively.

In the first step (Figure 7.2), a preliminary necessity hypothesis is selected based on the available sources of information (Section 7.4), and is embedded in a preliminary necessity theory by defining focal unit, concepts, (direction of) the hypothesis, and theoretical domain. Concepts are defined such that they are measurable and the hypothesis becomes testable. After this step the hypothesis is theory-grounded and testable.

In the second step (Figure 7.4) a thought experiment is conducted in which the preliminary necessity theory is challenged and possibly revised. After this step the hypothesis is also non-trivial and plausible.

In the third step (Figure 7.6) a detailed causal explanation for the hypothesis is given (Chapter 2). This final step makes the hypothesis complete. The formal hypothesis is now ready to be tested with empirical data.

Below, each step is discussed in more detail and illustrated with an example of the preliminary hypothesis that ‘An internet connection is necessary for attending an online meeting’.

7.5.1 Step 1: Define a preliminary necessity theory

The first step transforms the preliminary hypothesis into a preliminary necessity theory. This is a theory that has measurable concepts, a hypothesis describing the expected necessity relationship between the concepts, and a description of the theory’s focal unit and theoretical domain (Chapter 3). However, this theory is not yet challenged in a thought experiment, which is the topic of Step 2. Figure 7.2 shows the hypothesis development tool for Step 1 of the development process.

Hypothesis development tool for Step 1 of the development process of a formal necessity hypothesis: Defining a preliminary necessity theory.

Figure 7.2: Hypothesis development tool for Step 1 of the development process of a formal necessity hypothesis: Defining a preliminary necessity theory.

To begin, a preliminary hypothesis is formulated based on existing knowledge (Section 7.4). This hypothesis is then elaborated into a preliminary necessity theory by precisely defining the four constituent elements of a necessity theory:

Precise definition of focal unit. The focal unit is the entity of the theory to which the theory applies (e.g., person, team, country, project, etc.).

Sometimes, defining the focal unit is simple. For example, the Theory of Planned Behavior (TPB, Ajzen, 1991) is a theory about individual behavior. The focal unit is evident: ‘individual’ or ‘person’. Other times, the focal unit may not be immediately clear, for example, when theorizing the relationship between parent height and child height (Galton, 1886).⁵⁴ The focal unit is neither parent nor child but ‘parent-child pair’. The focal unit is expressed as a singular noun, not a plural noun. Specification is important because cases to be sampled or selected for testing the hypothesis must be specific units/examples of the focal unit (e.g., a person, a parent-child pair). Additionally, the concepts of the theory are characteristics of the focal unit.

Precise definition of the concepts. The concepts are the varying characteristics that are relevant for the theory. For a necessity theory, the concepts are condition $X$ and outcome $Y$. The outcome often represents something either desirable or undesirable. Desirable outcomes include, health, well-being, performance, innovation, success, etc. Undesirable outcomes may be sickness, risk, disease, failure, etc. The condition is the concept that must be present to have the outcome.

In the TPB example the condition is Intention for the behavior (of a person), and the outcome is Performance of the behavior (of a person). In the body height example the condition is Parent body height (of a parent-child pair) and the outcome is Child body height (of a parent-child pair).

Precise specification of the hypothesis. Once the concepts are defined, the hypothesis can be formulated as $X$ is necessary for $Y$. If the condition is desirable, the hypothesis could be formulated using one of the enabling terms of Table 7.1 to indicate that the necessary condition supports the outcome. If the condition is undesirable, the hypothesis could be formulated using one of the constraining terms of Table 7.1 to indicate that the absence of the necessary condition prevents the outcome.
Usually, theories in the social sciences are semantic, such that levels of its concepts are only expressed in binary terms and qualitatively (e.g., in terms of absence/presence, or low/high), without giving exact levels. For example, the presence of $X$ is necessary for the presence of $Y$, or high $X$ is necessary for high $Y$. This is the implicit meaning of $X$ is necessary for $Y$. The conceptual model of this hypothesis is shown in Figure 3.6-top-left. This figure shows a specification of the direction of necessity as or high-high (Section 3.5): presence/high $X$ is necessary for presence/high $Y$. Other possible directions are absence/low $X$ is necessary for presence/high $Y$ (; low-high), presence/high $X$ is necessary for absence/low $Y$ (; high-low), and absence/low $X$ is necessary for absence/low $Y$ (; low-low) (see the other conceptual models in Figure 3.6). The specified direction of the hypothesis defines the expected empty corner: which corner of an $XY$-plot is expected to be empty if the hypothesis holds (Figures 3.7 and 3.8).

For the TPB example, the preliminary hypothesis is the ’presence of intention for the behavior is necessary for the presence of performance of the specific behavior. For the body height example, the preliminary hypothesis is a high parent body height is necessary for a high child body height. In both hypotheses, the direction is high-high.

Precise definition of the theoretical domain. The theoretical domain specifies the universe of units (cases) of the focal unit to which the hypothesis is expected to apply. This universe can be defined in various ways, such as temporally (e.g., a specific time period), geographically (e.g., a region, country, or global context), demographically (e.g., age group, profession), or conceptually (e.g., type of organization or system). The specified theoretical domain determines which cases must be sampled (from a population in the theoretical domain) or selected (directly from the theoretical domain).

For the TPB example, the theoretical domain is ‘All adults in the world’. For the body height example, the theoretical domain is ‘All adult parent-child pairs in the world, when children have reached adulthood’.

After the preliminary hypothesis is embedded in theory, the first question (Q1) of Figure 7.2 checks if precise definitions are available for four elements of the theory: focal unit, concepts, hypothesis and theoretical domain.

➔ Q1: Are all four elements of the necessity theory precisely defined? When all elements of a necessity theory are specified, the hypothesis is considered to be embedded in theory.

➔ Q2: Are the $X$ and $Y$ concepts measurable? This question checks whether the concepts can be transformed into measurable variables. When concepts cannot be measured directly, substitute measurements (proxies) can be considered, such as measuring an informant’s perception or belief about an otherwise unmeasurable behavior or event, although this approach may compromise measurement validity. NCA accepts any meaningful, valid, and reliable scores of $X$ and $Y$ as input (Chapter 8). When the $X$ and $Y$ concepts of the necessity theory can be operationalized into measurable variables, the hypothesis is considered testable.

7.5.1.1 Example Step 1

The hypothesis development tool for Step 1 (Figure 7.2) can be applied to the example about the necessity of an Internet connection for attending an online meeting. First the preliminary necessity hypothesis that is formulated based on expert knowledge is selected.

Select preliminary hypothesis: Internet connection is necessary for Online meeting attendance.

Next, the hypothesis is embedded in theory.

Elaborate hypothesis into theory:

➔ Q1: Are all four elements of the necessity theory precisely defined? - Focal unit: ‘Person’. A person may or may not have an Internet connection and the person may or may not attend an online meeting.

Condition concept $X$: Internet connection, defined as ‘the communication pathway between a device such as a computer or smartphone and the global network known as the Internet’. This concept can be absent (no Internet connection) or present (Internet connection).
Outcome concept: $Y$: Online meeting attendance. This concept is defined as ‘being part of a real-time, multimedia interaction among multiple participants’. This concept can also be absent (no Online meeting attendance) or present (Online meeting attendance).
Direction of the hypothesis: the presence of an Internet connection is necessary for the presence of an Online meeting attendance (high-high).
Theoretical domain: All persons in the world

Answer: $\rightarrow$ yes.

➔ Q2: Are the concepts Internet connection ($X$) and Online meeting attendance ($Y$) measurable? The presence or absence of a person’s Internet connection, and the presence or absence of a person’s online meeting attendance can be measured in different ways such as by analyzing the platform and network information, user activity logs, observations, questionnaires, and other data sources. This makes the hypothesis testable. $\rightarrow$ yes.

Figure 7.3 shows the conceptual model and the expected empty corner for this preliminary theory. The preliminary hypothesis predicts that all persons that do not have an Internet connection also do not attend an online meeting, and that all persons that attend an online meeting have an Internet connection.

Conceptual model and $XY$-table with expected empty corner when the necessity proposition: having an Internet connection is necessary for Online meeting attendance.

Figure 7.3: Conceptual model and $XY$-table with expected empty corner when the necessity proposition: having an Internet connection is necessary for Online meeting attendance.

7.5.2 Step 2: Conduct thought experiments

The second step of the development process of a formal necessity hypothesis consists of conducting thought experiments with the preliminary theory including its hypothesis, which is now elaborated into theory and is testable. Figure 7.4 shows Step 2 of the development process.

Figure 7.4: Step 2 of the development process of a formal necessity theory: using thought experiments to challenge the preliminary theory.

A thought experiment in NCA is a mental evaluation to check the validity of the necessity theory and its hypothesis. Since a necessity hypothesis has only one $X$ and one $Y$, the prediction of the necessity theory is straightforward: the outcome is absent if the condition is absent, and the condition is present if the outcome is present. Therefore it is usually easy to evaluate the hypothesis mentally. The basic idea is to evaluate if the expected empty corner is indeed empty. Assuming that presence/high $X$ is necessary for presence/high $Y$, the expected empty corner is the upper-left corner.

A thought experiment can help to check if the preliminary theory may be trivial or implausible. Testing trivial or implausible hypotheses with empirical data is not interesting and a waste of resources. Furthermore, if the thought experiment shows that many counterexamples exist (the expected empty space is not empty) the hypothesis should be rejected. With a few counterexamples, further evaluation is needed and the theory may be refined by redefining the concepts or the theoretical domain, such that the (revised) hypothesis of the revised theory is plausible. Three thought experiments can be done (references to the questions in Figure 7.4 are given in brackets):

Thought experiment for non-triviality (Q3, Q4).
Thought experiment for plausibility (Q5, Q6).
Thought experiment for counterexamples and exceptions (Q7, Q8, Q9).

7.5.2.1 Non-triviality

A trivial necessity hypothesis is a hypothesis that is expected to hold, but is not informative because the condition or the outcome is nearly (constant) in the situation of interest (i.e., in the theoretical domain where the hypothesis is supposed to hold). Triviality applies when the condition is virtually always present, or the outcome is virtually always absent. The first type of triviality where the condition is constant (always present) and the outcome varies, is depicted in Figure 3.11-bottom-left. An example is the hypothesis that oxygen ($X$) is necessary for attending an online meeting ($Y$). This hypothesis holds because when an online meeting is held ($Y$ is present), oxygen ($X$) is present and needed for the participants to have the meeting. The hypothesis is trivial because oxygen is virtually never absent. The second type of triviality where the outcome is constant (always absent) and the condition varies, is depicted in Figure 3.11-top-left. An example is the hypothesis that having good food during lifetime ($X$) is necessary for a human to reach age 125. The necessity claim of this hypothesis is reasonable, but it is trivial because reaching age 125 is rare.

The situation of a constant condition that is always absent and an outcome that varies (Figure 3.11-bottom-right) and the situation of a constant outcome that is always present and a condition that varies (Figure 3.11-top-right) both result in a clear rejection because the outcome can be present when the condition is absent. In these situations the necessary condition hypothesis is immediately rejected, not because it is trivial but because it is not necessary.

Therefore, there are two questions to be raised and answered in a thought experiment for checking triviality of the theory and its hypothesis (Figure 7.4):

➔ Q3. Do cases exist with the outcome? This question checks if the outcome is virtually always absent. If not, the theory is trivial.

➔ Q4. Do cases exist without the condition? This question checks if the condition is virtually always present. If not, the theory is trivial.

If either of these questions is answered negatively, the theory and its hypothesis can be considered trivial and not worth testing with empirical data.

7.5.2.2 Plausibility

A plausible hypothesis is a non-trivial hypothesis (both the condition and outcome can vary) that cannot be rejected in a thought experiment. The goal of this thought experiment is to consider condition and outcome together to find supportive cases for the hypothesis, and in particular possible counterexamples that challenge the hypothesis. This thought experiment for plausibility resembles an empirical test of a necessity hypothesis using a small number of cases (Section 8.3.2).

To find supportive cases, two questions can be asked:

➔ Q5. Do all imaginable cases with the outcome have the condition? If the answer yes, this is an indication that the hypothesis may hold.

➔ Q6. Do all imaginable cases without the condition lack the outcome? If the answer yes, this is another indication that the hypothesis may hold.

If the answer to both questions is affirmative, all cases from the theoretical domain that can be imagined are supportive cases for the hypothesis. When only supportive cases and no counterexamples can be imagined, the hypothesis is plausible. If the answer to one or both questions is negative, counterexamples are identified. These counterexamples need further evaluation before a decision can be made about the plausibility of the hypothesis.

7.5.2.3 Counterexamples and exceptions

When counterexamples outnumber supportive cases, it is clear that the hypothesis needs to be rejected. However, when there is a limited number of counterexamples compared to supportive cases an additional evaluation is needed. Therefore, when counterexamples exist, the following question is answered first:

➔ Q7: Are counterexamples uncommon If counterexamples are not uncommon, the theory should be rejected. If counterexamples are uncommon, they need further evaluation before the analyst makes a decision to reject or keep the hypothesis, possibly after refinement of the theory or adoption of the typicality perspective (see below).

The further analysis consists of considering the possibility of incorporating or excluding counterexamples in the theory by redefining the condition $X$, the outcome $Y$, or the theoretical domain. Broadening the condition $X$ or narrowing the outcome $Y$ can make counterexamples become supportive cases within the redefined theory. Narrowing the theoretical domain excludes the counterexamples from the redefined theory.⁵⁵

Therefore, the analyst has three options to handle counterexamples: (1) broadening the condition, (2) narrowing the outcome, and (3) narrowing the theoretical domain. It is the analyst’s choice which option to adopt. Option 1 broadens the ways in which the condition can allow the outcome. Often, the broader condition is a higher-order concept that integrates the original concept and the counterexample concept. By broadening the condition, counterexamples that previously displayed an absence of the condition now display a presence of the condition, and become supportive cases under the refined theory. However, this risks making the condition too broad and weakening its explanatory power: almost any facilitating mechanism could then be considered part of the condition. Option 2 narrows the outcome to a more specific form, reducing the scope of what is being explained. With the redefined outcome concept, the exception becomes a supportive case that does not violate the hypothesis. However, this risks excluding cases that are theoretically relevant. Option 3 restricts the theoretical domain by focusing only on a subset of cases where necessity holds. With a new definition of the theoretical domain, the exception should not have been selected for testing the hypothesis because the claim of the hypothesis is only valid in a defined theoretical domain. While this approach preserves the original concepts and hypothesis, it reduces the generalizability of the theory. Nevertheless, it seems reasonable to adopt option 3 since a theoretical domain is seldom homogeneous and universal. In option 3, the original hypothesis is maintained, but the theoretical domain is narrowed. This strategy preserves the preliminary hypothesis, with more clearly specifying the scope of its application, while no known counterexamples exist.

If any of the options for handling counterexamples is selected, it is assumed that the redefined theory holds for all imaginable cases selected from the theoretical domain.⁵⁶

If counterexamples exist the following question needs to be answered:

➔ Q8. Can counterexamples be handled by redefining concepts/domain? If the answer is yes, the theory is adjusted, so that there are no more counterexamples. After the counterexamples are handled by redefining the necessity theory (concepts or domain) it is assumed that the hypothesis holds for all imaginable cases of the theoretical domain, such that the testing of the hypothesis can start with a deterministic view on necessity (Section 2.4.1).

If the answer to Q8 is no, it may be that (some) counterexamples remain. The analyst has now two options to proceed:

Adopting the deterministic perspective of necessity, single counterexamples reject the hypothesis (Section 2.4.1).
Adopting the typicality perspective of necessity. The counterexamples are rare and cannot be explained or handled. They are considered exceptions that are not driven by necessity and do not reject necessity the hypothesis (Section 2.4.3). If counterexamples are not rare, the hypothesis should be rejected. Note the difference between exceptions and noise (Section 4.5.4).

Adopting either perspective is a matter of analyst judgment. The deterministic perspective may be adopted when strict logical necessity is required without exceptions, when the theoretical domain contains few cases, or when logical rigor is prioritized. The typicality perspective may be adopted when it is recognized that rare exceptions can exist that cannot be explained with necessity logic.

Therefore, if counterexamples remain the following question needs to be answered:

➔ Q9. Is the typicality perspective adopted? If the answer is no, the theory’s hypothesis is rejected; if yes, it is plausible and (rare) exceptions are allowed.

7.5.2.4 Example Step 2

The development tool for Step 2 (Figure 7.4) can be applied to the example about the necessity of an Internet connection for attending an online meeting. First, the preliminary necessity theory from Step 1 is selected. The first question for the triviality check is:

➔ Q3. Do people exist who attend an online meeting? In 2021, about 500 million people daily had online meetings. Since then the number of daily users presumably has considerably increased $\rightarrow$ yes.

The second question of the triviality check is:

➔ Q4. Do people exist without an Internet connection? In 2025, about one-third of the world population had no access to the Internet $\rightarrow$ yes.

Since both questions are answered affirmatively, the hypothesis is non-trivial for the selected theoretical domain (all people in the world). To check for the plausibility of the hypothesis, another two questions are raised:

➔ Q5. Do all imaginable persons attending an online meeting have an Internet connection?

➔ Q6. Do all imaginable persons without Internet connection not attend an online meeting? The answer to both questions is no. It is possible that persons attending an online meeting at home or in an organization in a Local Area Network (LAN). These persons attend an online meeting without being connected to the Internet. Both questions Q5 and Q6 are therefore answered negatively, indicating that counterexamples exist. Trying to find counterexamples, even if they seem uncommon, is an essential part of conducting a thought experiment to test plausibility of a necessity hypothesis $\rightarrow$ no.

Since counterexamples exist, the next step is to evaluate if they are uncommon:

➔ Q7: Are people with LAN connections uncommon? For the online meeting example, the number of persons attending an online meeting with a LAN connection compared to the number of persons attending an online meeting with an Internet connection is relatively small. Attending an online meeting with a LAN connection is relatively uncommon, suggesting further analysis of these cases is needed to refine the theory $\rightarrow$ yes.

➔ Q8: Can counterexamples be handled by redefining concepts/domain? The three options to handling counterexamples are:

Broadening the condition means that the original definition of Internet connection (the communication pathway of devices such as computer or smartphone to the global network known as Internet) is broadened such that it incorporates LAN networks. The concept Internet connection could be broadened to the concept Digital network connection defined as the communication pathway of devices such as computer or smartphone to a digital network. ‘Digital network’ now includes Internet and LAN.

Narrowing the outcome means that the original definition of the outcome $Y$ (Online meeting attendance: being part of a real-time, multimedia interaction among multiple participants) is narrowed such that it incorporates the exception. For example, Online meeting attendance could be narrowed to Public online meeting attendance defined as being part of a real-time, multimedia interaction among multiple participants from the general public, excluding private online meetings (that use LAN) from the definition. For the cases that were previously counterexamples, the condition ‘Internet connection’ is absent (though LAN is present), and the redefined outcome concept ‘Public online meeting attendance’ is also absent (because the LAN meeting is considered a private meeting).

The third way to handle an exception is to exclude counterexamples from the theory by narrowing the theoretical domain. In the online meeting example, narrowing the theoretical domain means that the original definition, ‘all persons in the world’, would be revised such that it excludes people that use a LAN network in a private meeting. For example, the domain All persons in the world could be narrowed to All persons who are geographically distant from each other, which would exclude in a local gathering (home, organization) from the theoretical domain.

In summary, for the example the three possible ways to handle the exception by redefining concepts/domain are:

Option 1 (broadening condition):

Hypothesis: Digital network connection is necessary for Online meeting attendance. Focal unit: Person. Condition $X$: Digital network connection: the communication pathway of devices such as computer or smartphone to a digital network. Outcome $Y$: Online meeting attendance: real-time, multimedia interaction among multiple participants. Theoretical domain: All persons in the world.

Option 2 (narrowing the outcome):

Hypothesis: Internet connection is necessary for Public online meeting attendance. Focal unit: Person. Condition $X$: Internet connection: the communication pathway of devices such as computer or smartphone to the global network known as Internet. Outcome $Y$: Public online meeting attendance defined as a real-time, multimedia interaction among multiple participants from the general public. Theoretical domain: All persons in the world.

Option 3 (narrowing the theoretical domain):

Hypothesis: Internet connection is necessary for Online meeting attendance. Focal unit: Person. Condition $X$: Internet connection: the communication pathway of devices such as computer or smartphone to the global network known as Internet. Outcome $Y$: Online meeting attendance: real-time, multimedia interaction among multiple participants. Theoretical domain: All persons who are geographically distant from each other.

It is supposed that the second option is selected for further development in Step 3 (Section 7.5.3.1) $\rightarrow$ yes.

7.5.3 Step 3: Provide causal explanation

In the previous step, the preliminary necessity theory including its hypothesis was either rejected or accepted during a thought experiment, possibly after revision of concepts or theoretical domain, or by adopting the typicality perspective. If accepted, the next step is to explain why $X$ is necessary for $Y$. This involves developing a causal explanation. During the formulation of the preliminary necessity theory including its hypothesis and during the process of conducting thought experiments, the analyst implicitly used a causal reasoning about necessity. In the last step before empirical testing of the hypothesis, the causal reasoning is explicitly articulated and further deepened. Since causality is a human interpretation rather than a directly observable phenomenon (Chapter 2), the analyst must construct a plausible and coherent narrative to justify the proposed necessity-based causal link.

Thus, describing the causal explanation is a creative process that can be based on insights obtained during the thought experiments, and can be enriched with causal reasoning from the academic literature, from practice, and from the analyst’s own experience (Sections 7.4.1 and 7.4.2).

The causal narrative can be inspired by answering three questions that are positioned in the $XY$-plot shown in Figure 7.5:

Explaining why $X$ is an enabler for $Y$ (why the presence of $X$ allows the presence of $Y$).
Explaining why $X$ is a constraint for $Y$ (why the absence of $X$ leads to the absence of $Y$).
Explaining why there is no substitute for the absence of $X$ (why no other paths without $X$ can lead to the outcome $Y$).

The first question builds on conventional causal reasoning—exploring how one factor allows an outcome and in this way helps to produce it. The second and third, however, go beyond standard approaches: they introduce the need to reason counterfactually and to explore the uniqueness of $X$ in the causal structure. These perspectives encourage creative and rigorous thinking about necessity and irreplaceability.

One core idea of necessity causality is that the absence of $X$ causes the absence of $Y$. More precisely, if the necessity cause is reduced or removed, the present or high outcome should disappear. This thought experiment resembles a true ‘loss of function’, see the necessity experiment discussed in Section 8.2.4.

$Questions for explanation why $X$ is necessary for $Y$. $\neg X$ and $\neg Y$ mean absence of $X$ and $Y$.$

Figure 7.5: Questions for explanation why $X$ is necessary for $Y$. $\neg X$ and $\neg Y$ mean absence of $X$ and $Y$.

The three guiding questions are part of Step 3 of the hypothesis development tool (Figure 7.6).

Figure 7.6: Hypothesis development tool for Step 3 of the development process of a formal necessity hypothesis: Adding a causal explanation.

➔ Q10. Is it explained why $X$ enables $Y$? The accepted hypothesis from the thought experiments is first considered from the perspective that $X$ is an enabler for $Y$. To explain why a condition enables an outcome, the starting point is the presence of $X$. The explanation then focuses on how this presence can plausibly contribute to producing the outcome, possibly through a chain of necessity (Sections 3.3 and 4.6.4). Common narratives about the causal mechanisms through which $X$ contributes to $Y$ can support this explanation. While such narratives are used to describe probabilistic sufficiency where the presence of $X$ increases the likelihood of $Y$, the same reasoning can be used to illustrate $X$’s role as a contributor: when $X$ is present, $Y$ may occur, but it is not guaranteed; $Y$ may still be absent. The contributor becomes only an enabler when also the next two questions apply: if $X$ is absent, $Y$ will not occur.

➔ Q11 Is it explained why $X$ constrains $Y$? Now $X$ is considered from the perspective of being a constraint for $Y$. To explain why a condition constrains an outcome, the starting point is the absence of $X$. The explanation then focuses on how this absence disrupts the causal mechanism through which $X$ contributes to the outcome as described in the answer to Q10, thereby preventing the outcome from occurring.

➔ Q12. Is it explained why there is no substitute for the absence of $X$? But even if $X$ is not present, it may be that its absence is compensable by another factor or path. Therefore, the hypothesis is considered from the perspective of the possibility of substitutes. To explain why there is no substitute for the absence of a condition, the starting point is again the absence of $X$. The explanation then focuses on whether any alternative path (mechanism) could plausibly exist to still produce the outcome. If no viable substitutes exist, the absence of $X$ guarantees the absence of $Y$. If an alternative path is discovered that was not identified in Step 2, the analyst may return to Q7 of step 2.

➔ Q13. Are temporal aspects considered? If Q10-Q12 are answered affirmatively, the three components (enabler, constraint, substitute) provide the basis for a narrative for explaining why $X$ is a necessary cause for $Y$. In addition to the explanation that $X$ is essential to produce $Y$, and that its absence cannot be substituted, also three temporal aspects need to be considered. First, one fundamental requirement for establishing causality is temporal order: $X$ must precede $Y$. If this sequence is not self-evident, the expected temporal ordering should be explicitly stated and justified. Doing so helps to address the risk of reverse causality (Section ??). Second, it is important to clarify whether $X$ is necessary for the onset of $Y$ or for its continuation. For instance, a match is necessary to ignite a fire (onset), but not to keep it burning (continuation). In contrast, oxygen is necessary for both the onset and continuation of the fire. This distinction can be incorporated into the causal explanation or into the definition of the outcome concept itself. Third, the necessity of $X$ for $Y$ may be time-dependent. That is, $X$ may only be necessary during certain time periods. For example, in the past, attention to sustainability may not have been necessary for a company’s success, whereas it may be essential today. This time-sensitive aspect of necessity can be included either in the causal narrative or in the specification of the theoretical domain.

After providing the temporal aspects, the hypothesis is complete. The formal hypothesis is now fully specified and is ready for empirical testing:

Hypothesis: $X$ is necessary for $Y$. (add direction if needed)

Focal unit = …

Concept $X$ = …

Concept $Y$ = …

Theoretical domain = …

Causal explanation = …

7.5.3.1 Example Step 3

The hypothesis development tool for Step 3 (Figure 7.6) can be applied to the example about the necessity of an Internet connection for attending an online meeting. The accepted hypothesis after the thought experiments is ‘Internet connection is necessary for Public online meeting attendance’.

➔ Q10. Is it explained why Internet connection enables Online meeting attendance? First, it is evaluated if an Internet connection can be considered as an enabler for Public online meeting attendance. This enabling role can be explained as follows: the presence of a communication pathway makes it possible for a participant to connect to a platform and to communicate remotely. The platform provides the necessary technical infrastructure for video, audio, and chat functionalities that allow a public online meeting to occur.⁵⁷ $\rightarrow$ yes.

➔ Q11 Is it explained why Internet connection constrains Public online meeting attendance? Next, it is considered if an Internet connection is a constraint for a public online meeting. The constraining role of an Internet connection can be explained as follows. Without a communication pathway it is also not possible to connect to a digital platform, such that participants are unable to connect remotely. The essential infrastructure for real-time communication is missing. As a result, the conditions required for a public online meeting are not met, and the person cannot attend the meeting. The absence of the communication pathway thus blocks the occurrence of the outcome $\rightarrow$ yes.

➔ Q12. Is it explained why there is no substitute for the absence of $X$? Next, it is considered if, when a digital platform is absent, other technologies for example, email or satellite communication could serve as substitutes. Email is an asynchronous communication tool that may support collaboration in a broader sense, but does not enable the real-time interaction required for a public online meeting. Satellite communication may offer connectivity for specific groups or in remote areas, but does not facilitate general public online meetings. If no substitute exists in the theoretical domain, with the absence of an Internet connection the outcome cannot be achieved, confirming that $X$ is irreplaceable in this context $\rightarrow$ yes.

➔ Q13. Are temporal aspects considered? An online meeting cannot begin without a working Internet connection already in place. Thus, the Internet connection must exist before or at least at the onset of the meeting. The Internet connection is necessary both for the onset of the online meeting (to initiate the call or session), and the continuation of the meeting (to maintain the live connection throughout). The necessity of an Internet connection for an online meeting is time-dependent in a technological sense: Before Internet-based communication tools existed (e.g., before the 1990s), an Internet connection was not necessary — online meetings didn’t exist as a concept, so only after 1990 the hypothesis is non-trivial. In the future, alternative technologies might make traditional Internet connections obsolete and become substitutes $\rightarrow$ yes.

If so, the hypothesis is now a formal necessity hypothesis, and is ready to be tested with empirical data.

Chapter 8 Data

8.1 Summary of this chapter

This chapter describes the steps between the hypothesis and data analysis that are needed to obtain empirical data for testing a formal necessity hypothesis. The hypothesis discussed in the previous chapter (Chapter 7) guides which data should be collected. The hypothesis has precise definitions of (1) the focal unit, (2) the concepts $X$ and $Y$, (3) the direction of necessity and a causal explanation, and (4) the theoretical domain where the hypothesis is supposed to hold. Empirical data refer to observed $x$- and $y$-scores (values or levels) of variables that represent the concepts $X$ and $Y$. These scores are measured on cases selected from the theoretical domain. To collect high-quality empirical data for testing the necessity hypothesis, three key decisions must be made: (1) selection of study design, which is the type of study (observational study, longitudinal study, case study, or experimental study discussed in Section 8.2), (2) sampling or selection of cases from the theoretical domain: random sampling and purposive case selection (Section 8.3), and (3) conducting the measurement to obtain valid and reliable $x$ and $y$ scores (e.g., qualitative or quantitative data; archival or new data) for each case (Section 8.4). Section 8.5 discusses the format of a useful dataset.
While NCA does not impose new requirements on data collection and treats empirical data as input to NCA rather than as part of NCA, some choices must be tailored to fit the logic of necessity. This applies, for example, to the study design when a necessity experiment is done (Section 8.2.4.1), or to the use of purposive case selection in a small-n qualitative study (Section 8.3.2).

8.2 Study design

Study design refers to the type of study that is done to test the hypothesis. This book distinguishes between the observational study, which is a study that analyzes a large number of cases in the natural environment (e.g., large-n quantitative study), the longitudinal study, which is an observational study where cases are studied at different time points, the case study, which is an observational study with a small number of cases in the natural environment (e.g., small-n qualitative study), and the experimental study, which is a study in which the condition $X$ is manipulated and the outcome $Y$ is observed. This section discusses each of these four study designs in the context of NCA.

8.2.1 Observational study

The most common design of an NCA study is the observational study. The analyst observes the phenomena of interest in the natural setting without manipulating $X$ and $Y$. Such studies are often cross-sectional, meaning that $X$ and $Y$ are observed at a single point in time. The observational study is usually associated with a large-n quantitative study where many cases are sampled (Section 8.3.1), and scores of $X$ and $Y$ are numeric (Section 8.4.1).

The popularity of this design in NCA can be explained not only by the popularity of quantitative studies in general, but also because the method is not vulnerable to omitted variable bias. This is one reason in regression-based studies for conducting randomized experiments (see Section ??). The absence of omitted variable bias allows findings from an observational NCA study to be readily interpreted causally without taking special measures to control for other variables, assuming that an adequate sample (Section ??) and a solid necessity theory (Chapter 7) are available. Thus, the observational study is particularly useful when the hypothesis is well-developed with a plausible causal explanation that also specifies the temporal aspect (first $X$ then $Y$) with a proper sample that adequately covers the population of interest. NCA does not impose new requirements for designing a good observational study.

8.2.2 Longitudinal study

In this book, a longitudinal study (common name in statistics) and a panel study (common name in econometrics) are treated as synonyms, and the term longitudinal study is used. A longitudinal study is an observational study in which the same cases are followed over time and $X$ and $Y$ are measured at different time points.

When a necessity hypothesis includes an expectation about differences of necessity over time, or when the analyst is unsure about the possibility of reverse causality (see Section ??), a longitudinal study may be appropriate. It may be that the analyst expects that necessity exists at one time point but not at another or that the necessity between time points differs in strength (effect size).

Three types of longitudinal study are discussed here. First, it may be that data at different time points are available, but that the analyst is not interested in time per se. Then these data can be pooled together, resulting in a design that corresponds to the single cross-sectional observational study (e.g., Jaiswal & Zane, 2022). Second, if the analyst is interested in time trends, NCA can be applied for each time point separately. This corresponds to a multiple cross-sectional observational study. Third, if the analyst wants to avoid a reverse causality interpretation of the findings, $X$-scores are selected from a point in time before the $Y$-score is selected. Such a time-lagged study ensures that the condition $X$ existed before the occurrence of the outcome $Y$. NCA does not impose new requirements for designing a good longitudinal study.

Below, the longitudinal study is illustrated with an example of the hypothesis that a country’s economic prosperity is necessary for a country’s average life expectancy:

Focal unit: Country.
Condition concept $X$: Economic prosperity.
Outcome concept: $Y$: Live Expectancy.
Hypothesis: A high level of Economic prosperity is necessary for a high level of Live expectancy (high-high).
Theoretical domain: All countries in the world.

Economic prosperity captured by log GDP per capita and Life expectancy by Life expectancy at birth, expressed in years. The data are archival data from the World Bank. The data span a period of 64 years between 1960 and 2023, for 211 countries. Data are available for 11,020 country-years.

8.2.2.1 Pooled data

$XY$-plot of Economic prosperity and Life expectancy with two ceiling lines (Pooled World Bank data from 1960 to 2023 for all countries/years).

Figure 8.1: $XY$-plot of Economic prosperity and Life expectancy with two ceiling lines (Pooled World Bank data from 1960 to 2023 for all countries/years).

In the first illustration, the data are pooled to test the hypothesis. Figure 8.1 shows the $XY$-plot. Each data point is a country-year combination. The plot shows a clear empty space in the upper-left corner as expected. This suggests that a necessity relationship exists between Economic prosperity and Life expectancy⁵⁸. This indicates that a certain level of life expectancy is not possible without a certain level of economic prosperity. For example, for a country’s life expectancy of 75 years, an economic prosperity of about 3 ($10^3$ = 1,000 equivalent US dollars) is necessary. Such a level of economic prosperity is not sufficient as many cases with an Economic prosperity level of 3 have a lower level of Life expectancy⁵⁹.

8.2.2.2 Time trends

In the second illustration, the data for the different time points are used to test expected time trends regarding the necessity of high Economic prosperity for high Life expectancy⁶⁰. Table 8.1 shows the NCA parameters for each separate year.⁶¹

Table 8.1: NCA parameters per year.
Year	Ceiling intercept	Ceiling slope	Effect size	-value
1960	2.2	19.8	0.19	0.001
1961	18.8	15.0	0.18	0.001
1962	28.8	11.6	0.19	0.001
1963	27.5	12.2	0.19	0.001
1964	27.7	12.3	0.18	0.001
1965	38.2	8.7	0.21	0.001
1966	37.6	9.0	0.20	0.001
1967	41.4	7.7	0.22	0.001
1968	42.3	7.5	0.22	0.001
1969	43.3	7.2	0.22	0.001
1970	35.4	10.0	0.19	0.001
1971	36.5	9.6	0.19	0.001
1972	53.5	4.8	0.20	0.001
1973	56.3	4.2	0.19	0.001
1974	58.3	3.8	0.19	0.001
1975	38.9	9.0	0.19	0.001
1976	40.3	8.7	0.18	0.001
1977	51.1	5.7	0.19	0.001
1978	52.2	5.5	0.18	0.001
1979	52.6	5.5	0.18	0.001
1980	56.1	4.6	0.17	0.001
1981	45.5	7.6	0.17	0.001
1982	45.7	7.6	0.17	0.001
1983	45.0	7.8	0.16	0.001
1984	46.9	7.4	0.16	0.001
1985	47.6	7.3	0.16	0.001
1986	49.1	6.9	0.16	0.001
1987	50.7	6.5	0.15	0.001
1988	51.2	6.5	0.15	0.001
1989	50.2	6.8	0.15	0.001
1990	56.5	5.2	0.14	0.001
1991	58.6	4.7	0.14	0.001
1992	59.1	4.7	0.13	0.001
1993	58.3	4.9	0.13	0.001
1994	57.4	5.2	0.13	0.001
1995	56.7	5.4	0.13	0.001
1996	56.1	5.6	0.13	0.001
1997	54.7	5.9	0.13	0.001
1998	57.1	5.4	0.12	0.001
1999	56.4	5.6	0.12	0.001
2000	56.0	5.7	0.12	0.001
2001	55.8	5.8	0.12	0.001
2002	56.6	5.6	0.12	0.001
2003	57.5	5.5	0.11	0.001
2004	56.7	5.7	0.11	0.001
2005	57.0	5.7	0.11	0.001
2006	57.9	5.5	0.11	0.001
2007	59.1	5.3	0.10	0.001
2008	31.9	13.1	0.11	0.001
2009	59.0	5.5	0.09	0.001
2010	58.0	5.7	0.10	0.001
2011	57.1	5.9	0.10	0.001
2012	55.5	6.3	0.10	0.001
2013	54.4	6.6	0.10	0.001
2014	54.4	6.6	0.10	0.001
2015	55.8	6.3	0.10	0.001
2016	57.0	6.0	0.10	0.001
2017	57.0	6.0	0.10	0.001
2018	57.5	6.0	0.09	0.001
2019	57.8	5.9	0.09	0.001
2020	49.7	7.8	0.11	0.001
2021	51.3	7.4	0.11	0.001
2022	51.3	7.5	0.10	0.001
2023	48.9	8.2	0.10	0.001

Figure 8.2: Animation with countries per year with the ceiling line in red.

Figure 8.2 is an animated figure showing the time trends. In the figure, each data point is a country. The size of a data point refers to the country’s population size, and the color to the geographic region to which it belongs, e.g., yellow for the Sub-Saharan region, and violet for East Asian and Pacific region.

The interpretation of time trends focuses on the change of necessity effect size, ceiling line intercept and ceiling line slope over time. Table 8.1 and Figure 8.3-top-left show the clear trend that necessity effect size reduces over time. This indicates that, over the years, Economic prosperity has become less of a bottleneck for high Life expectancy. Figures 8.3-top-right and 8.3-bottom show that this decrease of effect size is due to the increase of the intercept (upward moving ceiling line) and decrease of the slope (ceiling line flattens).

Time trends of NCA parameters for the necessity of Economic prosperity for Life expectancy. Top-Left: necessity effect size. Top-Right: ceiling intercept. Bottom: ceiling slope.

Figure 8.3: Time trends of NCA parameters for the necessity of Economic prosperity for Life expectancy. Top-Left: necessity effect size. Top-Right: ceiling intercept. Bottom: ceiling slope.

Figure 8.4 compares the $XY$-plots for the first (1960) and last (2023) year in the dataset and shows that the effect size in 1960 was higher than in 2023.

Figure 8.4: $XY$-plot of Economic prosperity and Life expectancy for 1960 (Left) and 2023 (Right).

NCA’s statistical difference test for paired observations (Section ??) shows that the 0.08 difference between the effect sizes of 1960 and 2023 (0.19 and 0.11, respectively) is statistically significant ($p$ < 0.001).

In 1960 the maximum life expectancy for an Economic prosperity of 3 ($10^3$ = 1000 equivalent US dollars) was about 60 years, and in 2023 it was about 75 years. In 2023 compared to 1960, the countries are more clustered near the ceiling, indicating that the difference in Life expectancy between countries diminished, but that a hard border of maximum possible Life expectancy for a given Economic prosperity level remains.

Socio-economic policy practitioners could make new interpretations from these necessity time-trends. For example, in the past, economic prosperity was a major bottleneck for high life expectancy. Currently, most countries are economically prosperous enough for having the possibility of a decent life expectancy of 75 years; if these countries do not reach that level of life expectancy, other factors than economic prosperity are the reason for it. If the trend continues, in the future Economic prosperity may not be a limiting factor for high Life expectancy, because the empty space in the upper-left corner disappears.

8.2.2.3 Time-lagged study

The observed empty space in the upper-left corner could also be the results of reverse causality. In reverse causality $Y$ existed before $X$. An empty space in the upper-left corner could then also indicate that a low level of $Y$ is necessary for a low level of $X$. This would mean that a low level of Life expectancy is necessary ($Y$) for a low level of Economic prosperity $X$. When forward necessity makes sense, the argument for reverse causality is often weaker but cannot be excluded. With a time-lagged study, reverse causality is less likely when an empty space in the upper-left corner is observed while the $X$-score is obtained at time point ($T_1$) before the time point ($T_2$) of the $Y$-score. The lag should be long enough for $Y$ to develop in the presence of $X$ and other causal factors. For example, by using a 5-year time-lag the analysis with Economic prosperity in 2018 and Life expectancy in 2023 has a similar empty space in the upper-left corner (Figure 8.5).

Figure 8.5: $XY$-plot of Economic prosperity in 2018 and Life expectancy in 2023.

The differences between Figures 8.5 and 8.4-right are minor.

8.2.3 Case study

The case study investigates the phenomena in a small number of cases in their natural environment. Such a design is associated with a small-n qualitative study that allows a rich description of the phenomena. In such a study, relationships are often described in words rather than numerically.

NCA can be applied to a case study design in two ways. First, NCA’s methodology provides the causal logic to describe phenomena. Case studies are often used to explore complex causal relationships within specific contexts, and NCA complements this by introducing the idea that some factors may be necessary (but not sufficient) for an outcome to occur. In this way, NCA helps case investigators to move beyond traditional probabilistic sufficiency cause-effect explanations toward identifying the minimal requirements that must be present for a phenomenon to manifest.

Second, NCA’s data analytic method provides the tools to analyze qualitative data within or across cases. In multiple-case designs, coded evidence from interviews, documents, or observations can be transformed into ordinal or categorical indicators of conditions and outcomes. These can then be used as input scores for NCA to identify which conditions consistently act as enablers or bottlenecks across cases. The results can be interpreted to explain how and why these conditions function as necessary conditions.

If a necessity hypothesis already exists, the case study design can be used to test this hypothesis with the small number of cases. Case studies explicitly designed for testing necessity hypotheses are rare, but implicitly many studies do test such a hypothesis given the way the cases are selected (e.g., Fujita & Kusano, 2020; Harding et al., 2002, see Section 8.3.2 for more details). However, testing necessity hypothesis as part of an existing case study design, possibly originally aimed at exploration and theory building, offers a new opportunity for many qualitative studies (e.g., Novales et al., 2025).

When testing a necessity hypothesis with a small number of cases selected from the theoretical domain (deductive case study), a rich description of the context is not required, although it is not objectionable either. The only requirement is that the qualitative data can be scored into levels of the condition and the outcome (e.g., in terms of low, medium, high). Because necessity is independent of other variables, scoring the concepts $X$ and $Y$ is enough for testing the hypothesis. Given the small number of cases and their non-random selection, an estimation of $p$-values is not feasible. However, the reduced data requirements often allow the analyst to a dive deeper into the cases for better understanding the necessity mechanisms, or selecting more than a few cases, bringing the design closer to an observational study (Section 8.2.1). Case selection for testing necessity hypotheses is discussed in Section 8.3.

8.2.4 Experimental study

In an experimental study design, $X$ is first manipulated and afterwards the effect on $Y$ is observed, such that the required causal direction is ensured: first $X$ then $Y$. This excludes the possibility of reverse causality that $Y$ occurs before $X$ (Section ??). Whereas uncertainty about the causal direction could be handled with a time-lagged longitudinal study, where the condition is measured at $T_1$ and the outcome at $T_2$, such a study can only detect necessity if the outcome has developed during this time period. To have more control, the necessity experiment is an alternative approach for establishing the causal direction when the analyst is uncertain about reverse causality. With this goal in mind, the experiment is useful for testing a necessity hypothesis in the situation that reverse necessity is theoretically plausible.

The experimental setup for a necessity experiment differs from the traditional probabilistic sufficiency experiment is several ways. Whereas in a common experiment $X$ is manipulated (increased) to produce the presence of $Y$ on average, in a necessity experiment $X$ is manipulated (decreased) to produce absence of $Y$ (assuming a high-high hypothesis or more precisely: assuming that for a desired level of $y = y_c$, $x$ must be equal to or larger than $x_c$).

The necessity experiment starts with cases with high outcome ($y ≥ y_c$) and with high condition ($y ≥ x_c$), then the condition is removed or reduced ($x < x_c$), and it is observed if the outcome has disappeared or reduced ($y < y_c$). This contrasts the traditional “average treatment effect” experiment in which the research starts with cases with low outcome ($y < y_c$) and with low condition ($x < x_c$), then the condition is added or increased ($x ≥ x_c$), and it is observed if the outcome has appeared or increased ($ ≥ y_c$).

The necessity experiment is conducted as follows:

Obtain a representative sample from the population.
Check the expected empty corner.
Select the entire sample or part of it.
Divide this sample randomly into two groups (treatment and control group).
Remove ($x \rightarrow 0$) or reduce ($x \rightarrow low$) the condition in the treatment group.
Observe: (1) whether the effect size does not disappear; and (2) whether the outcome reduction in the treatment group is larger than the outcome reduction in the control group. If both are true, necessity is supported.

This necessity experiment ensures the causal direction (first $X$, then $Y$). The randomization between groups ensures that the group constitutions are similar, and that possible time effects (effect of other disappearing necessary conditions) are controlled for. The possible disappearance/reduction of the outcome due to other necessary conditions happens in both groups; the treatment group must have more outcome disappearance/reduction than the control group.

Below several types of necessity experiments depending on whether $X$ and $Y$ are dichotomous or continuous are discussed.

8.2.4.1 Necessity experiment: dichotomous $X$ and dichotomous $Y$

$$XY$-table of the necessity experiment before (Left) and after (Right) manipulation. $X$ = Condition (0 = no; 1 = yes). $Y$ = Outcome (0 = no; 1 = yes). \caseabsent{} = control group. × = treatment group.$

Figure 8.6: $XY$-table of the necessity experiment before (Left) and after (Right) manipulation. $X$ = Condition (0 = no; 1 = yes). $Y$ = Outcome (0 = no; 1 = yes). = control group. × = treatment group.

For dichotomous variables the experiment to evaluate a high-high necessity hypothesis the steps are conducted as follows:⁶²

Obtain a representative sample from the population.
Observe that expected corner is empty (corner 1).
Select the cases of corner 2. The experiment is done with these cases. In Figure 8.6-left it is assumed that 100 cases are available for the experiment.
Randomly assign these cases into two groups:
- Treatment group (symbol ×).
- Control group (symbol ).
Apply the treatment:
- Treatment group: Remove the condition: $x\rightarrow$ 0.
- Control group: Keep the condition: $x = 1$.
Observe the outcome $Y$ in both groups (Figure 8.6-right):
- Treatment group: number of cases that move to corner 1.
- Treatment group: number of cases that move to corner 3.
- Control group: number of cases that move to corner 4.
- Control group: number of cases that remain in corner 2.

If all treatment cases have moved from corner 2 to corner 3, but not to corner 1, and if not all control cases have moved from corner 2 to corner 4 (because another necessary condition disappeared during the manipulation) there is evidence that $X$ is necessary for $Y$. If treatment cases have moved from corner 2 to corner 1, this indicates that $Y$ can occur without $X$ and that $X$ is not necessary for $Y$. If control cases have moved from corner 2 to corner 4, this suggests that there is a time effect. This is not a problem for concluding that $X$ is necessary for $Y$ as long as not all control cases have moved to corner 4.⁶³

8.2.4.2 Necessity experiment: dichotomous $X$ and continuous $Y$

Figure 8.7 illustrates the situation when the condition is dichotomous and the outcome is continuous. Before the manipulation the condition is present ($x$ = 1). The $y$-values include the maximum value of the outcome ($y$ = 1). Necessity is supported when after the manipulation ($x \rightarrow 0$) the maximum value in the treatment group is reduced to a value of $y = y_c$, resulting in an empty upper-left corner of the $XY$-plot. The necessary condition can be formulated as $x = 1$ is necessary for $y \geq y_c$. The necessity effect size is the empty area divided by the scope. In this example the effect size = 0.18.

$$XY$-plot of the necessity experiment with a continuous outcome and a dichotomous condition before (Left) and after (Right) manipulation. $X$ = Condition (0 = no, 1 = yes). Y = Outcome between 0 and 1. \caseabsent{} = control group. × = treatment group. $y_c$ is the ceiling value of $Y$ for $x$ = 0.$

Figure 8.7: $XY$-plot of the necessity experiment with a continuous outcome and a dichotomous condition before (Left) and after (Right) manipulation. $X$ = Condition (0 = no, 1 = yes). Y = Outcome between 0 and 1. = control group. × = treatment group. $y_c$ is the ceiling value of $Y$ for $x$ = 0.

8.2.4.3 Necessity experiment: continuous $X$ and dichotomous $Y$

Figure 8.8 illustrates the situation when the condition is continuous and the outcome is dichotomous. Before the manipulation, the condition is present: $x \geq x_{high}$, where $x_{high}$ is a value above the presumed necessity level $x = x_c$. The treatment reduces the value of $X$ of all treatment cases. Some cases may end below $x = x_{high}$ but above $X = x_{c}$. Since the bottleneck value $x_c$ is not reached, these cases have $y$ = 1. Other treatment cases have new $x$-values below the necessity value $x$ = $x_c$. The latter group of cases will then end up in the lower-left corner with $y = 0$. Necessity is supported when after manipulation no cases with $Y = 1$ exist with $x < x_c$. In other words, all cases with $x < x_c$ should be located in the lower-left corner. The necessary condition can be formulated as $x \geq x_c$ is necessary for $y = 1$.

$$XY$-plot of the necessity experiment with a continuous condition and a dichotomous outcome before (Left) and after (Right) manipulation. $X$ = Condition (0 = no, 1 = yes). $Y$ = Outcome between 0 and 1. \caseabsent{} = control group. × = treatment group. $x_c$ is ceiling value of $x$-value for $y$ = 1. $x_{high}$ is the minimum $x$-value for selecting cases with $y$ = 1.$

Figure 8.8: $XY$-plot of the necessity experiment with a continuous condition and a dichotomous outcome before (Left) and after (Right) manipulation. $X$ = Condition (0 = no, 1 = yes). $Y$ = Outcome between 0 and 1. = control group. × = treatment group. $x_c$ is ceiling value of $x$-value for $y$ = 1. $x_{high}$ is the minimum $x$-value for selecting cases with $y$ = 1.

8.2.4.4 Necessity experiment: continuous $X$ and continuous $Y$

In a necessity experiment with a continuous condition and a continuous outcome, the starting point is the existence of an empty area in the upper-left corner of the $XY$-plot (Figure 8.9-left). The manipulation consists of reducing the level of the condition in the treatment group but not in the control group. Necessity is supported when after the manipulation no cases enter the empty zone. This is shown in Figure 8.9-right, where the necessity effect size before manipulation (0.25) stays intact after manipulation.

$$XY$-plot of a necessity experiment with a continuous condition and outcome before (Left) and after (Right) manipulation (reducing $X$). Effect size remains after manipulation. \caseabsent{} = control group. × = treatment group.$

Figure 8.9: $XY$-plot of a necessity experiment with a continuous condition and outcome before (Left) and after (Right) manipulation (reducing $X$). Effect size remains after manipulation. = control group. × = treatment group.

However, when cases enter the empty zone, the necessity is reduced (smaller effect size) or rejected (no empty space). Figure 8.10-right shows an example of a reduction of the necessity effect size from 0.25 to 0.15.

$$XY$-plot of a necessity experiment with continuous conditions and outcome before (Left) and after (Right) manipulation (reducing $X$). Effect size reduces after manipulation. \caseabsent{} = control group. × = treatment group.$

Figure 8.10: $XY$-plot of a necessity experiment with continuous conditions and outcome before (Left) and after (Right) manipulation (reducing $X$). Effect size reduces after manipulation. = control group. × = treatment group.

Figure 8.11 displays an example of an experiment in which a necessity effect disappears after experimental manipulation. In such cases, the necessary condition should be rejected.

$$XY$-plot of an experiment to challenge necessity with continuous condition and outcome before (Left) and after (Right) manipulation of reducing $X$. Effect size disappears after manipulation. \caseabsent{} = control group. × = treatment group.$

Figure 8.11: $XY$-plot of an experiment to challenge necessity with continuous condition and outcome before (Left) and after (Right) manipulation of reducing $X$. Effect size disappears after manipulation. = control group. × = treatment group.

In situations with a poorly covered sample, before manipulation the necessary condition may not be visible as an empty space in the upper-left corner because only cases are selected where the conditions are satisfied. After manipulation, the new scope (empirical or theoretical) is extended and a hidden empty space becomes visible. (Figure 8.12).

$$XY$-plot of a necessity experiment with a continuous condition and outcome before (Left) and after (Right) manipulation (reducing $X$). Effect size appears after manipulation. \caseabsent{} = control group. × = treatment group.$

Figure 8.12: $XY$-plot of a necessity experiment with a continuous condition and outcome before (Left) and after (Right) manipulation (reducing $X$). Effect size appears after manipulation. = control group. × = treatment group.

8.3 Sampling and selection of cases

After the choice of the type of study design, cases must be obtained from the theoretical domain where the necessity hypothesis is supposed to hold. The cases are used to test the hypothesis. In general it is a good strategy to select as many cases as possible. However, necessity theory may be broadly applicable in the theoretical domain consisting of many cases. If not all cases can be selected, a thoughtful selection must be done. NCA offers two types of case selection: large-n sampling for a quantitative study and small-n case selection for a qualitative study.

8.3.1 Large-n sampling

The goal of large-n sampling for a quantitative NCA study is to have a sample that is representative of a defined population from the theoretical domain. Usually, different populations where the cases have some relevant common characteristic (e.g., from the same country) could be selected. Since the theoretical domain may not be homogeneous, different populations could have different effect sizes, although the hypothesis claims that overall necessity exists for all cases. The selection of the population itself is not guided by statistical principles and can often be done for convenience. However, after the population is selected from the theoretical domain, random sampling of a large sample helps to ensure that the population is well-covered in the sample and that findings based on the sample can be statistically generalized to the population (inferential statistics). A possible approach to achieve this is to have a sampling frame (list of cases) of the population and to select cases randomly from this list. This ideal sampling approach contrasts with the more common approach of convenience sampling where cases are selected from the population for the analyst’s convenience, for example because the analyst has easy access to the cases. With such a sampling procedure, the sample may not be representative of the population, which can lead to biased effect size estimates and invalid $p$-values. The decision on the size of the sample depends on available resources and other practical factors. The minimum required sample size can be guided by the expected power of the study. Power refers to the minimum required sample size such that an existing relevant necessity effect size can be detected with high probability (Section ??).

8.3.2 Small-n case selection

It is possible to test a necessity hypothesis with a small number of cases, even with one case (e.g., Dul & Hak, 2008). In qualitative studies, usually a small number of cases is selected with a certain purpose in mind.⁶⁴ For testing a necessity hypothesis, the purpose of sampling is to have cases where the outcome is present ($Y = 1$) or the condition is absent ($X = 0$). After selecting such cases, it is observed if the condition is present ($X = 1$) or the outcome is absent ($Y = 0$), respectively. If the condition is absent in cases where the outcome is present, or the outcome is present in cases where the condition is absent, the necessary condition hypothesis is rejected for these cases. This type of purposive sampling in small-n studies is possible with a deterministic view on necessity (no exceptions) and when the conditions and the outcome have dichotomous scores (e.g., absent/present, low/high). Since according to the hypothesis necessity applies to each single case in the theoretical domain, a single case can be selected for the test. If necessity is rejected in that single case, the deterministic version of the necessity hypothesis is rejected. However, support for necessity based on only one or a few cases is often not convincing for a conclusion about the entire theoretical domain. Other cases that falsify the necessary condition hypothesis may not have been selected. Following replication logic, the confidence in a hypothesis increases when more cases do not falsify the hypothesis.

Purposive selection of cases for testing a necessity hypothesis was used implicitly in a study by Fujita & Kusano (2020) of highly controversial visits by Japanese Prime Ministers (PM’s) to the Yasukuni Shrine. Three necessary conditions were expected: a conservative ruling party, a government enjoying high popularity, and Japan’s perception of a Chinese threat. They tested these hypotheses by selecting all five cases where the outcome was present ($Y = 1$, PM’s who visited the shrine) from all 22 cabinets between 1986 and 2014. They found that the three conditions were present in all cases ($X = 1$) suggesting support for necessity, at least no rejection. Additionally, they selected the 17 cases where at least one potential necessary condition was absent ($X = 0$) and found that in these cases no visit occurred ($Y = 0$), giving further support for necessity.

In another small-n qualitative study, Harding et al. (2002) studied rampage school shootings in the USA. They first identified five potential necessary conditions: gun availability, cultural script (a model why the shooting solves a problem), perceived marginal social position, personal trauma, and failure of a social support system.⁶⁵ Subsequently, they tested the necessary conditions with two cases of school shooting ($Y = 1$) and found support for necessity (non-rejection) because the five conditions were present in these cases ($X = 1$).

Note that this approach of selecting a small number of cases for testing a necessity hypothesis is used in Chapter 7 as part of thought experiments to evaluate a preliminary necessity hypothesis based on existing knowledge (Section 7.5.2).

8.3.3 Replication

Since the necessity hypothesis is assumed to hold for all cases in the entire theoretical domain and the samples or selected cases that are used for testing the hypothesis are only a subset of the theoretical domain, replication studies with other cases are needed. Replication is re-testing the hypothesis with new data (Köhler & Cortina, 2023). A single shot study is not conclusive about a hypothesis in the entire theoretical domain. A next study should select cases from other parts of the theoretical domain, i.e. other samples from the sample population and samples from other populations for large-n sampling, or other cases for small-n case selection. Although a hypothesis can never be proven correct, the confidence in its correctness increases if repeated tests with different samples/cases fail to falsify the hypothesis.

8.4 Measurement

After the cases are sampled/selected, the $X$ and $Y$ characteristics of each case must be measured. In NCA the scores are often numeric (numbers, quantitative data) but also words or letters (qualitative data) can be used to represent levels of concepts (e.g., high or low). The goal of measurement is to obtain valid and reliable scores of the concepts $X$ (the conditions) and $Y$ (the outcome) that are defined in the necessity hypothesis. Scores are valid when they represent the value of a concept (measuring what you want to measure) and reliable when they are consistent under identical circumstances.

8.4.1 Quantitative data

Quantitative data are data expressed by numbers where the numbers are scores (values or levels) with a meaningful order and meaningful distance. Scores can have two levels (dichotomous, e.g., 0 and 1 ), a finite number of levels (discrete, e.g., 1,2,3,4,5) or an infinite number of levels (continuous, e.g., 1.8, 9.546, 306.22).

NCA can be used with any type of quantitative data. It is possible that a single indicator represents an $X$ or $Y$ concept. Then this indicator score is used to score the concept. For example, in a questionnaire study where a subject or informant is asked to score conditions and outcomes with seven-point Likert scales, the scores are one of seven possible (discrete) scores.

It is also possible that a construct that is built from several indicators is used to represent the concept of interest. Separate indicator scores are then summed, averaged or otherwise combined (aggregated), for example for scoring a formative index. It is also possible to combine several indicator scores statistically, for example by using the factor scores resulting from factor analysis, or construct scores of latent variables resulting from a measurement model.

8.4.2 Qualitative data

Qualitative data are data expressed by names, letters or numbers without a quantitative meaning. For example, gender can be expressed with names (male, female, other), letters (m, f, o) or numbers (1, 2, 3). Qualitative data are discrete and usually have no order between the scores (e.g., names of people) or if there is an order, the distances between the scores are unspecified (e.g., low, medium, high). Although with qualitative data a quantitative NCA (with the NCA software for R) is not possible, it is still possible to apply NCA in a qualitative way using visual inspection of the $XY$-table or plot.

When $X$ and $Y$ of cases are scored qualitatively, for example $X$ = A,B, or C, and $Y$ = no or yes, the corresponding $XY$-table has 6 cells where a case has a position in one of the cells. The hypothesis may be that B is necessary for yes. The analyst can inspect which $X$-score can reach that $Y$-score. When the logical order of the condition is absent, the position of the empty space can be in any upper cell (left corner, middle or right; assuming the hypotheses that high $X$ is necessary for high $Y$). When a logical order exists between the qualitative $X$-scores, the empty space is in the upper-left corner. In both cases the size of the empty space is arbitrary and no meaningful effect size can be calculated. However, when $Y$ is quantitative, the distance between the highest observed $Y$-score and the next-highest observed $Y$ score could be an indication of the constraint of $X$ on $Y$.

8.4.3 Set membership data

NCA can be used with set membership scores. This is often done when NCA is combined with QCA (Section ??). In QCA ‘raw’ variable scores are ‘calibrated’ into set membership scores with values between 0 and 1. A set membership score indicates to what extent a case belongs to the set of cases that have a given characteristic. In crisp set QCA the scores are dichotomous (0 or 1), and in fuzzy set QCA the scores are discrete or continuous between 0 and 1.

8.5 Dataset

After the measurements are done, the $X$- and $Y$-scores are integrated into a dataset. An appropriate format is having cases as rows and conditions and outcome as columns, where the first row (header) consists of the names of the conditions and outcome, and the first column the names of the cases.

Table 8.2 shows the dataset of the example used in Chapter 9 that is part of the NCA software.

Table 8.2: Example dataset nca.example from the NCA software in R.
	Individualism	Risk taking	Innovation performance
Australia	90	84	50.9
Austria	55	65	52.4
Belgium	75	41	75.1
Canada	80	87	81.4
Czech Rep	58	61	14.5
Denmark	74	112	116.3
Finland	63	76	173.1
France	71	49	77.6
Germany	67	70	109.5
Greece	35	23	12.0
Hungary	80	53	5.4
Ireland	70	100	62.3
Italy	76	60	19.7
Japan	46	43	171.6
Mexico	30	53	1.2
Netherlands	80	82	68.7
New Zealand	79	86	14.9
Norway	69	85	75.1
Poland	60	42	3.5
Portugal	27	31	11.1
Slovak Rep	52	84	3.5
South Korea	18	50	42.3
Spain	51	49	17.3
Sweden	71	106	184.9
Switzerland	68	77	149.7
Turkey	37	50	1.4
UK	89	100	79.4
USA	91	89	214.4

8.5.1 Archival data

NCA needs a dataset with $X$- and $Y$-values as input to empirically test the hypothesis that $X$ is necessary for $Y$. Often, new data on $X$ and $Y$ are gathered for conducting such a test. This requires a decision about the study design, the sampling/selection of cases, and the measurement of $X$ and $Y$. The quality of NCA depends on the quality of conducting these steps. However, it is also possible that an existing dataset is used. It may be that data on $X$ and $Y$ have already been tested for other purposes in previous studies. For example, a large number of datasets have been used with the goal of testing probabilistic sufficiency relationships (e.g., average effect of $X$ on $Y$ using regression analysis). If existing datasets contain $X$- and $Y$-scores of cases that are part of the theoretical domain, and if study design, sampling/case selection and measurement meet the quality criteria, the datasets from earlier studies may be useful for testing the necessity hypothesis. Given the trends towards open science, relevant datasets for testing necessity hypotheses are available, for example the World Bank dataset used in Section 8.2.2.When using archival data, the origins of the data should be evaluated and acknowledged, possible adaptations of $X$- and $Y$-scores should be justified and reported, and relevant earlier studies that use the same data should be evaluated and referenced to ensure accumulation of knowledge (Dul et al., 2024). In a multimethod study that combines NCA with other methods (Chapter 11), data that were collected for the other method may be re-used for NCA.

8.5.2 Data transformation

In quantitative (statistical) studies, data are often transformed before the actual analysis is done. For example, in machine learning, cluster analysis and multiple regression analysis, data are often standardized. The sample mean is subtracted from the original score and divided by the sample standard deviation to obtain the z-score: $X_{new} = (X — X_{mean}) / SD$. This allows, for example, a comparison of the regression coefficients of the variables, independently of the variable scale. Min-max normalization is another type of data transformation: $X_{new} = (X — X_{min}) / (X_{max} — X_{min})$. Such normalized variables have scores between 0 and 1. Transforming original scores into z-scores or min-max normalized scores are both examples of a linear transformation.

In principle, NCA does not need data transformation. The effect size is already normalized between 0 and 1 and allows comparison of effect sizes of different variables. Also, NCA does not make assumptions about the distribution of the data. Furthermore, interpretation of the results with original data is often easier than the interpretation of the results with transformed data as the original data may have a more direct meaning than the transformed data. If, nevertheless, the analyst wishes to transform data, a linear transformation (or in general an affine transformation) does not affect NCA’s estimation of the effect size or the $p$-value (Appendix D).

However a non-linear data transformation as is often done in quantitative (statistical) studies affects NCA’s estimation of effect size and $p$-value and is normally not acceptable for NCA. The transformation may change the cases that are used to establish the ceiling line, and this will affect the effect size. This can happen, for example, when data with a skewed distributed are log-transformed to obtain more normally distributed data. This is relevant when a statistical approach assumes normally distributed data, but this is not a requirement in NCA. Calibration in QCA is another example of a non-linear transformation. Original variable scores (‘raw scores’) are transformed into set membership scores, often using a logistic function (‘s-curve’). Calibration is needed in QCA as a set-theoretic method, but not in NCA, that can also handle raw scores.

Non-linear transformations should be avoided in NCA, unless there is a good reason for it. Such a good reason can be that a non-linear transformed original score is the valid representation of the concept of interest ($X$ or $Y$ in the necessity hypothesis). For example, the concept of a country’s economic prosperity may be expressed as the log-transformed GDP (Section 8.2.2.1). In QCA, the calibration process can justify the non-linear transformation of the original data. However, the general advice is to refrain from non-linear data transformation when the original data represent the concepts of interest, and only to transform data non-linearly when this is needed to validly capture the concept of interest.

Chapter 9 Data analysis

This chapter will be made available soon.

Chapter 10 Reporting

This chapter will be made available soon.

Chapter 11 Multimethod studies

This chapter will be made available soon.

Chapter 12 NCA in practice

This chapter will be made available soon.

Summary and personal reflection

Summary of the book

This book has introduced Necessary Condition Analysis (NCA) as a methodological approach that adds a distinct necessity perspective to the toolbox of empirical research and practical decision-making. NCA starts from a simple idea: some factors are not just helpful contributors to an outcome, they are indispensable. If a necessary condition is not present (at a required level), a desired outcome cannot occur, and no other factor can compensate for this absence. Conversely, for undesirable outcomes, certain stop factors can be kept below a threshold to make the outcome impossible.

Part I of the book set out the principles of NCA. It began in Chapter 2 with causality, clarifying how necessity logic (if not $X$, then not $Y$) differs from sufficiency logic (if $X$, then $Y$) and from configurational, probabilistic, or typicality-based perspectives. NCA extends classical binary necessity to a continuous view in which different levels of a condition enable different levels of an outcome, while still maintaining a clear zone in which cases are impossible, above a ceiling line in the $XY$-plot. The theory chapter (Chapter 3) then showed how necessity relationships can be formulated as parsimonious necessity theories, discussed directions of necessity, double necessity, and clarified what “necessary but not sufficient” means in a causal framework.

The mathematical foundations of NCA (Chapter 4) characterize necessity in geometric terms: bounded variables, ceiling lines, degrees of freedom, effect size, and several model fit measures such as ceiling accuracy, sharpness, and spread. Together, these measures describe how well the ceiling line separates the “impossible” empty corner from the feasible area, and how sharply the necessity pattern appears in the data.

The statistics chapter (Chapter 5) then explained how NCA uses permutation tests, simulations, and power analyses to test whether an observed necessity effect could have arisen by chance. It also showed how regression-based concepts such as mediation, moderation, omitted variable bias, and confounding are different or do not apply from the perspective of necessity logic and NCA.

The credibility chapter (Chapter 6) synthesized these foundations into a set of criteria for deciding whether a necessary condition claim is believable: theoretical support (a well-developed necessity hypothesis), practical relevance (a substantively meaningful effect size), statistical significance, robustness checks, and model fit. Together they provide a structured way to evaluate whether an observed ceiling pattern truly supports a necessity claim rather than reflecting a statistical artefact or a trivial result.

Part II translated these principles into concrete steps for conducting NCA studies. First, the hypothesis chapter (Chapter 7) described how to develop formal necessity hypotheses by drawing on explicit and tacit knowledge, building a preliminary necessity theory, performing thought experiments, and adding a causal explanation.

The data chapter (Chapter 8) discussed study designs (observational, longitudinal, case-based, experimental), sampling, case selection, and measurement for quantitative, qualitative, and set-membership data, as well as the use of archival data and data transformations.

The data analysis chapter (Chapter 9) presented NCA’s core analytic approach, which involves specifying the model bounding box and the ceiling line, visually inspecting the $XY$-plot, estimating the ceiling line, computing effect size and $p$-values, identifying and handling outliers, and assessing model fit. It then distinguished between necessity in kind (whether a condition is necessary at all) and necessity in degree (how much of a condition is required for a specific target outcome), operationalized via bottleneck tables. These bottleneck tables show which levels of conditions constrain particular target levels of the outcome, thus directly connect NCA results to potential action. Robustness checks (varying ceiling functions, scopes, thresholds, outlier decisions, and target outcomes) help to assess the stability of necessity claims.

The reporting chapter (Chapter 10) explained how to write up NCA studies for different purposes (empirical, theoretical, methodological introduction, and practical publications) and introduced the SCoRe checklist for high-quality reporting, covering theory, methods, data, analysis, results, and discussion.

The multimethod chapter (Chapter 11) positioned NCA within a broader landscape of methods and explained how NCA can be combined with regression-based approaches (MLR, SEM, PLS) and with QCA. Rather than treating methods as competing, the book advocates a causal-pluralist view where necessity, sufficiency, and probabilistic perspectives complement one another: they are first analyzed separately and then interpreted collectively using tools like NERT, NEST and BIPMA. Each method answers a different causal question and combining them allows a richer understanding of the same phenomenon.

The final chapter (Chapter 12) translated NCA into practice. Here, necessary conditions become levers for change. NCA helps to identify must-have factors for desirable outcomes and stop factors for preventing risks. Its threshold logic and bottleneck areas make it possible to design interventions that target those cases that cannot reach a target outcome unless a specific condition is improved, or that will remain protected from an undesired outcome as long as a risk factor stays below its threshold. The chapter introduced concepts such as intervention effort, effectiveness, waste and efficiency, and illustrated how NCA-based interventions can be more targeted and cost-effective than interventions guided by average-effect logic.

Taken together, the book shows that NCA is not a replacement for existing methods, but a complementary approach that adds a missing causal lens: necessity. It invites analysts to move beyond the question “What works on average?” and to also ask “What is indispensable?” For researchers, NCA encourages the development of necessity theories and provides tools to test them rigorously. For practitioners, NCA offers a way to detect and remove bottlenecks in real systems, whether in education, business, healthcare, public policy, or any other field. With its conceptual framework, analytic tools, reporting standards, software support, and practical guidance, the book aims to enable thoughtful, high-quality use of NCA in both academic and applied work, and to stimulate further development of necessity-based thinking in the years to come.

Reflections on a decade of NCA

The publication of this book coincides with the tenth anniversary of Necessary Condition Analysis. The first formal exposition of NCA appeared in the journal Organizational Research Methods in 2016, but the idea itself required another decade of thinking, experimenting, and refining before it was ready to enter the scientific world. My own path toward NCA was far from linear. Trained first in mechanical engineering, then in biomedical engineering, I worked in applied research and consultancy in the field of human factors and ergonomics, and then moved into academia in business and management in the social sciences. I did not begin my career in the fields where NCA would ultimately take root.

I held managerial and administrative positions. These experiences gave me a deep appreciation for practical decision-making: understanding constraints, identifying what is truly essential, and distinguishing what merely helps from what must be present. With my strong connections to engineering and medical sciences, I was accustomed to environments where replication, empirical validation, and evidence-based practice were indispensable. Predictions had to be precise and actionable.

Entering the social science field late and as an outsider was, in some ways, a disadvantage. I lacked the traditional background and networks. But it was also an advantage: I came without disciplinary blinders, free to be curious, to ask naive questions, and to challenge established assumptions. When I moved into the social sciences, I found a landscape overwhelmed by theories and concepts, often complex, difficult to interpret, and hard to translate into action. I saw a surprising tolerance for drawing strong conclusions from single-shot studies. The lack of replication, limited availability of high-quality data, and the growing complexity of models left me wondering how decisions could be made with confidence.

At the same time, it became clear to me that this situation was not due to negligence. The social sciences are young compared to engineering and medicine, and the scale of funding available for research is simply not comparable. Despite these constraints, the social sciences have achieved great advancements. Real-life problems involve people, decisions, organizations, and institutions, and the problems cannot be understood and solved without the insights of the social sciences. Addressing real-world challenges requires the combined strengths of sciences working together.

In this context, I sensed that something essential (literally, something necessary) was missing. The deterministic logic of necessity, which can perfectly predict the absence of an outcome and is foundational in how real systems function, offered a way out. Although the idea emerged during my encounter with the social sciences, it applies equally in engineering and medicine. Indeed, it is now taking hold in those fields as well. I often wondered how it was possible that the necessity perspective of NCA had not been developed earlier.

Before 2016, I also took time to think carefully about dissemination. I reflected on how new ideas spread and where resistance may lie: among gatekeepers, reviewers, editors, and entrenched methodological traditions. I knew that for NCA to have a chance, it should not be perceived as a competitor to established methods but as a complementary lens that adds something genuinely new. I also knew that it would need openness, transparency, and accessibility. From the start, my principle was that NCA should be freely available: free software, free materials, free support, and freely shared ideas. Dissemination meant planting seeds among doctoral students, researchers, and practitioners, and letting those seeds grow where they found fertile ground.

The years following the first publication were a period of extremes. On one side were strong negative reactions, sometimes surprisingly intense. There were science trolls (yes, they exist) who attacked not only the method but also its intentions. There were editors who feared that using NCA was “naive”, reviewers who issued strong opinions without knowing the method, and peers who advised others not to engage with the method. Some speculated that “if applied in medicine, people will die.” Others insisted that NCA should first prove itself in the top-tier econometrics journal before it could be taken seriously. And there were cynical remarks about miraculous empty spaces or trivial data transformations. These moments were not easy, but in areas where NCA has been accepted, they are largely behind us now. As Schopenhauer observed, new ideas first encounter ridicule, then opposition, and finally acceptance as self-evident.

Yet there were equally strong and uplifting experiences. Brave editors such as James LeBreton were willing to give the method a fair chance. Scholars like Herman Aguinis immediately saw the potential of necessity thinking, and many colleagues contributed enthusiastically (as recognized in the Acknowledgements). Even some early critics who once dismissed the method later became among the strongest supporters once they understood what NCA actually offers. A few even speculated wildly that the invention merited a Nobel Prize or could be a panacea for all methodological challenges (it is not!).

Alongside these positive developments, I also encountered weaker applications: rushed implementations, superficial readings, and a kind of “science populism” when people present NCA with great certainty while missing its underlying logic, sometimes even giving incorrect answers. Others tried to monetize the method, which only strengthened my commitment to keep NCA free and openly available.

Even with this turbulence, the growth of NCA has been remarkable. More scholars have used the method than I ever imagined, and many have helped push it forward. I am proud of what has been achieved: the development of concepts, terminology, mathematical foundations, statistical procedures, and tools that previously did not exist. Much had to be invented from scratch like names for ceiling lines, new concepts like the bottleneck table, or new measures of fit, and many other components that now form the method’s core.

Why has NCA not yet been more widely adopted in practice, despite its potential? One reason is academic incentives: researchers are encouraged to move quickly to the next publication, leaving limited time for sustained engagement with practical application. Many who use NCA strive for both academic rigor and practical relevance, but the system rarely rewards that combination. Nevertheless, I remain optimistic. With growing experience, more applications in practice, and continued methodological development, NCA is becoming increasingly mature, and its practical value is gradually being recognized.

Looking forward, I see three important directions:

• Broader uptake in academia, including in fields such as the medical sciences, where NCA can help prevent undesired outcomes by identifying and eliminating indispensable single conditions, as well as in disciplines that still focus primarily on explaining why outcomes occur on average.

• Further deepening of the method. Once a method is established and accepted, the following decades allow for refinement, extension, and specialization.

• More real-world applications demonstrating how NCA can guide action, decision-making, and practical interventions. Practitioners who currently rely solely on average-effect logic can benefit from necessity logic as well.

At the same time, there are no shortcuts in NCA. Applying NCA requires methodological competence, respecting the fundamentals of NCA: understanding its deterministic necessity logic (with possible exceptions), its mathematical and statistical characteristics, and its assumptions, without slipping into probabilistic or conventional statistical thinking. Founders and early methodologists of other methodological approaches such as SEM, QCA, and grounded theory have warned in their own traditions that users should refrain from mechanical application, neglected assumptions, and overclaiming results. I have no illusion that NCA is immune to these tendencies, or that this book can fully prevent them. NCA, too, will be (and from the very beginning already has been) misinterpreted, routinized, overclaimed, and at times misused.

I therefore call upon NCA users to respect its principles: build strong theoretical support with explicit necessity hypotheses; avoid tweaking analyses to “find” necessity (since non-necessity can be equally informative); and report approaches, results and conclusions transparently. These principles are operationalized in the SCoRe checklist for analysts, authors, readers, peers, editors, reviewers and others to assess and enhance the quality of NCA applications.

Finally, on a personal note, with thankfulness, many people have joined me along the way, contributing to the development and growth of NCA. But only one person has witnessed every effort, doubt, frustration, and joy, and has supported me unconditionally throughout the entire journey. For that, Renata, I am deeply grateful.

Part III. Additional materials

Appendix A Nomenclature and glossary

A.1 Nomenclature

Symbol	Meaning
$\alpha$	Threshold value for statistical significance
$a$	Slope of a linear function
$b$	Intercept of a linear function
$C$	Size of the ceiling zone; a point on the ceiling line $(x_c, y_c)$
$ca$	Model fit metric ceiling accuracy
$cp$	Model fit metric complexity
$d$	Effect size
$df$	Degrees of freedom of a necessity model
$ex$	Model fit metric exceptions
$f$	General symbol for a mathematical function
$F$	Feasible area
$ft$	Model fit metric fit
$h$	Horizontal segment of a ceiling line
$H$	Hypothesis
$i$	Index for an observation or of a segment of a ceiling line
$j$	Index for a variable
$J$	Number of necessary conditions
$k$	Number of outliers in a set of outliers
$lowP$	Number of cases in the lowP-zone
$lowS$	Number of cases in the lowS-zone
$max$	Maximum value of a given range
$medP$	Number of cases in the medP-zone
$medS$	Number of cases in the medS-zone
$min$	Minimum value of a given range
$n$	Sample size
$N$	Necessary condition (in NEST)
$nc$	Necessary condition; necessary cause (near arrow in causal graph)
$ns$	Model fit metric noise
$p$	$p$-value for a statistical significance test
$P$	Number of peers that define the ceiling line
$S$	Scope: the size of the bounding box
$sd$	Standard deviation
$sh$	Model fit metric sharpness
$sp$	Model fit metric spread
$su$	Model fit metric support
$T$	Time indicator
$v$	Vertical segment of a ceiling line
$X, Y, Z$	Variable names (capitals)
$x, y, z$	Variable values (lower case)

A.2 Glossary

This glossary defines important terms used throughout this book. Defined terms are printed in bold. When a definition contains another defined term, it is printed in italic. When a defined term appears more than once within a single definition, only its first occurrence is highlighted.

# above. An NCA parameter indicating the number of cases above the ceiling line.

Absence/low value of condition/outcome. A value of a condition or outcome that is close to its minimum. Also see Presence/high value of condition/outcome, Direction (of a necessary condition).

Absolute inefficiency. The area of the bounding box where the necessary condition does not constrain the outcome and the outcome is not constrained by the necessary condition. Also see Relative inefficiency, Condition inefficiency, Outcome inefficiency.

Academia. The world of higher education and research, like universities, focusing on theory, discovery, and teaching. Also see Practice.

Accuracy. See Ceiling accuracy, $p$-value accuracy.

Adjacent corner. A corner in the bounding box next to the corner of interest. Also see Opposite corner, Expected empty corner.

Alternative hypothesis. A statistical concept describing a competing claim to the null hypothesis. Also see Formal necessity hypothesis.

Analyst. A scholar or practitioner who uses NCA or other methodological approaches.

Approximate permutation test. See Permutation test.

Binary logic. Two-valued logic where statements can only be true or false. Also see Conditional logic, Causal logic, Necessity logic, Sufficiency logic.

BIPMA Bottleneck Importance Performance Map Analysis. An NCA tool for a giving practical advice based on SEM results NCA results based in single bottleneck cases. Also see NERT, IPMA, cIPMA.

Bivariate analysis. A statistical analysis with two variables. Also see Multiple bivariate analyses, $XY$-plot, $XY$-table.

Boolean logic. See Binary logic.

Bottleneck case. A case without the necessary level(s) of the condition(s), such that it is unable to achieve the target outcome. Also see Non-bottleneck case, Ignored case, Irrelevant case, Single bottleneck case, Multiple bottleneck case.

Bottleneck distance. The gap between a case’s condition value ($x$) and target condition value ($x_c$).

Bottleneck table. A tabular representation of the ceiling line showing which values of the condition(s) is/are necessary for a particular value of the outcome.

Bounding box. The rectangle defined by observed or theoretical limits of the condition and the outcome. Also see Scope, Empirical scope, Theoretical scope, Tight bounding box.

c-accuracy. See Ceiling accuracy.

C-LP. Ceiling - Linear Programming. A linear ceiling line based on minimization of heights of the CE-FDH peers by linear programming. Also see CE-FDH, CR-FDH, CR-VRS, QR.

Case. An instance of a focal unit.

Case selection. The selection of one or a small number of cases from a set of cases for inclusion in a small-n study. Also see Sampling.

Case study. A study design in which one or a small number of cases is selected for a small-n study. Also see Observational study, Longitudinal study, Experimental study.

Causal explanation. A story that connects causes to effects.

Causal interpretation. The causal explanation of a relationship judged as credible based on extra-data considerations such as domain knowledge, simplicity, stability, and the plausibility of underlying assumptions. Often data allow multiple interpretations.

Causal logic. The logic in which statements are causal relationships. Also see Binary logic, Conditional logic, Necessity logic, Sufficiency logic.

Causal narrative. See Causal explanation.

Causal perspective. The perspective selected by the analyst to theorize or infer causality from data. Also see Necessity perspective, Typicality perspective, Sufficiency perspective, Probabilistic sufficiency perspective.

Causal-pluralism study. A multimethod study that uses multiple causal perspectives and corresponding different methods with the same data. Also see Method-triangulation study, Separate studies, Theory-method fit.

Causal relationship. A relationship between two variable characteristics $X$ and $Y$ of a focal unit in which a value of $X$ (or its change) permits, or results in a value of $Y$ (or in its change).

Causal underdetermination. The idea that data are compatible with multiple distinct causal perspectives and explanations, so the causal structure cannot be uniquely identified from the data alone.

Cause. A variable characteristic $X$ of a focal unit of which the value (or its change) permits, or results in a value (or its change) of another variable characteristic $Y$. Also see Necessity cause, Sufficiency cause.

CB-SEM. Covariance-Based SEM. A SEM approach that estimates model parameters based on the covariance matrix. Also see PLS-SEM.

CE-FDH. Ceiling Envelopment - Free Disposal Hull. A stepwise linear ceiling line based on the free disposal hull. Also see Ceiling line, CR-FDH, CE-VRS.

CE-VRS. Ceiling Envelopment - Variable Returns to Scale. A piecewise linear ceiling line consisting of oblique line segments such that envelope is concave when the ceiling zone is in corner 1. Also see Ceiling line, CE-FDH, CR-FDH, CR-VRS.

Ceiling accuracy. A model fit metric expressing the extent to which cases are on or below the ceiling line.

Ceiling Envelopment - Free Disposal Hull. See CE-FDH.

Ceiling Envelopment - Variable Returns to Scale. See CE-VRS.

Ceiling line. The borderline within the bounding box between the area where cases are impossible (ceiling zone) and the area where cases are possible (feasible area), such that a point $(x, y)$ is on the ceiling if and only if, for any other point $(x', y')$ in the bounding box, it holds that if $x' < x$ and $y' > y$, then $(x', y')$ is in the ceiling zone and if $x' \ge x$ and $y' \le y$, then $(x', y')$ is in the feasible area. Also see CE-FDH, CR-FDH, C-LP, CE-VRS, CR-VRS, QR.

Ceiling outlier. An outlier in the ceiling zone. Also see Scope outlier.

Ceiling points. Points on or near the ceiling line. Also see MedP-zone, MedS-zone.

Ceiling Regression - Free Disposal Hull. See CR-FDH.

Ceiling zone. The area in a corner of the bounding box where points cannot exist, except for exceptions and noise. Also see Feasible area.

Chain (of) necessity. See Necessity chain.

Chain necessity effect. The necessity effect of the first element in a necessity chain on the final outcome (the extent to which the initial condition constrains the maximum attainable level of the chain’s endpoint through the intermediate necessary links).

Chained necessity. See Chain necessity effect.

cIPMA. Combined Importance Performance Map Analysis. An extended version of IPMA that includes NCA results by considering a condition’s number of bottleneck cases. Also see BIPMA.

Combined Importance Performance Map Analysis. See cIPMA.

Complete hypothesis. A necessity hypothesis with a plausible causal explanation about why the condition is necessary for the outcome. Also see Formal necessity hypothesis, Trivial hypothesis, Testable hypothesis, Plausible hypothesis.

Completeness. See Complete hypothesis.

Complexity. A model fit metric expressing the number of parameters that are needed for describing the necessity model (degrees of freedom - 4). Also see Model specification.

Concept. The varying characteristic of a focal unit of a theory. Also see Condition, Outcome.

Conceptual model. A visual representation of a hypothesis of a necessity theory in which the condition and outcome are presented by rectangles and the relationship between them by an arrow. The arrow originates in the condition and points to the outcome and represents the causal direction with the letters nc near it to express the direction of the necessary condition. Also see Formal necessity hypothesis.

Condition. A varying characteristic $X$ of a focal unit of which the value (or its change) permits, or results in a value (or its change) of another varying characteristic $Y$ (which is called the outcome). Also see Necessary condition, Sufficient condition, Outcome, Predictor (variable).

Condition inefficiency. The area of the bounding box where the condition does not constrain the outcome. Also see Absolute inefficiency, Relative inefficiency, Outcome inefficiency.

Conditional logic. If-then statements that can only be true or false. Also see Binary logic, Causal logic, Necessity logic, Sufficiency logic.

Confounder. A term used in regression-based analysis indicating the theoretical role of a concept as a common cause of two other concepts and that may account for part of the observed relationship between them. Also see Mediator, Moderator.

Contingency table. See $XY$-table.

Continuous necessity. A necessity relationship in which the condition and the outcome can have infinite numbers of levels (values). Also see Dichotomous necessity, Discrete necessity.

Contrast test. A permutation test in NCA in which for each case the labels $X_1$ and $X_2$ are switched with 50/50 probability. Also see Null test, Independent test, Paired test.

Control variable. A variable that is added in a regression-based data analyses for improving the prediction of the outcome and avoiding biased estimation of regression coefficients. Also see Predictor (variable), Predicted variable.

Convenience sample. A sample in which the instances are selected for convenience of the analyst. Also see Random sample, Purposive case selection.

Corner 1. Upper-left corner of the $XY$-plot or bounding box.

Corner 2. Upper-right corner of the $XY$-plot or bounding box.

Corner 3. Lower-left corner of the $XY$-plot or bounding box.

Corner 4. Lower-right corner of the $XY$-plot or bounding box.

Corner of interest. The corner of the bounding box implied by the necessity hypothesis. Also see Formal necessity hypothesis, Corner 1, Corner 2, Corner 3, Corner 4.

Counterexample. A case that is inconsistent with the necessity hypothesis. Also see Exception, Outlier, Noise.

Covariance-Based SEM. See CB-SEM

CR-FDH. Ceiling Regression Free Disposal Hull. A linear ceiling line based on a trend line through the CE-FDH peers. Also see C-LP, QR, CR-VRS.

Credibility. The trustworthiness of conclusions about necessity identified from data. Also see Statistical credibility, Empirical credibility.

Crisp-set Qualitative Comparative Analysis (csQCA). See csQCA.

CR-VRS. Ceiling Regression Variable Returns on Scale. A linear ceiling line based on a trend line through the CE-VRS peers. Also see CR-FDH, C-LP, QR.

csQCA. Crisp-set Qualitative Comparative Analysis. A variant of QCA in which all conditions and outcomes are expressed as binary (0/1) set membership scores, indicating full non-membership (0) or full membership (1) in a set. Also see fsQCA.

$\mathbf{d}$. See Effect size.

Data. Recordings of evidence generated in the process of data collection. Also see Measurement, Qualitative data, Quantitative data, Longitudinal data.

Data analysis. The interpretation of scores obtained in a study in order to generate the result of the study. Also see Qualitative data analysis, Quantitative data analysis.

Data collection. The process of identifying and selecting one or more objects of measurement, extracting evidence of the value of the relevant variable properties or characteristics from these objects, and recording this evidence. Also see Data, Measurement, Score, Dataset.

Data Generation Process. See DGP.

Dataset. A collection of scores obtained from data collection.

Degrees of freedom. The minimum number of parameters that are needed for describing a necessity model. See also Model specification, Complexity.

Deterministic necessity. Necessity from the deterministic perspective. Also see Typicality necessity.

Deterministic perspective. A causal perspective taken by the analyst that a cause always influences an effect without exceptions. Also see Probabilistic perspective, Typicality perspective.

DGP. Data Generation Process. The hypothetical mechanism that produces the data that are observed.

Dichotomous necessity. A necessity relationship in which the condition or the outcome has only two levels (values). Also see Discrete necessity, Continuous necessity.

Direct effect. A term used in regression-based analyses indicating the effect of $X$ on $Y$ without going through an intermediate variable (mediator). Also see Indirect effect, Total effect, Moderator.

Direction (of a necessary condition). An indication whether the condition and outcome in the necessity hypothesis are absent or present. Also see nc, Formal necessity hypothesis.

Discrete necessity. A necessity relationship in which the condition or the outcome has a finite number of levels (values). Also see Dichotomous necessity, Continuous necessity.

Domain. See Theoretical domain.

Effect size. The magnitude of the constraint that a necessary condition poses on the outcome expressed as the size of the ceiling zone relative to the size of the bounding box (scope). Also see $p$-value.

Effect size threshold. The value of the effect size selected by the analyst for evaluating the necessity hypothesis. Also see Formal necessity hypothesis, Statistical significance threshold.

Effective case. An intervention case that is in the bottleneck area (bottleneck case) before the intervention and removed from it after the intervention. Also see Ineffective case, Irrelevant case, Ignored case.

Embedded necessity theory. A necessity theory with one or more necessity relationships and on or more other relationships. Also see Pure necessity theory.

Emerging theory. A theory that is not (yet) broadly accepted in academia or practice. Also see Established theory.

Empirical credibility. The trustworthiness of the necessity hypothesis, the study design, the data, and the data analysis used to draw conclusions about necessity. Also see Formal necessity hypothesis, Statistical credibility, Effect size, $p$-value, Robustness, Model fit.

Empirical data. Data from the real world. Also see Simulation data

Empirical scope. The bounding box (or its area) defined by empirically observed minimum and maximum values of the condition and the outcome. Also see Theoretical scope, Scope.

Empty area. See Ceiling zone.

Empty corner. See Ceiling zone.

Empty space. See Ceiling zone.

Established theory. A theory that is broadly accepted in academia or practice. Also see Emerging theory.

Evidence-based intervention. An intervention that is based on empirical data.

Exception. A case in the lowP-zone of the ceiling zone that is not the result of error and is accepted by the analyst as a rare counterexample of necessity. Also see Outlier, Noise, Typicality perspective.

Exceptions. A model fit metric expressing the number of cases in the lowP-zone of the ceiling zone that the analyst considers an exception.

Expected empty corner. The area in the bounding box that is predicted to be empty according to the necessity hypothesis. Also see Formal necessity hypothesis, Ceiling zone.

Experimental study. A study design in which the condition is manipulated and the outcome is observed. Also see Observational study, Longitudinal study, Case study.

Expert knowledge. Tacit knowledge of a scholar or practitioner. Also see Explicit knowledge.

Explicit knowledge. Documented knowledge in the literature from academia and practice to be used for formulating a formal necessity hypothesis. Also see Tacit knowledge, Expert knowledge.

Extension. A horizontal or vertical line segment added to the endpoints of a polyline that connects peers to envelop the feasible area.

Falsification. The view that theories and hypotheses cannot be proven true, but can only be proven false.

Feasible area. The area within the bounding box below the ceiling line for corner 1 or corner 2, or above the ceiling line (“floor line”) for corner 3 or corner 4. Also see Ceiling zone.

Fiss chart. A solution table of QCA with specific notation for presence or absence of a condition, and for core or peripheral conditions. Also see Solution table, NEST.

Fit. A model fit metric expressing the effect size of a selected ceiling line as percentage of the effect size of the CE-FDH ceiling line.

Floor line. A ceiling line in corner 3 or 4.

Focal unit. The unit of a theory or hypothesis. Examples are ‘employee’, ‘team’, ‘company’, ‘country’. Also see Theory, Theoretical domain.

Formal necessity hypothesis. A necessity relationship that is theory-grounded, testable, non-trivial, plausible and complete. Also see Preliminary necessity hypothesis.

Formal necessity theory. A necessity theory in which the necessity hypotheses are formal necessity hypotheses. Also see Preliminary necessity theory.

Fragility. The absence of robustness.

fsQCA. Fuzzy-set Qualitative Comparative Analysis. A variant of QCA in which conditions and outcomes are expressed as fuzzy set membership scores ranging from 0 to 1. Also see csQCA.

Fuzzy-set Qualitative Comparative Analysis. See fsQCA.

High-high necessity. A necessity relationship where the presence/high value of the condition is necessary for the presence/high value of the outcome (‘+ nc +’). Also see nc, Low-high necessity, High-low necessity, Low-low necessity, direction.

High-low necessity. A necessity relationship where the presence/high value of the condition is necessary for the absence/low value of the outcome (‘+ nc -’). Also see nc, High-high necessity, Low-high necessity, Low-low necessity, direction.

Hypothesis. A statement about the relationship between concepts or variables. Also see Hypothesis testing, Necessity hypothesis, Formal necessity hypothesis.

Hypothesis testing. The procedure of making a decision about the credibility of a hypothesis. Also see Statistical credibility, Empirical credibility, Null hypothesis test, Thought experiment.

Ignored case. An non-intervention case that is in the bottleneck area (bottleneck case) before and after the intervention. Also see Irrelevant case, Effective case, Ineffective case.

Importance Performance Map Analysis. See IPMA.

Independent test. A permutation test in which from a pool of two groups cases are randomly reassigned to two groups of the same sizes. Also see Null test, Contrast test, Paired test.

Indirect effect. A term used in regression-based analyses indicating the effect of $X$ on $Y$ through a mediator. Also see Direct effect, Total effect, Moderator.

Ineffective case. An intervention case that is in the bottleneck area (bottleneck case) before and after the intervention. Also see Effective case, Irrelevant case, Ignored case.

Inefficiency. A set of NCA parameters related to the extend to which the necessary condition covers the range of the condition and the outcome. Also see Absolute inefficiency, Relative inefficiency, Condition inefficiency, Outcome inefficiency.

Infeasible area. See Ceiling zone.

Influential case. A case that has a large influence on the necessity effect size when removed. Also see Outlier.

Informant. A person who is the object of measurement for a variable and who is knowledgeable about that variable and informs the analyst about it. Also see Subject.

Instance of a focal unit. A single case.

Intervention. An external action applied to a case in order to increase or decrease the condition value with the intention to remove the case from the bottleneck area (for a desired target outcome) or move it into the bottleneck area (for an undesired target outcome).

Intervention case. A case to which an intervention is applied. Also see Non-intervention case, Effective case, Ineffective case, Irrelevant case.

Intervention design. The approach to collect and analyze data and displaying crucial results using the bottleneck area.

Intervention effectiveness. The number of treated bottleneck cases that have left the bottleneck area (effective cases) as a percentage of intervention cases.

Intervention efficiency. The amount of intervention effort on intervention cases that was effective in removing cases from the bottleneck area. It is the complement of intervention waste and is expressed in percentages.

Intervention effort. The magnitude of change in the condition value produced by the intervention.

Intervention goal. The intended result of an intervention specified as desired or undesired outcome and possibly by a specific target outcome.

Intervention group. A group consisting of intervention cases.

Intervention strategy. The approach to establish the team for the intervention, set the goals and the target groups, evaluate drivers and barriers, and formulate hypotheses. See also Intervention design.

Intervention waste. The amount of intervention effort that does not contribute to moving cases out of or into the bottleneck area. Also see Undershoot waste, Overshoot waste.

Irrelevant case. An intervention case that is not in the bottleneck area (non-bottleneck case) before (and after) the intervention. Also see Ignored case, Effective case, Ineffective case.

IPMA Importance Performance Map Analysis. A tool for giving practical advice based on SEM results. Also see cIPMA, BIPMA.

Iso-purity line. See Iso-P line.

Iso-P line. A line representing points with the same purity value. Also see Iso-S line, Model fit, NCA ribbon.

Iso-solidity line. See Iso-S line.

Iso-S line. A line representing points with the same solidity value. Also see Iso-P line, Model fit, NCA ribbon.

Large-n study. A study with a large number of cases. n stands for the number of cases. Also see Small-n study.

Likert scale. A rating scale in the format of a limited number of points (e.g., 5 or 7) that can represent a person’s response. Also see Informant, Subject.

Logic. See Binary logic, Causal logic, Conditional logic, Necessity logic, Sufficiency logic.

Longitudinal data. Scores of condition and outcome that are measured at multiple time points. Also see Quantitative data, Qualitative data, Set membership data.

Longitudinal study. A study design with data collection at multiple time points. Also see Observational study, Case study, Experimental study.

Low-high necessity. A necessity relationship where the absence/low value of the condition is necessary for the presence/high value of the outcome (‘- nc +’). Also see nc, High-high necessity, High-low necessity, Low-low necessity, direction.

Low-low necessity. A necessity relationship where the absence/low value of the condition is necessary for the absence/low value of the outcome (‘- nc -’). Also see nc, High-high necessity, Low-high necessity, High-low necessity, direction.

Low-purity zone. See lowP-zone.

Low-solidity zone. See lowS-zone.

lowP-zone. The area of the ceiling zone far from the ceiling line. Also see lowS-zone, medP-zone.

lowS-zone. The part of the feasible area far from the ceiling line. Also see lowP-zone, medS-zone.

Measurement. The process in which scores are generated for data analysis. Also see Data, Measurement validity, Measurement reliability.

Measurement reliability. The degree of precision of a score when the measurement is repeated. Also see Measurement, Measurement validity.

Measurement validity. The extent to which procedures of data collection and of scoring can be considered to meaningfully capture the ideas contained in the concept of which the value is measured. Also see Measurement, Measurement reliability.

Mediator. A term used in regression-based analyses indicating the theoretical intermediate role of a concept between two other concepts. Also see Moderator, Confounder, Necessity chain.

Medium-purity zone. See medP-zone.

Medium-solidity zone. See medS-zone.

medP-zone. The area of the ceiling zone close to the ceiling line. Also see lowP-zone, Iso-P line.

medS-zone. The part of the feasible area close to the ceiling line. Also see lowS-zone, Iso-S line.

Membership score. See Set membership data.

Method-triangulation study. A multimethod study that uses a single causal perspective and corresponding different methods with the same or different data. Also see Causal-pluralism study, Separate-studies, Theory-method fit.

Min-max normalization. A linear transformation that maps values from an original range [min,max] to a chosen range (e.g., 0-1, or 0-100). Also see Standardization.

Model fit. The extent to which a model captures the pattern in the data. Also see Complexity, Fit, Ceiling accuracy, Noise, Exceptions, Support, Spread, Sharpness, Purity, Solidity.

Model specification. The process of selecting the functional form of the ceiling line and the bounding box for chosen condition(s) and outcome.

Moderator. A term used in regression-based analyses indicating the theoretical role of a concept as qualifier of the relationship between two other concepts. Also see Mediator, Confounder, Subgroup ceiling line.

Monte Carlo simulation. A computational method that uses repeated random sampling to approximate the behavior of a system or the value of a quantity. Also see Power.

Multimethod study. A hypothesis-testing study that uses different methods for testing one or more hypotheses with the same or a different causal perspective and with the same or different data. Also see Method-triangulation study, Causal-pluralism study, Separate studies, Theory-method fit.

Multiple bivariate analyses. A series of bivariate analyses. Also see $XY$-plot, $XY$-table.

Multiple bottleneck case. A case that is a bottleneck case for multiple conditions.

Multiple NCA. The application of NCA with the same outcome and different conditions, where each condition-outcome pair is analyzed with NCA.

Natural distribution. The distribution describing how a system behaves under its unaltered, observable conditions, without forced manipulation.

nc. An abbreviation of ‘necessary condition’ or ‘necessity cause’ placed near the arrow in a conceptual model and preceded and followed by ‘+’ or ‘-’ to indicate the direction of the necessary condition. Also see high-high necessity, low-high necessity, high-low necessity, low-low necessity.

NCA. Necessary Condition Analysis. A methodological approach developed by Jan Dul that uses a necessity causal logic (methodology) for identifying necessary conditions from data (method). Also see NESS, QCA, SSC.

NCA approach. see NCA.

NCA method. The part of NCA concerned with data analysis and empirical testing. Also see NCA methodology.

NCA methodology. The part of NCA concerned with causal logic and theory. Also see NCA method.

NCA parameter. A quantity to evaluate a necessary condition. Also see Ceiling zone, Effect size, # above, Model fit, Inefficiency.

NCA Ribbon. The zone in the bounding box consisting of the medP-zone and medS-zone. Also see Purity, Solidity.

NCA statistical test. One of NCA’s permutation tests to estimate the $p$-value. Also see Null test, Contrast test, Independent test, Paired test.

NCA-Extended Solution Table. See NEST.

Necessary condition. A cause that must exist in order for the outcome to exist. Also see Sufficient condition.

Necessary Condition Analysis. See NCA.

Necessary condition hypothesis. See Necessity hypothesis, Formal necessity hypothesis.

Necessary condition in degree. See NiD.

Necessary condition in kind. See NiK.

Necessary Element of a Sufficient Set. See NESS.

Necessity cause. See Necessary condition.

Necessity chain. A sequence of linked necessary conditions where each condition is required for the next outcome in the sequence to be possible. Also see Chain necessity effect.

Necessity corner. See Expected empty corner.

Necessity hypothesis. A hypothesis about a necessity relationship. Also see Preliminary necessity hypothesis, Formal necessity hypothesis.

Necessity-in-degree. See NiD.

Necessity-in-kind. See NiK.

Necessity logic. A causal logic describing a necessity relationship. Also see Binary logic, Conditional logic, Sufficiency logic.

Necessity model. The ceiling line and the bounding box representing necessity-in-degree. Also see Model specification.

Necessity perspective. The causal perspective on necessity selected by the analyst. In NCA the Deterministic perspective or the Typicality perspective can be selected. Also see Sufficiency perspective, Probabilistic sufficiency perspective.

Necessity relationship. A causal relationship in which the cause always, probably or typically enables or constrains the outcome. Also see Necessity hypothesis, Formal necessity hypothesis, Deterministic necessity, Typicality necessity.

Necessity theory. A theory consisting of at least one necessity hypothesis, with specified focal unit, condition(s) and outcome(s), and a defined theoretical domain. Also see Preliminary necessity theory, Formal necessity theory.

NESS. Necessary Element of a Sufficient Set. A tool developed by Richard W. Wright that argues that an event is a legal cause of an outcome if and only if the event was a necessary element of a set of conditions that together were sufficient for the outcome to occur. Also see NCA, QCA, SCC.

NEST. NCA-Extended Solution Table. A solution table to with an extra column representing the condition’s minimum required necessity level according to NCA. Also see Fiss chart.

NHST. Null Hypothesis Statistical Test. A procedure for deciding whether sample data provide enough evidence to reject a null hypothesis. Also see Null test, Contrast test, Independent test, Paired test, Alternative hypothesis, Formal necessity hypothesis.

NiD. Necessity-in-degree. A necessary condition that is quantitatively formulated as level $x$ of $X$ is necessary for level $y$ of $Y$. Also see NiK.

NiK. Necessity-in-kind. A necessary condition that is qualitatively formulated as $X$ is necessary for $Y$. Also see NiD.

Noise. A model fit metric expressing the number of cases in the medP-zone of the ceiling zone as a percentage of all cases in the bounding box. Also see, Counterexample, Exception, Outlier.

Non-bottleneck case. A case with satisfied target condition level. Also see Bottleneck case, Necessity-in-degree, Bottleneck table.

Non-intervention case. A case to which an intervention is not applied. Also see Intervention case, Ignored case.

Non-trivial hypothesis. A hypothesis where both the absence of the condition and the presence of the outcome are possible; otherwise it is trivial. Also see Formal necessity hypothesis.

Normal distribution. A continuous probability distribution in which values cluster around a mean in a symmetric, bell-shaped pattern, defined by its mean and standard deviation. Also see Uniform distribution, Truncated normal distribution, Skewed distribution.

Normalization. See Min-max normalization.

Null hypothesis. The statistical concept describing the claim of no (nil) effect or relationship in the population. Also see Null test, Contrast test, Independent test, Paired test, Alternative hypothesis.

Null hypothesis test. See NHST.

Null test. A permutation test in which $Y$ is randomly shuffled across cases while keeping $X$ fixed. Also see Contrast test, Independent test, Paired test.

Observational study. A study design in which variables are observed in the real life context without manipulation by the analyst. Also see Longitudinal study, Case study, Experimental study.

Omitted variable bias. The estimation error that is made in a regression-based analysis when a variable is omitted from the regression model specification.

Opposite corner. A corner in the bounding box diagonally across the corner of interest. Also see Adjacent corner.

Outcome. The varying characteristic $Y$ of a focal unit of which the value (or its change) is the result of, or is permitted by a value (or its change) of another varying characteristic $X$ (which is called the condition). Also see Predicted variable.

Outcome inefficiency. The area of the bounding box where the outcome is not constrained by the condition. Also see Absolute inefficiency, Relative inefficiency, Condition inefficiency.

Outlier. A point (case) in the bounding box that is considered to be ‘far away’ from the other points (cases) and has a large influence on the effect size if removed. Also see Ceiling outlier, Scope outlier, Counterexample, Exception, Noise, LowP-zone, MedP-zone.

Overall ceiling line. A ceiling line that applies to the total group of cases. Also see Subgroup ceiling line.

Overshoot waste. Intervention waste caused by too much intervention effort beyond what is required for reaching the target condition. Also see Undershoot waste.

$\mathbf{p}$-value. The probability of obtaining a result that is greater than or equal to the observed result when the null hypothesis is true. Also see Permutation test, NCA statistical test.

$\mathbf{p}$-value accuracy. The estimated difference between the exact $p$-value and the estimated $p$-value Also see Permutation test, NCA statistical test

$\mathbf{p}$-value threshold. See Statistical significance threshold.

Paired test. A permutation test in which for each case it is randomly decided whether to switch the labels. Also see Null test, Contrast test, Independent test.

Partial Least Squares SEM. See PLS-SEM.

Peer. A point in the $XY$-plane that is used to define a ceiling line. Also see CE-FDH, CE-VRS, CR-FDH, CR-VRS, C-LP.

Permutation test. A statistical test that builds an empirical null distribution by repeatedly randomly permuting the observed $X$-$Y$ data to break any systematic relationship, recomputing NCA’s effect size each time, and then comparing the original effect size to this distribution to obtain the $p$-value. Also see Null test, Contrast test, Independent test, Paired test.

Plausible hypothesis. A hypothesis for which virtually no cases are reasonably expected in the ceiling zone. Also see Formal necessity hypothesis, Trivial hypothesis, Testable hypothesis, Complete hypothesis.

Plausibility. See Plausible hypothesis.

PLS-SEM. Partial Least Squares SEM. A component-based SEM approach that estimates model parameters to maximize explained variance in the predicted variables. Also see CB-SEM.

Population. The set of instances of a focal unit defined by one or a small number of criteria and that is defined in the theoretical domain. Also see Sample.

Potential necessary condition. A condition that is suggested to be necessary based existing knowledge and is a candidate for formulating a necessity hypothesis. Also see Preliminary necessity hypothesis, Formal necessity hypothesis, Expert knowledge, Tacit knowledge, Explicit knowledge.

Potential outlier. A case that has a large influence on the effect size if removed. Also see Outlier.

Power. The probability that a statistical test correctly rejects the null hypothesis when a specific alternative hypothesis is true. Also see $p$-value, Statistical credibility.

Practice. The real-world application of knowledge, like business, policy, or services, focusing on action, outcomes, and impact. Also see Academia, Practitioner.

Practitioner. An investigator, a data analyst, a data scientist, etc. active in practice. Also see Analyst, Scholar.

Predicted variable. Name used for outcome in the context of a regression-based analysis. Also see Predictor (variable).

Predictor (variable). Name used for a condition in the context of a regression analysis. Also see Predicted variable.

Preliminary necessity hypothesis. A “best guess” necessity hypothesis based on existing sources of knowledge about potential necessary conditions and to be developed toward a formal necessity hypothesis.

Preliminary necessity theory. A necessity theory in which the necessity hypotheses are not yet formal necessity hypotheses. Also see Formal necessity theory.

Presence/high value of condition/outcome. A value of a condition or outcome that is close to its maximum. Also see Absence/low value of condition/outcome, Direction (of a necessary condition).

Probabilistic sufficiency perspective. The probabilistic perspective selected by the analyst that a cause probably produces an effect. Also see Necessity perspective Deterministic perspective, Typicality perspective, Sufficiency perspective.

Proposition. A statement about the relationship between concepts of a theory. Also see Hypothesis.

Pure necessity theory. A necessity theory with only necessity relationships. Also see Embedded necessity theory.

Purity. A measure of closeness of a point in the ceiling zone to the ceiling. Also see Solidity, Model fit, NCA ribbon.

Purposive case selection. A selection of cases from the theoretical domain in which instances are selected purposefully (e.g., instances with a certain value of the outcome). Also see Convenience sample, Random sample.

Qualitative Comparative Analysis. See QCA.

QCA. A comparative method developed by Charles Ragin that analyzes combinations of conditions (configurations) that are sufficient for the outcome using set theory. Also see fsQCA, csQCA, NCA, NESS, SCC.

QR. Quantile Regression. A linear (pseudo) ceiling line obtained through a regression method that estimates the relationship between $X$ and a chosen conditional high quantile of $Y$. Also see Ceiling line, **CR-FDH, CR-VRS, C-LP*.

Qualitative data. Scores expressing in words or letters the extent to which a case has a property or characteristic. Also see Quantitative data, Longitudinal data, Set membership data.

Qualitative data analysis. Identifying and evaluating a pattern in qualitative data.

Quantitative data. Scores expressing in numbers the extent to which a case has a property or characteristic. Also see Qualitative data, Longitudinal data, Set membership data.

Quantitative data analysis. Generating and evaluating a pattern in quantitative data. Also see Qualitative data analysis.

R. A programming language and environment for statistical computing and graphics, used to import, manage, analyze, and visualize data.

Random sample. A sample in which instances have the same probability of being selected from the population into the sample. Also see Convenience sample, Purposive case selection.

Rating scale. A method in which a person assigns a value to an object. Also see Informant, Subject.

Relative inefficiency. The total area of the bounding box where the necessary condition does not constrain the outcome and the outcome is not constrained by the necessary condition, expressed as percentage of the scope. Also see Absolute inefficiency, Condition inefficiency, Outcome inefficiency.

Replication. Conducting a test of a hypothesis in another instance, or in another group or population of instances of the focal unit.

Ribbon. See NCA ribbon.

Robustness. The extent to which the conclusions of a study remain essentially unchanged when the analyst makes other plausible choices (e.g., model specification, evaluation criteria). Also see Fragility.

Robustness check. An analysis where the original analysis is re-run using other plausible choices by the analyst to conclude if the results are robust or fragile.

Sample. A set of instances selected from a population of the theoretical domain. Also see Convenience sample, Random sample, Purposive case selection.

Sampling. The process of drawing a sample. Also see Sampling frame.

Sampling frame. A list of all instances of a population. Also see Random sample.

SatP. Saturated purity zone. The feasible area including the ceiling line where Purity = 1.

SatS. Saturated solidity zone. The feasible area including the ceiling line where Solidity = 1.

Saturated purity zone. See SatP.

Saturated solidity zone. See SatS.

Scatter plot. See $XY$-plot

SCC. Sufficient-Component Cause model. A conceptual framework in epidemiology (also called causal pie) developed by Kenneth J. Rothman, that explains how an outcome occurs when a complete set of component causes collectively forms a sufficient cause. Also see NCA, NESS, QCA.

Scholar. A researcher active in academia. Also see Practitioner, Analyst.

Scope. The area of the bounding box. Also see Empirical scope, Theoretical scope.

Scope outlier. An outlier on the bounds of the bounding box.

Score. A value assigned to a condition or outcome based on data or expert knowledge. Also see Quantitative data, Longitudinal data. Set membership data.

SCoRe checklist. The Strengthening (theoretical rigor), Conducting (data & analysis quality), and Reporting (transparency) checklist. A tool for authors, editors and reviewers to evaluate the quality of an NCA study and its reporting.

SEM. Structural Equation Modeling. A model consisting of a measurement model specifying how indicators relate to the underlying constructs (“latent variables”) and a structural model specifying the (average) relationships among the constructs. (latent variables). Also see PLS-SEM, CB-SEM.

Sensitivity. See TPR.

Separate studies. Studies that use multiple causal perspectives and corresponding different methods with different data. Also see Method-triangulation study, Causal-pluralism study, Theory-method fit.

Set membership data. Scores expressing the extent to which a case belongs to a set. Also see Qualitative data, Quantitative data, Longitudinal data

Sharpness. A model fit metric expressing the difference in density of cases in the lowS-zone and cases in the lowP-zone.

Significance. See Statistical significance, Substantive significance.

Simulated data. Data that are fabricated, for example, for a Monte Carlo simulation. Also see Empirical data.

Single bottleneck case. A bottleneck case for only one condition. Also see BIPMA.

Skewed distribution. A distribution with a longer tail on one side (right-skewed or left-skewed). Also see Truncated normal distribution, Uniform distribution, Normal distribution.

Small-n study. A study with one or a small number of cases. The letter n stands for the number of cases. Also see Large-n study.

Solidity. A measure of closeness of a point in the feasible area to the ceiling. Also see Purity, Model fit, NCA ribbon.

Solution table. A table presenting the results of a QCA sufficiency analysis as the configurations (combinations of conditions) that consistently led to the outcome (pass the frequency and consistency thresholds set by the analyst) together with their consistency and coverage information. Also see QCA, Fiss chart.

Specificity. See TNR.

Spread. A model fit metric expressing the evenness of the $X$ positions of the cases in the medS-zone and on the ceiling line.

Spurious relationship. An observed relationship that seems causal but is more plausibly explained by another model.

Standardization. A linear transformation that centers and scales values using the sample mean $\mu$ and standard deviation $\sigma$, producing values in standard deviation units. Also see Normalization.

Statistic. A number computed from a sample that summarizes the evidence against the null hypothesis. Also see Effect size.

Statistical credibility. The trustworthiness of conclusions about necessity identified from data when necessity is absent or present in the population. Also see Empirical credibility, TPR, TNR.

Statistical generalization. The statement that the study results that are obtained in a sample of a population also apply to the population from which the sample is drawn.

Statistical significance. The meaningfulness of the effect size from a statistical perspective. Also see $p$-value, Substantive significance.

Statistical significance threshold. The $p$-value ($\alpha$) selected by the analyst for evaluating the formal necessity hypothesis. Also see Effect size threshold.

Structural Equation Modeling. See SEM.

Study. An academic research activity or a project in practice.

Study design. A category of procedures for selecting or generating one or more instances of a focal unit as well as for analyzing the data that are observed or generated in the selected or generated instance or instances. Also see Observational study, Longitudinal study, Experimental study, Case study.

Subgroup ceiling line. A ceiling line that applies to a subgroup of the total group of cases. Also see Overall ceiling line.

Subject. A person who is the object of measurement and an instance of the focal unit of the theory. Also see Informant.

Substantive significance. The meaningfulness of the effect size from a practical perspective. Also see Statistical significance.

Sufficiency cause. See Sufficient condition.

Sufficiency logic. A causal logic describing a sufficiency relationship. Also see Sufficiency relationship, Necessity logic.

Sufficiency perspective. The causal perspective on sufficiency selected by the analyst. Also see Necessity perspective, Probabilistic perspective sufficiency.

Sufficient condition. A cause that always results in an outcome. Also see Necessary condition.

Sufficient-Component Cause model. See SCC.

Support. A model fit metric expressing the number of cases in the medS-zone of the feasible area including the cases on the ceiling line as a percentage of all cases in the bounding box.

Tacit knowledge. Undocumented knowledge in academia and practice to be used for formulating a formal necessity hypothesis. Also see Explicit knowledge, Expert knowledge.

Target condition. The level of a condition that is necessary for the target outcome. Also see Necessity-in-degree.

Target group. The group of cases for which an intervention goal is set.

Target outcome. A desired or undesired level of the outcome set by the analyst that is to be achieved or to be avoided. Also see Target condition, Necessity-in-degree.

Test. Determining whether a hypothesis is rejected or not rejected (supported) in an instance or in a group or population of instances selected from the theoretical domain.

Test statistic. See Statistic.

Testable hypothesis. A hypothesis with measurable variables (variables that can have scores). Also see Formal necessity hypothesis, Trivial hypothesis, Testable hypothesis, Plausible hypothesis, Complete hypothesis.

Testability. See Testable hypothesis.

Theoretical domain. The universe of instances of a focal unit of a theory or hypothesis where the theory or hypothesis is supposed to hold. Also see Necessity theory, Formal necessity hypothesis.

Theoretical justification. The availability of a formal necessity hypothesis when evaluating a necessity relationship.

Theoretical scope. The bounding box (or its area) defined by specified minimum and maximum values of the condition and the outcome. Also see Empirical scope, Scope.

Theorizing. Development of a formal (necessity) hypothesis.

Theory. A (set of) hypotheses (propositions) regarding the relationships between the varying characteristics (concepts) of a focal unit, and the description why the relations exist (causal explanation) in a theoretical domain. Also see Formal necessity hypothesis.

Theory-grounded hypothesis: A hypothesis that is embedded in a necessity theory. Also see Formal necessity hypothesis.

Theory-in-use. A more or less consistent set of beliefs in practice about the world. Also see Theory, Formal necessity hypothesis.

Theory-method fit. The situation that the causal perspective is aligned with the data analysis method.

Thought experiment. The mental evaluation to check the validity of the necessity theory and its hypothesis. Also see Hypothesis testing, Formal necessity hypothesis.

Tight bounding box. The pair of bounding box and ceiling line when the ceiling zone holds the single corner of interest.

TNR. True Negative Rate (specificity). The ability of an approach to correctly identify that necessity does not exist in the population. Also see TPR, Credibility, Statistical credibility, Empirical credibility, Formal hypothesis. Effect size, $p$-value, Robustness, Model fit.

Total effect. A term used in regression-based analyses indicating the combined effect of the direct effect and the indirect effect of $X$ on $Y$. Also see Direct effect, Indirect effect.

Total intervention effort. The sum of intervention efforts applied to the intervention cases.

TPR. True Positive Rate (sensitivity). The ability of an approach to correctly identify that necessity exists in the population. Also see TNR, Credibility, Statistical credibility, Empirical credibility, Formal hypothesis. Effect size, $p$-value, Robustness, Model fit.

Treated bottleneck cases. Bottleneck cases that received the intervention.

Trivial hypothesis. A hypothesis in which the absence of the condition or the presence of the outcome are not possible. Also see Formal necessity hypothesis, Testable hypothesis, Plausible hypothesis, Complete hypothesis.

Triviality. See Trivial hypothesis.

True Negative Rate. See TNR.

True Positive Rate. See TPR.

Truncated normal distribution. A normal distribution that is restricted to a range. Also see Uniform distribution, Skewed distribution

Typicality necessity. Necessity from the typicality perspective. Also see Deterministic necessity.

Typicality perspective. A lens taken by the analyst that a cause always influences an effect, but that there can be a rare exceptions. Also see Necessity perspective, Sufficiency perspective, Probabilistic sufficiency perspective.

Undershoot waste. Intervention waste caused by insufficient intervention effort such that cases do not move from or into the bottleneck area. Also see Overshoot waste.

Uniform distribution. A distribution in which all values within a specified range are equally likely. Also see Truncated normal distribution, Normal distribution, Skewed distribution.

Unit square. A special case of a bounding box in which the condition and the outcome are both limited to the interval [0,1]. Also see Min-max normalization.

Variable. The operationalized concept (varying characteristic of a focal unit of a hypothesis).

Visual inspection. The procedure by which patterns are discovered or compared by looking at the scores or a graphical representation of the scores.

$\mathbf{XY}$-plot. A graphical representation of the relationship between condition and outcome in a coordinate system with cases shown as points. Also see $XY$-table.

$\mathbf{XY}$-table. A matrix representation of the relationship between condition and outcome with the number of cases shown in the cells. Also see $XY$-plot.

Appendix B Software

This chapter will be made available soon.

Appendix C Publications

This chapter will be made available soon.

Appendix D Affine transformations

This appendix shows that affine transformations preserve the NCA effect size $d$.

The following definitions are used:

Let $\Omega \subseteq \mathbb{R}^2$ be the data in the $XY$-plane.
Let $S$ be the scope defined by:

\[ S = [X_{\min}, X_{\max}] \times [Y_{\min}, Y_{\max}] \]
Let $C \subseteq S$ be the ceiling zone or empty area (above the ceiling line).

The effect size in NCA is defined as:

\[ d = \frac{\text{Area}(C)}{\text{Area}(S)} \]

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a general differentiable and invertible transformation:

\[ f(x, y) = (u, v) = (f_1(x), f_2(y)) \]

The Jacobian determinant $J_f(x, y)$ indicates how an infinitesimal area around point $(x, y)$ is transformed:

\[ J_f(x, y) = \begin{vmatrix} \frac{df_1}{dx} & 0 \\ 0 & \frac{df_2}{dy} \end{vmatrix} = f_1'(x) \cdot f_2'(y) \]

The area of a transformed region $R$ becomes:

\[ \text{Area}(f(R)) = \iint_R |J_f(x, y)| \, dx\,dy \]

So the transformed effect size becomes:

\[ d' = \frac{\iint_{{C}} |J_f(x, y)| \, dx\,dy}{\iint_{{S}} |J_f(x, y)| \, dx\,dy} \]

For the affine case:

If $f_1(x) = ax + b$ and $f_2(y) = cy + e$, then:

\[ f_1'(x) = a, \quad f_2'(y) = c \quad \Rightarrow \quad J_f(x, y) = ac = \text{constant} \]

So: \[ d' = \frac{ac \cdot \text{Area}({C})}{ac \cdot \text{Area}({S})} = d \]

Thus, affine transformations preserves the effect size.

Appendix E Simulations for TPR and TNR

This chapter will be made available soon.

Appendix F Necessity and the traditional experiment

This chapter will be made available soon.

Appendix G Correlation by necessity

This chapter will be made available soon.

Appendix H Demonstration NCA with PLS-SEM

This chapter will be made available soon.

Appendix I Bottleneck distance table

This chapter will be made available soon.

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This chapter will be made available soon.

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This book uses specific terminology, such as condition and outcome, that is explained in the glossary of Appendix A.↩︎
In this book the general term for a user of NCA or other methodological approaches is analyst. The analyst can be an academic scholar or a practitioner. The general term for an academic research activity or a project in practice is study.↩︎
In previous publications I have inconsistently used the terms ‘approach’, ‘methodology’ and ‘method’ in the context of NCA. In this book, I follow the common distinction between methodology, referring to the philosophical and theoretical framework, and method referring to the specific techniques, tools, or procedures used to collect and analyze data. The combination of the two parts is called approach or methodological approach.↩︎
The NCA methodology and method can be used in any type of investigation that aims to describe causal relationships. Although usually quantitative data (numbers) are used in NCA, NCA can also be used with qualitative data (words, symbols, see Chapter 8) as long as scores (values, levels) for the condition and the outcome are available.↩︎
The dissemination of NCA’s core paper (Dul, 2016b) has been welcomed by Bergh et al. (2022): “The significant expansion of an unknown method along with its usability (including an illustration, user-friendly recommendations, and a software tool) makes the article a gold standard for a contribution via transfer” (p. 1840).↩︎
Journals ranked in Clarivate’s Journal Citation Reports.↩︎
Most misconceptions relate to a lack of awareness about the principles of NCA. Example misconceptions are: “NCA should specify the full data generation process (DGP)” (given its purpose, NCA specifies only a part of the DGP), “NCA is just a statistical method” (NCA is a broad methodological approach that combines theory, mathematics (geometry) and uses statistical tools), “NCA’s statistical test is a test of H1” (NCA’s statistical tests are null hypothesis tests, testing H0), “NCA is the necessity analysis of QCA” (the necessity analysis of NCA and that of fuzzy-set QCA are fundamentally different), “Necessary conditions according to NCA are the same as INUS conditions” (INUS conditions are local necessary parts of a sufficient configuration that produces the outcome; NCA considers overall necessary conditions for the outcome), “NCA cannot test sufficiency” (NCA can test the absence of $X$ being necessary for absence of $Y$, which is equivalent to $X$ is sufficient for $Y$), etc. Several misconceptions are addressed in specific publications (Dul et al., 2019; Dul, Vis, et al., 2021; Dul, 2022).↩︎
A positive or negative conclusion about necessity depends on the analyst’s judgement of many elements, summarized in NCA’s SCoRe checklist for conducting and reporting an NCA study (Chapter 10).↩︎
In this book, it is assumed that hypothesis formulation is done before data collection. The reasons are discussed in Chapter 7.↩︎
If $X$, then $Y$ means: if we see $X$, we also see $Y$, and if we add $X$, we get $Y$.↩︎
If not $X$, then not $Y$ means: if we do not see $X$, we also do not see $Y$, and if we remove $X$, we do not see $Y$ anymore.↩︎
Causality itself cannot be observed, see Section 2.7 and Chapter 7.↩︎
See for example QCA’s use of the consistency score < 1 to allow imperfect sufficiency.↩︎
Note the difference between Exception and Noise as discussed in Section 4.5.4.↩︎
Necessity-in-kind (NiK) refers to the qualitative statement that $X$ is necessary for $Y$ without specifying the level of $X$ and $Y$ other than presence/absence or low/high. This contrasts NCA’s necessity-in-degree (NiD) where specific levels of $X$ and $Y$ are specified.↩︎
Formally, PN is defined as: $PN = P(Y_{X=0} = 0 \mid X = 1, Y = 1)$.↩︎
Until recently, I was not aware of the existence of the typicality perspective on causality. What I previously called a ‘probabilistic view’ on necessity causality (Dul, 2020) refers to what I now (Dul, 2024a) call the ‘typicality perspective’ of necessity causality (based on cardinality). The current description ‘probabilistic perspective’ of causality refers to the use of probabilities to describe causality, but this is not part of NCA.↩︎
Also in fuzzy set QCA (fsQCA) the necessary condition in the sufficiency solution is binary, see Section ??.↩︎
Section ?? discusses model fit in the context of NCA.↩︎
An important element of parsimony is that theories should be simple: “… frustra fit per plura, quod potest fieri per pauciora.” (it is pointless to do with more what can be done with fewer) (Ockham, 1323).↩︎
“Causas rerum naturalium non plures admitti debere, quam quae & vera sint & earum phaenomenis explicandis sufficiunt.” (The causes of natural things should not be multiplied beyond what is true and sufficient to explain their appearances) (Newton, 1687).↩︎
“It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” (Einstein, 1934).↩︎
“We do not seek highly probable theories, but explanations; and the best explanations are often simple explanations.” (K. Popper, 1959).↩︎
“Simplicity is not an optional accessory in human thought and action; it is a necessity.” (Simon, 1957).↩︎
See also Section 3.5 for various possible directions of necessity and probabilistic sufficiency relationships and their notation. It is possible that $X$ is necessary for $Y$ with or without being also probabilistically sufficient for $Y$. NCA only considers the necessity role of the concept. Assuming a double role of a concept is commonplace in theories that are analyzed both by NCA and a regression-based model (see Section ??).↩︎
In this framework it is also assumed that the three necessary conditions are jointly sufficient. Referring to the origins of the AMO model, it has recently been analyzed as a pure necessity theory (Hauff et al., 2021; Tuuli & Rhee, 2021; Van Rhee & Dul, 2017) although AMO studies until then (re)formulated the original necessity relationships as probabilistic sufficiency relationships and used regression analyses for testing them (e.g., Siemsen et al., 2008).↩︎
In another example, Eccarius & Chen (2024) added Trust to the TPB model as a necessary antecedent of the drivers of Intention.↩︎
These are examples of ‘causal pluralism’ studies where both perspectives are studied at the same time, see Chapter 11.↩︎
Also, NCA’s null hypothesis tests for estimation of the $p$-value do not make assumptions about distributions (non-parametric tests, see Section ??). However, when statistical simulations are done with NCA such as in Monte Carlo simulations for conducting NCA’s power analysis, the analyst must assume variable distributions including the shape. Any bounded distribution could be selected, for example uniform or truncated normal (Chapters 5 and 6).↩︎
A theoretical scope may apply, for example, when points are obtained from scales with meaningful minimum and maximum values, such as percentage scales, or Likert scales.↩︎
A proof of the acceptability of an affine transformation including linear transformation is presented in Appendix D.↩︎
Although in mathematics the term line is usually reserved for a straight line and curve for a non-straight one, line is used here in the broader, graphical sense to refer to both. Specific shapes of ceiling lines are distinguished with adjectives, such as linear, stepwise linear, or concave piecewise linear.↩︎
For a low-high hypothesis (the absence/low value of $X$ is necessary for the presence/high value of $Y$) the expected empty corner is the upper-right corner (corner 2), etc.↩︎
When a ceiling line has increasing and decreasing parts as for a parabolic ceiling, multiple analyses can be done for each increasing and decreasing part separately. For the increasing part, the empty area is in the upper-left corner, whereas it is in the upper-right corner for the decreasing part. With a parabola-shaped ceiling, an optimum (rather than just low or high) level of $X$ is necessary for a high level of $Y$ (Section 3.7).↩︎
A hypograph is the set of all points that lie on or below the graph of a function.↩︎
For better visibility, the plot can be produced by the NCA software in R as follows: library(NCA); set.seed(123); data <- nca_random (50, 0.3, 1); model <- nca_analysis(data, "X", "Y", ceilings = c("ce_fdh", "cr_fdh", "c_lp", "ce_vrs", "cr_vrs", "QR")); nca_output(model, summaries = FALSE, pdf = TRUE).↩︎
It is also possible as an option in the NCA software that the analyst selects a custom line using a theoretical intercept and the slope.↩︎
Early NCA work proposed quantile regression and stochastic frontier analysis as statistical ceiling estimators (Dul, 2016b; Goertz et al., 2013). Since these approaches lower the ceiling line when more cases appear below the ceiling they are no longer recommended. Stochastic frontier analysis has not been adopted in practice of NCA and is deprecated in the NCA software. Quantile regression remains available, but is rarely used in NCA (for an exception see, Chen, 2026).↩︎
A low fit value (e.g., < 80%) suggests that the selected ceiling line does not closely follow the boundary.↩︎
A ceiling line can be selected theoretically or estimated from the data. A theoretically selected ceiling line means that the parameters of the ceiling equation are selected a priori, for example, based on previous studies (e.g., in replication studies). It may be that the ceiling line is too low when applied to a new dataset. This would result in having points in the ceiling zone. An estimated ceiling line means that the parameters of the ceiling equation are estimated from the current data. For example, when the CR-FDH technique (Section 4.4) is used, normally some points will be in the ceiling zone.↩︎
For the dichotomous situation Ragin (1987) suggested at least 80% of points should be in the feasible area for support of necessity. In a prototype version of NCA, Dul et al. (2010) suggested at least 95% should be in the feasible area. A lower ceiling accuracy score than for example 95% classifies several cases as not being compatible with strict necessity, which requires further investigation of these cases or a reconsideration of the selection of the ceiling line.↩︎
The conic segment arises from calculations with areas.↩︎
For example, not more than 5% of points may be tolerable Noise for a medium-purity zone corresponding to 0.9 iso-purity (in this zone the points have a purity of 0.9 $\leq$ purity < 1).↩︎
Exceptions should be rare (e.g., ~ 0 in the ceiling zone corresponding to purity < 0.9) to justify a typicality causal perspective. No strict rules exist for what number should be considered as “rare”. What matters is that, unlike noise, an exception-case represents a different causal process not captured by necessity, which may be studied separately.↩︎
Various other distance measures such as horizontal, vertical, or perpendicular distance to the ceiling line could be used as a proximity measure, but each has limitations. When using horizontal or vertical distance of point $P$ to the ceiling, the variation in the other direction is ignored, which is problematic when the ceiling line is non-linear. Perpendicular distance could solve this but this distance measure loses its orthogonality under affine transformations (e.g., normalization) of the scope. To address these issues, a proximity measure based on the area between the point’s horizontal and vertical distances and the ceiling line is used. Assuming that the ceiling line is non-decreasing, this area-based measure has several desirable properties: continuity meaning that it varies smoothly with the point’s location, monotonicity meaning that the value increases as the point moves farther from the ceiling line, and invariance meaning that it remains invariant under affine transformations of the scope.↩︎
For example, at least 5% of points could be considered acceptable support with a medium-solidity zone corresponding to 0.8 iso-solidity (in this zone the points have a solidity of 0.8 $\leq$ solidity $\leq$ 1.).The proposed suggestion for iso-solidity is less strict than that for iso-purity. The reason is that purity < 1 violates strict necessity. However, solidity < 1 does not violate necessity. Solidity < 1 implies that the single necessary condition is only less informative for sufficiency. If solidity = 1 all points are on or above the ceiling, which makes the necessary conditions also sufficient. However, assessing the necessity of a single condition is not a goal of NCA, which focuses on single factors being necessary. Low solidity may affect sharpness, see Section 4.5.6.↩︎
For good model fit the sharpness should be for example at least 0.8 such that from all points in the ribbon 80% is on the ‘correct’ side.↩︎
The specification of the bounding box is tacitly assumed for simplification.↩︎
Data are from the Maungawhau volcano in Auckland, New Zealand and were obtained from the package .↩︎
When subsamples are drawn it is possible that the ceiling for the subgroup is lower than the ceiling of the total group, but the ceiling can never be higher. This subgroup ceiling may represent a different necessity theory with a different theoretical domain, see Section ?? and Chapter 7.↩︎
Factors that are not important on average may still be necessary (Table ??).↩︎
Elements of such a reasoning can be found in Tynan et al. (2020). In this study the necessity-in-kind analysis showed that class attendance is necessary for high grades ($d$ = 0.28, $p$ < 0.001), and the subsequent necessity-in-degree analysis found that students needed to attend at least 50% of classes to achieve a grade of 90% or higher.↩︎
In the context of formulating a formal hypothesis for testing, the term ‘hypothesis’ instead of ‘proposition’ (Chapter 3) is used to describe the relationship between the theory’s concepts.↩︎
This example is discussed in Section ??.↩︎
Narrowing the condition $X$ weakens the theory because it applies to fewer cases from reality. Broadening the outcome $Y$ and broadening the theoretical domain adds new counterexamples.↩︎
It is, of course, possible that during empirical testing, counterexamples emerge that were not anticipated during the thought experiment. In the NCA method, counterexamples are empirically defined as uncommon cases in the expected empty space, far from the ceiling, see Section 4.5.4. Such findings result in an empirical rejection of deterministic necessity. After reporting this result, the analyst may then reflect on whether the original deterministic interpretation of necessity was overly strict. However, adopting the typicality perspective after the results are known, and suggesting that this causal perspective was adopted before the results were known, should be avoided (HARKING, see Hollenbeck & Wright, 2017).↩︎
However, the mere presence of the Internet connection does not guarantee that the meeting will be attended. Other conditions must also be met, such as the participant’s willingness to join, availability, and the scheduling of the meeting itself. Thus, the Internet connection enables the possibility of the outcome (a public online meeting), but it does not ensure its occurrence. Ensuring the outcome would mean that corner 4 is also empty. However, the hypothesis is only about necessity, and does not make a statement whether or not the condition is also sufficient. For the hypothesized necessity, corner 4 is not relevant.↩︎
The NCA analysis is done with the C-LP ceiling line (Section 4.4), which is a linear ceiling line that has no observations above it. The effect sizes for the pooled data with the two ceiling techniques are 0.10 (p < 0.001) and 0.14 (p < 0.001), respectively.↩︎
There is also a well-known positive average relationship (imaginary line through the middle of the data), but this average trend is not considered here. Furthermore, the lower-right corner is empty as well. This suggests that a low level of Economic prosperity is necessary for a low level of Life expectancy. However, this low-low necessity relationship was not claimed in the hypothesis that is tested here. The necessity of low $X$ for low $Y$ can be evaluated with the corner = 4 argument in the nca_analysis function of the NCA software (Appendix B)↩︎
The time trend expectation can be included in the causal explanation of the hypothesis, see Section 7.5.3.↩︎
Only the C-LP ceiling technique is used. For making the NCA results comparable per year, the theoretical scope for each year is the scope of the pooled data (based on the absolute minima and maxima of empirically observed Economic prosperity and Life expectancy in the data).↩︎
Convention in this book for corner numbers: 1 = upper-left; 2 = upper-right; 3 = lower-left; 4 - lower-right.↩︎
Note that an odds ratio (OR) is often used to estimate the average effect when $X$ and $Y$ are dichotomous. The odds ratio summarizes how much more likely (in terms of odds) the outcome is to be absent in the treatment group (where $X$ is removed) compared to the control group (where $X$ is retained). The mathematical formula for the odds ratio in $OR = \frac{c3 / c1}{c4 / c2} = \frac{c3 \cdot c2}{c1 \cdot c4}$ where $c1$, $c2$, $c3$, and $c4$ refer to the number of cases in corner 1, corner 2, corner 3, and corner 4, respectively. After the treatment, $c2$ is the number of cases in the control group with outcome success, $c4$ is the number of cases in the control group with outcome failure (spontaneous decay), $c1$ is the number of cases in the treatment group with outcome success (success remains after removing $X$), and $c3$ is the number of cases in the treatment group with outcome failure. With an observed data pattern of an empty space in corner 1, the odds ratio is infinite, regardless of the distribution of the cases in the control group: $(50/0)/(10/40) = \infty$. This situation is called “perfect separation” and is considered a problem for average effect statistical analysis. A common advice is to remove the $X$ from further analysis. However, $X$ is a very important predictor of the outcome: the perfect predictor ensures absence of the outcome when $X$ is absent: a necessary condition.↩︎
For example typical cases that represent a wider population of cases, deviant cases that are unusual, or maximum variation cases to ensure maximum variation of a certain characteristic.↩︎
They identified these five potential necessary conditions not by using existing knowledge (Section 7.2) but by first conducting an inductive case study. The authors selected two cases where the outcome was present (school shooting) and observed common factors. The two cases were different from the cases used for testing the hypothesis. Identifying “important” factors in this way has a long tradition in explorative and theory-building case study designs. For example, Znaniecki (1934) introduced ‘analytic induction’ for finding causal relationships. This method identifies common characteristics in cases where the outcome is present, thus identifies potential necessary conditions (Katz, 2001; W. S. Robinson, 1951).↩︎

Symbol	Meaning
\(\alpha\)	Threshold value for statistical significance
\(a\)	Slope of a linear function
\(b\)	Intercept of a linear function
\(C\)	Size of the ceiling zone; a point on the ceiling line \((x_c, y_c)\)
\(ca\)	Model fit metric ceiling accuracy
\(cp\)	Model fit metric complexity
\(d\)	Effect size
\(df\)	Degrees of freedom of a necessity model
\(ex\)	Model fit metric exceptions
\(f\)	General symbol for a mathematical function
\(F\)	Feasible area
\(ft\)	Model fit metric fit
\(h\)	Horizontal segment of a ceiling line
\(H\)	Hypothesis
\(i\)	Index for an observation or of a segment of a ceiling line
\(j\)	Index for a variable
\(J\)	Number of necessary conditions
\(k\)	Number of outliers in a set of outliers
\(lowP\)	Number of cases in the lowP-zone
\(lowS\)	Number of cases in the lowS-zone
\(max\)	Maximum value of a given range
\(medP\)	Number of cases in the medP-zone
\(medS\)	Number of cases in the medS-zone
\(min\)	Minimum value of a given range
\(n\)	Sample size
\(N\)	Necessary condition (in NEST)
\(nc\)	Necessary condition; necessary cause (near arrow in causal graph)
\(ns\)	Model fit metric noise
\(p\)	\(p\)-value for a statistical significance test
\(P\)	Number of peers that define the ceiling line
\(S\)	Scope: the size of the bounding box
\(sd\)	Standard deviation
\(sh\)	Model fit metric sharpness
\(sp\)	Model fit metric spread
\(su\)	Model fit metric support
\(T\)	Time indicator
\(v\)	Vertical segment of a ceiling line
\(X, Y, Z\)	Variable names (capitals)
\(x, y, z\)	Variable values (lower case)