I Bottleneck distance table

In an NCA-based intervention, the focus is on removing cases from, or moving cases into, the bottleneck area. For an effective and efficient intervention, effort should correspond to the distance of the case from the threshold level \(x_c\) of the condition that is needed for the target outcome \(y_c\). The bottleneck distance is \(x - x_c\) such that a non-negative distance indicates that the case is in the non-bottleneck area (\(x \ge x_c\)) and a negative distance indicates that the case is in the bottleneck area (\(x < x_c\)). When the outcome is desired the goal of the intervention is to remove a case from the bottleneck area by increasing its \(x\)-value. For an undesired outcome, the goal is to move the case from the non-bottleneck area to the bottleneck area by decreasing its \(x\)-value.

Table I.1 shows the bottleneck distances for the illustrative example discussed in Section 12.7.2. Here the outcome is desired, so the table only shows the cases that are in the bottleneck area (the black dots in Figure 12.6). A case can be a bottleneck case for one or more conditions. The 38 cases have 64 bottlenecks (i.e., the number of times cases fall in a condition’s bottleneck area).

To obtain an ideal intervention, all bottlenecks must be resolved. This means that they must receive just enough individualized intervention effort. That is, the case’s \(x\)-value must increase by an amount corresponding to the bottleneck distance, such that the cases can leave the bottleneck area and are no longer constrained by this condition to achieve the target outcome. With such an intervention, the case becomes an effective case after intervention. If all bottleneck cases are treated that way for all conditions, the effectiveness and efficiency of the intervention are both 100%.

The table is organized by the total bottleneck distance from low to high. The case’s total bottleneck distance is the sum of the case’s bottleneck distances. This assumes that the levels of the conditions and the intervention efforts are comparable, which may not always be correct. For example, it is possible that an intervention effort on one condition is more difficult or more costly than the same intervention effort on another condition.

Table I.1: Bottleneck distances
Case Perceived usefulness Perceived ease of use Compatibility Emotional value Number of bottlenecks Total bottleneck distance
10 0 1 0
42 −0.02 1 −0.02
115 −0.02 1 −0.02
109 −0.03 1 −0.03
62 −0.18 1 −0.18
8 −0.2 −0.02 2 −0.22
111 −0.2 −0.02 2 −0.22
2 −0.26 1 −0.26
68 −0.28 1 −0.28
119 −0.28 1 −0.28
143 −0.28 0 2 −0.28
169 −0.28 1 −0.28
125 −0.3 1 −0.3
61 −0.3 −0.03 2 −0.33
151 −0.34 1 −0.34
173 −0.02 −0.33 2 −0.35
163 −0.45 1 −0.45
138 −0.57 1 −0.57
31 −0.65 −0.02 2 −0.66
141 −0.67 1 −0.67
50 −0.68 1 −0.68
78 −0.92 1 −0.92
96 −0.95 −0.03 2 −0.98
90 −1.02 1 −1.02
52 −1.3 0 2 −1.3
122 −0.35 −0.99 2 −1.33
146 −0.02 −0.35 −0.99 3 −1.35
103 −1.2 −0.34 2 −1.54
158 −0.57 0 −0.99 3 −1.56
168 −1.35 −0.33 2 −1.67
88 −1.99 1 −1.99
14 −0.65 −1.35 2 −2
121 −0.37 −1.99 2 −2.36
95 −1.1 −1.35 2 −2.44
9 −1.55 −1.35 2 −2.9
47 −0.02 −1.35 −1.99 3 −3.35
29 −1.92 −1.35 −1.02 3 −4.29
19 −1.92 −0.68 −1.35 −1.99 4 −5.94