Appendix B: Derivation of regression analysis predictions
Using the definitions of parameter weights specified in the text and following Eq. 1, the probability of choosing X over Y can be described as a function of the difference between evk (X) and evk (Y),
where mXY,j are match variables indicating whether X and Y match on dimension j. For example, mXY,C = 2 if the cause feature is present in X but absent in Y, −2 if it is present in Y but absent in X, and 0 if X and Y both have the cause feature or both do not have it. Table 2 presents how diffk (X, Y) simplifies for each of the seven test items. For example, because evk (10xx) = wC − wE − wH, and evk (01xx) = −wC + wE − wH, then diffk (10xx, 01xx) = 2wC − 2wE for test pair TC.
The probability of choosing X over Y can now be obtained by passing diffk (X, Y) through a logistic function,
Equation B2 predicts a choice probability in favor of X of close to 1 when diffk (X, Y) ≫ 0, close to 0 when diffk (X, Y) << 0, and close to .5 when diffk (X, Y) ≅ 0.
Typicality effect only. Table 2 presents a set of weights that exemplifies the parameter constraints for an effect of typicality with no causal status, coherence, or relational centrality, namely, wC = 1, wE = 1, wN = 1, and wH = 0. For example, for test item TA, evk (11xx) = 2 and evk (00xx) = −2, which results in diffk = 4. Applying Equation B2, this value of diffk yields a choice probability choicek = .98; that is, X should be strongly preferred over Y. In addition, there should be no preference between objects with the same number of typical features. For example, for test item TC, evk (10xx) = evk (01xx) = 0, diffk = 0, and thus choicek = .50. Note that these predictions and those presented below hold not just for the example parameter values shown in Table 2 but for any parameters that satisfy the constraints.
Typicality + causal status effect only. Table 2 presents the values of diffk for a set of w weights that exemplifies the presence of typicality and causal status effects but the absence of coherence and relational centrality, namely, wC = 2, wE = 1, wN = 1, and wH = 0. For test item TC, these parameters yield evk (10xx) = 1 and evk (01xx) = −1, and thus diffk = 2 and choicek = .88; that is, X should be chosen over Y. The predictions for the remaining test items are generated in a similar manner. For TD and TF, diffk = 1 and choicek = .73, so X should be chosen over Y for both items. Conversely, for test items TE and TG, the Y alternatives should be preferred because the Xs are missing the cause feature, resulting in diffk = −1 and choicek = .27.
Typicality + coherence effect only. Table 2 shows the w weights that reflect a presence of typicality and coherence effects but the absence of causal status and relational centrality, namely wC = 1, wE = 1, wN = 1, and wH = 1. For test item TC, both X (10xx) and Y (01xx) a choice probability of .50 is indicated by evk(10xx) = evk(01xx) = −1 and diffk = 0. In test item TD, the choice of Y over X is indicated by the values evk(10xx) = −1, evk(xx10) = 0, and diffk = −1. Y should be chosen over X in test item TE for the same reason.
Typicality + relational centrality effect only. Table 2 shows example parameter values for an effect of typicality and relational centrality in the absence of causal status and coherence effects; namely wC = 2, wE = 2, wN = 1, and wH = 0. For TD and TE, a choice probability of 0.5 is indicated by evk (10xx) = evk (01xx) = −1 and diffk = 0. For TF, evk (11xx) = 4 and evk (xx11) = 2, and thus diffk = 2 and choicek = .88; that is, X should be chosen over Y. For TG, evk (00xx) = −4 and evk (xx00) = 2, and thus diffk = −2 and choicek = .12; that is, Y should be chosen over X.