A note on false positives and power in G × E modelling of twin data.

a Full bivariate moderation model for a twin pair as proposed by Purcell (2002). Ac, Cc, and Ec are the variance components common to the moderator M and the trait T; Au, Cu, and Eu are the variance components unique to T. All latent variables have unit variance. Path loadings for M are denoted by am, cm, and em. The loadings of the cross-paths connecting M to T consist of parts that are unrelated to moderator M, i.e., ac, cc, and ec, and parts that depend on M via weights β ac, β cc, and β ec. Similarly, the loadings of the paths unique to T consist of parts that are unrelated to M, i.e., au, cu, and eu, and parts that depend on M via weights β a, β c, and β e. b Univariate moderation model for a twin pair as proposed by Purcell (2002). All latent variables have unit variance. Path loadings for T consist of parts that are unrelated to moderator M, i.e., a, c, and e, and parts that depend on M via weights β a, β c, and β e. M is also included in the means model of T, where β 0 denotes the intercept and β 1 the regression weight of T on M

a–c Bivariate models in which the correlation of ~.24 between moderator M and trait T either exclusively runs via A (a), exclusively via C (b), or exclusively via E (c). The variance of both trait T and moderator M are for 40% due to A, for 30% to C, and for 30% due to E. All latent variables have unit variance

PP-plots for the univariate moderation model in Simulation study 1, in which T and M are correlated exclusively via A (upper part), exclusively via C (middle part), or exclusively via E (lower part). Deviations from the 45° line show whether the use of the regular χ2(1) test would result in conservative (above the line) or liberal (below the line) decision. % hits denotes the percentage of likelihood-ratio tests smaller than the critical value 3.84 (i.e., significant given α = .05). A hit rate of .05 is expected given α = .05, and given that moderation effects were absent in the data. Hit rates outside the .04–.06 range should be considered incorrect (i.e., significantly too low or too high)

PP-plots for the extended univariate moderation model in Simulation study 2, in which T and M are correlated exclusively via A (upper part), exclusively via C (middle part), or exclusively via E (lower part). Deviations from the 45° line show whether the use of the regular χ2(1) test would result in conservative (above the line) or liberal (below the line) decision. % hits denotes the percentage of likelihood-ratio tests smaller than the critical value 3.84 (i.e., significant given α = .05). A hit rate of .05 is expected given α = .05, and given that moderation effects were absent in the data. Hit rates outside the .04–.06 range should be considered incorrect (i.e., significantly too low or too high)