The results of these simulations are summarized in Table 4. When moderation on the cross paths is absent (Baseline: β ac = β cc = β ec = 0), the false positive rate of the extended univariate model is correct, while the false positive rate of the full bivariate model is deflated for β au and β cu. The false positive rate of the full bivariate model, however, remains largely unchanged when moderation on the cross paths is introduced. In contrast, the false positive rate of the extended univariate model becomes considerably inflated, especially for β a and β c. Clearly, when the covariance between M and T is not stable across levels of M, the sample-wise regression coefficients β 0,MZ, β 1,MZ, β 2,MZ, β 0,DZ, β 1,DZ, and β 2,DZ in Eqs. 8 and 9 do not sufficiently accommodate the varying association between M and T. As a result, the remaining moderation, which is actually located on the cross paths, is now picked up in the extended univariate moderation model as if it were located on the unique
