To investigate whether the extended univariate moderation model results in the correct false positive rate of 5%, we re-analyzed the data-sets created under Simulation study 1 using the extended univariate moderator model. Again we tested the significance of moderator parameters β a, β c and β e. As these parameters were simulated to be zero, we expected the χdiff2 to be χ(1) distributed across the 2000 datasets, independent of whether T and M were correlated via A, via C, or via E. In addition, these same data sets were analyzed using the full bivariate moderation model (as depicted in Fig. 1a) in which all 17 parameters were estimated freely (Note that we did not use the full bivariate moderation model to analyze the data with rM1,M2 = 1 or rM1,M2 = 0 because in practice one would never choose a bivariate parameterization under these circumstances).
