It is important to realize that the bivariate moderation model considers the joint distribution of M and T, while the univariate moderation model considers moderation of the variance decomposition of T conditional on M. With M included in the means model of T, the univariate moderation model does not allow further investigation of the nature of the covariance between M and T but specifically focuses on the question whether the decomposition of the variance unique to T depends on M. Entering M in the means model of T to allow for a main effect is believed to effectively remove from the covariance model any (genetic) effects that are shared between trait and moderator (Purcell 2002, p. 563). In essence, the variance common to M and T is partialled out, and the moderator effects of M are modeled on the residual variance of T, T′, i.e., the variance of T that was not shared with M. As a result, the effects that M has on the variance decomposition of the residual T′ are believed to be independent of (i.e., not due to) any (unmodeled) (genetic) correlation between M and T (Purcell 2002, p. 563).
