In this paper, we showed that the univariate moderation model proposed by Purcell (2002) produces (highly) inflated false positive rates if the moderator M is correlated between twins, and M and T are correlated as well. We investigated an extension of this model as a solution to this problem, and conclude that the extended univariate moderation model works well, but only if moderation on the covariance between M and T is absent. Moderation of the covariance between M and T is, however, not accommodated adequately in the extended univariate moderation model, and as a result, moderation of the covariance is picked up as moderation on the variance components unique to T. In the absence of moderation of the covariance between M and T, the extended univariate moderation model is actually more powerful than the full bivariate moderation model, but in the presence of moderation of the covariance between M and T, the extended univariate moderation model detects moderation of the variance components unique to T, as such misspecifying the actual location of the moderation.
