However, although M1 is indeed unrelated to the for-M1-corrected residual \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{T}}_{1}^{\prime } $$\end{document}, this residual \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{T}}_{1}^{\prime } $$\end{document} is not necessarily uncorrelated to the moderator M2 of the co-twin. In this paper, we first show that non-zero semi-partial correlations between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{T}}_{1}^{\prime } $$\end{document} and M2 can result in a considerable increase in false positive moderation effects on variance components A and C, especially if the correlation between T and M runs fully (or predominantly) via E (rather than via A and/or C). We subsequently study whether a simple extension of the univariate moderation model prevents this increase of false positive rate. In the first part of this paper, we focus on illustrations and simulations in which the correlation between trait T and moderator M runs either exclusively via A, or via C or via E. Although these settings may be considered quite special, they conveniently simplify the explanation of the
