In this section we aim to investigate whether we can solve the problems that can result from non-zero semi-partial correlations by extending the univariate moderation model. An obvious solution is to extend the means model such that the trait value of twin 1 is not only corrected for the moderator value of twin 1, but also for any residual association to the moderator value of the co-twin, as this would result in a residual \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_{1}^{\prime \prime } $$\end{document} (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_{2}^{\prime \prime } $$\end{document}) that is uncorrelated to both M1 and M2. Taking into account the way regression coefficients in a multiple regression model with two predictors are calculated, it is easy to show that the parameters in the means models should generally also differ across zygosity.
