Clearly, as a result of partialling out M1 from T1, the semi-partial correlation 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 is lower than the correlation between T1 and M2. However, the semi-partial correlation 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 is often not equal to zero: especially if the correlation between T1 and M2 was zero to begin with (i.e., if T and M are correlated via E), the semi-partial correlation 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 is quite large and negative. Estimated across an entire study sample (while weighing for the MZ/DZ ratio), these non-zero semi-partial correlations can be quite considerable (e.g., in the case that T and M are correlated via E), and are likely to cause problems in the univariate moderation model. After all, these non-zero semi-partial correlations, whether positive or negative, will somehow need to be accommodated in the model. Considering the univariate moderation model as depicted in
