## DV predicted by two IVs, vs. triangular pyramid

-- Diagram from Wiki

It is easier to imagine relation in three spatial vectors by their angles, than by their correlations. For standardized $DV$ $Y=\left(y_{1},y_{2},\dots,y_{N}\right)^{\tau}$ and $IV$s $X_{1}=\left(x_{1,1},x_{2,1},\dots,x_{N,1}\right)^{\tau}$, $X_{2}=\left(x_{1,2},x_{2,2},\dots,x_{N,2}\right)^{\tau}$, cosines of three angles of the triangular pyramid determinate the correlation matrix, thus, all statistics of the regressions $Y=\beta_{1}X_{1}+\beta_{2}X_{2}+\varepsilon$ and $Y=\beta_{1}X_{1}+\varepsilon$ . Unexpected but imaginative results on the impact of introducing $X_{2}$ are --

1. Both $IV$s are nearly independent of $DV$. Togethor they predict $DV$ almost perfectly ($\angle YX_{1}=\angle YX_{2}=89^{\circ}$ and $\angle X_{1}X_{2}=177.9^{\circ}$).

2. Both $IV$s are almost perfectly correlated with $DV$. Togethor, one of the regressive coefficient is significantly negative ($1^{\circ}$, $0.6^{\circ}$ and $0.5^{\circ}$ respectively).

3. Redundancy (Cohen, Cohen, West, & Aiken, 2003) increases to full and then decreases to zero and even negative ($\angle YX_{1}=60^{\circ}$, $\angle YX_{2}=45{}^{\circ}$ and $\angle X_{1}X_{2}$ closes from $90^{\circ}$ to $45^{\circ}$ then to $15^{\circ}+\epsilon$ ).

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Cohen, J., Cohen, P., West, S. G., & Aiken, L. S.  (2003). Applied multiple regression/correlation analysis for the behavioral sciences(3rd ed.) Mahwah, NJ: Lawrence Erlbaum Associates.