## Different corr(s) of different IV scopes with same regression coef

$Y=\alpha+\beta X+\varepsilon,\;\varepsilon\sim N\left(0,\sigma^{2}\right)$

With $\alpha,\beta, \sigma$ known in the linear relationship, can the correlation in the scatter plot of Y against X be estimated from the linear formula?

You may recall in Hierarchical Linear Model class, the scopes of the W dramatically impact the regression coefficients of F~W in the following R demo (hlm.jpg). While this time the regression coefficient has been fixed to a known $\beta$. So the scopes of X would never impact the regression coefficient. However, it proved that the correlation r could range from zero to unit (or -1) according to the variance of X in the final close form $r=\frac{\beta\mbox{Var}\left(X\right)}{\mbox{Std}\left(Y\right)\mbox{Std}\left(X\right)}=\beta\frac{\mbox{Std}\left(X\right)}{\sqrt{\beta^{2}\mbox{Var}\left(X\right)+\sigma^{2}}}$.

Let me quote as the final words from Cohen (1994; p.1001; Where the role of IV is replaced by that of DV within typical contexts like ANOVA) --

... standardized effect size measures, such as d and f, developed in power analysis (Cohen, 1988) are, like correlations, also dependent on population variability of the dependent variable and are properly used only when that fact is kept in mind.

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Cohen, J. (1994). The earth is round (p<.05). American Psychologist, 49, 997-1003.

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Compare to the following case: different corr(s) of different IV scopes with hierarchical regression coefficients --

## “Effect Size” — same data, different interpretations

Just a short R-script note to embody the 3-page-paper of Rosenthal & Rubin (1982).

Table 1. (p. 167) listed a simple set-up. There was a between-subject treatment. Control group includes 34 alive cases and 66 dead cases. Treatment group includes 66 alive cases and 34 dead cases. The question is what is the percentage of the variance explained by the nominal IV indicating the group?

The authors pointed out that one may interpret the data result as death rate was reduced by 32% while the other may interpret the same as 10.24% variance was explained. Let's demo it more dramatically to imagine just 4% explained variance would reduce death rate by 20%.

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Rosenthal, R. & Rubin, D. B. (1982). A simple, general purpose display of magnitude of experimental effect. Journal of Educational Psychology, 74, 166-169.