# 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 --

## 2 thoughts on “Different corr(s) of different IV scopes with same regression coef”

1. For your question at the beginning, Cohen has gave a clear answer “Unlike regression coefficients, correlations are subject to vary with selection as researchers change populations” in the paper.
Another simple way to understand it: $$r=\sqrt{b_{y.x}b_{x.y}}$$, from the formula it’s clear that correlations cann’t be determined only by $$b_{y.x}$$ ,another expression of $$\beta$$. As regression coefficient,$$b_{y.x}$$ just represents the effect From X to Y, correlations emphasize a mutual effect , however. So correlations equal to the geometric means of regression coefficients $$b_{y.x}$$ and $$b_{x.y}$$, which include the effect from X to Y as well as the effect from Y to X

2. My emphasis is that Ecological Fallacy or hierarchical structure is not the essential reason for the simplest dependence of corr on scopes. In the missing data class, the corr dependence topic was also discussed briefly. In that case, some simple missing patterns were just changes in IV scope in a probability way.

The mutual formula of r is interesting. It means changes in X scope affect $$b_{x.y}$$, given the fact it has nothing to do with $$b_{y.x}$$ without Ecological Fallacy.