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From   Richard Williams <[email protected]>
To   [email protected]
Date   Wed, 29 Oct 2003 08:31:21 -0500

At 05:09 AM 10/29/2003 +0000, Clive Nicholas wrote:
unlikely to vote Labour and vice versa. Because this overlap is carried
forward to the computation of R^2, R^2 has been upwardly biased.
Thanks, but I'm afraid I still don't follow. If the beta coefficients were all zero, R^2 would be zero. Further, while the intercorrelations of the Xs may affect how large R^2 is, I don't see how that causes R^2 to be "upwardly biased", i.e. just because something causes R^2 to be bigger doesn't mean that it becomes biased towards a larger value. I'm aware of various consequences of multicollinearity, e.g. large standard errors, large confidence intervals, increased likelihood of saying a coefficient does not differ from zero when it really does. But, I don't remember ever hearing "upwardly biased R^2" as a problem. But that doesn't mean I couldn't have missed it! But multicollinearity does not cause regression coefficients to be biased (wildly variable from one sample to the next, maybe, but not biased) so I am not sure why it would cause R^2 to be biased.

What I might say instead is, suppose you have two populations. In both populations, the effects of the Xs on Y are identical. But, in one population, the Xs are much more highly correlated with each other than they are in the other population. This will likely cause the R^2 to differ between the 2 populations. If you just compared R^2 between the two populations and not the actual coefficients, you could get a very misleading idea of the differences between the two populations. These kinds of ideas are discussed in my "Evils of R^2" handout at

Richard Williams, Associate Professor
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