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Re: st: Concession

From   Richard Williams <[email protected]>
To   [email protected]
Subject   Re: st: Concession
Date   Thu, 30 Oct 2003 18:44:24 -0500

Dave makes very good points. I was objecting to the use of the term "biased" but had overlooked the fact that the term "inflated" probably wasn't on the mark either. However I will point out that the following is not always true:

>Logic dictates that the explanatory power of a model
> will decrease as the correlation among predictors increases because
> there's less independent information being added to the system.

>Compare the R^2 from a three
> variable equation in which Y is correlated with X1 at .2, with X2 at .2,
> and X1
> and X2 are correlated at .9, to the same equation with the correlation
> between
> X1 and X2 now at .2.  The R^2 for the latter equation (~.067) is much
> larger
> than the R^2 for the former (~.042).
A counter-example is when suppressor effects are present, e.g. X1 is negatively correlated with Y, X2 is positively correlated with Y, and X1 and X2 are positively correlated with each other. So, for example, R12 = .1, RY1 = -.4, RY2 = .4. Then R^2 = .356. Now, increase R12 to .4, and R^2 = .533.

So, terms like R^2 being "biased" and "inflated" by multicollinearity should be avoided, but it is safe to say that R^2 is affected by many things (including the correlations of the Xs with each other) and you should be very careful how you use it. While quibbling over some details, we all seem to be in agreement on that! Thanks for an interesting discussion.

Richard Williams, Associate Professor
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