Re: st: Re: adjusted r square

[Richard Williams <Richard.A.Williams.5@ND.edu>]

At 10:34 AM 2/20/2007, Ulrich Kohler wrote:
> However, as an aside: I do not find the arguments for the adjusted R2 very
> convincing. It is sometimes said that you have to be punished for including
> additional variables in a model. But why? Because the R2 increases? Why do I
> need to be punished for this? It is just a simple fact that I can explain
> more variance with an additional variable. Punishment and especially the
I don't think "punishment" is the original rationale for adjusted 
R^2, although that is often cited as one of its benefits.  Rather, 
R^2 is biased upwards, especially in small samples.  Adjusted R^2 
corrects for that.

McClendon discusses this in "Multiple Regression and Causal 
Analysis", 1994, pp. 81-82.

Basically he says that sampling error will always cause R^2 to be 
greater than zero, i.e. even if no variable has an effect R^2 will be 
positive in a sample.  When there are no effects, across multiple 
samples you will see estimated coefficients sometimes positive, 
sometimes negative, but either way you are going to get a non-zero 
positive R^2. Further, when there are many Xs for a given sample 
size, there is more opportunity for R^2 to increase by chance.

So, adjusted R^2 wasn't primarily designed to "punish" you for 
mindlessly including extraneous variables (although it has that 
effect), it was just meant to correct for the inherent upward bias in 
regular R^2.


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Richard Williams, Notre Dame Dept of Sociology
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