# Re: st: Re: adjusted r square

 From Richard Williams To statalist@hsphsun2.harvard.edu Subject Re: st: Re: adjusted r square Date Tue, 20 Feb 2007 19:29:26 -0500

```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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