Previously I've read that adjusted-$R^2$ is not a measure of fit. Recently, though, I wanted to substantiate that piece of knowledge by understanding the reason why but I couldn't find any substantive sources to back this up.

The Wikipedia article on it states "while $R^2$ is a measure of fit, adjusted $R^2$ is instead a comparative measure of suitability of alternative nested sets of explanators" but does not provide a citation.



Gary King provides a detailed, accessible explanation of arguments related to (mis)interpretations of $R^2$ here (starting on p. 675), including citations for arguments against using $R^2$ as a measure of goodness-of-fit. Essentially, there is nothing all too interesting about how spread out your points are from the regression line relative to the mean.

King writes that $R^2$ can be a guide in the case of comparing two equations with the same dependent variable but different independent variables. Adjusted $R^2$ imposes a penalty for including additional predictors. There are a number of alternative measures aimed at measuring goodness-of-fit, among them AIC and BIC, as @whuber points out.

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  • $\begingroup$ Thank you for answering me twice today, I really appreciate it. If I understand correctly, $R^2$ (adjusted or not) is not an absolute measure of goodness of fit, and can only be used for comparisons of equations with the same dependent variable, but different independent variables, on the same dataset. $\endgroup$ – Twilight Sparkle Sep 11 '13 at 20:52
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    $\begingroup$ That is correct. $R^2$ is a relative measure that bears only on the particular data set at hand; recall that is is, essentially, a measure of "gain" in terms of how much of the variance the model explains, measured against simply using the mean. Different samples have different means, so you'd be comparing apples to oranges. $\endgroup$ – Sycorax says Reinstate Monica Sep 12 '13 at 0:27

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