[r] What is the difference between Multiple R-squared and Adjusted R-squared in a single-variate least squares regression?

Note that, in addition to number of predictive variables, the Adjusted R-squared formula above also adjusts for sample size. A small sample will give a deceptively large R-squared.

Ping Yin & Xitao Fan, J. of Experimental Education 69(2): 203-224, "Estimating R-squared shrinkage in multiple regression", compares different methods for adjusting r-squared and concludes that the commonly-used ones quoted above are not good. They recommend the Olkin & Pratt formula.

However, I've seen some indication that population size has a much larger effect than any of these formulas indicate. I am not convinced that any of these formulas are good enough to allow you to compare regressions done with very different sample sizes (e.g., 2,000 vs. 200,000 samples; the standard formulas would make almost no sample-size-based adjustment). I would do some cross-validation to check the r-squared on each sample.

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