R2 and model fit. Because the coefficient of determination, R2, always increases when a new independent variable is added to a model, it is tempting to include many variables in the model in order to force R2 to be near 1. However, doing so reduces the number of degrees of freedom available for estimating s2, which adversely affects our ability to make reliable inferences. Suppose you want to use 20 psychological and sociological factors to predict a student’s standardized test score. You fit the model y = b0 + b1 x1 + g+ b20 x20 + e where y = test score and x1, x2,c, x20 are the psychological and sociological factors. Only 22 years of data 1n = 222 are used to fit the model, and you obtain R2

= .95. Test to see whether this impressive-looking R2 is large enough for you to infer that the model is useful—that is, that at least one term in the model is important in predicting test scores a = .01.