Coefficient of Determination

Abstract

As with simple regression, in the multiple linear regression model, we can interpret ! SYY " RSS SYY as the fraction of the variability in Y explained by including the terms u 1 , u 2 , … , u k-1 in the mean function (as compared to the constant mean function). In the multiple regression context, ! SYY " RSS SYY is denoted as R 2 (with capital R). R 2 is called the coefficient of (multiple) determination or (misleadingly) the squared multiple correlation. • R alone (unsquared) has no meaning in multiple regression. • By convention, we use small r for the sample correlation in simple regression. • In multiple regression, we can talk about correlation between two variables (i.e,, just two at once). • In particular, in multiple regression, r ij is often used to denote the sample correlation coefficient between terms u i and u j. • R 2 is sometimes used for comparing models. But caution is needed: o It only makes sense to use for comparing models that are in the same units (e.g., submodels of the same full model). o A submodel of a model will always have a smaller R 2 than the larger model. o As discussed above and below, many other considerations should be taken to account in selecting a model.