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  7.7            R² IN FIXED- REGRESSION

In (7.39), we have . Thus the corrected total sum of squares  can be partitioned as

                                                                                                  (7.53)

Where   is the regression sum of squares. From (7.37), we obtain , and multiplying this by   gives . Then   can be written as

                                       .                           (7.54)

In this form, it is clear that SSR is due to .

The proportion of the total sum of squares due to regression is               

                                                                                                                                (7.55)

which is known as the coefficient of determination or the squared multiple correlation. The ratio in (7.55) is a measure of model fit and provides an indication of how well the ’s predict .

The partitioning in (7.55) can be rewritten as the identity

                                                      = SSR + SSE,

which leads to an alternative expression for R²:

           

                                                     

                                                                                                                                (7.56)

The positive square root R obtained from (7.55) or (7.56) is called the multiple correlation coefficient. If the  variable were random, R would estimate a population multiple correlation (see Section (10.4)).

We list some properties of R² and R:

1.      The range of R² is 0 ≤ R²≤ 1. If all the ’s were zero, except , R² would be 0. (This event has probability 0 for continuous data.) If all the  values fell on the fitted surface, that is, if  then R² would be 1.