Multiple regression correlation coefficient (R2)

is a measure of the proportion of variability explained by, or due to the regression (linear relationship) in a sample of data (equal to the percentage of variance accounted for). It is a measure of the effect of X in reducing the uncertainty in predicting Y. The statistic is given by / s.s. (total) [i.e., ESS/TSS] where s.s. is the sum of squares. It is also called the coefficient of determination. When all observations fall on the fitted regression line, then s.s. (residual) = 0; and ESS=TSS; thus, . When the fitted regression line is horizontal , then s.s. (regression) = 0 and . The adjusted has a
different formula which takes into account the number of explanatory variables. The fit of a model should never be judged from the value (for example, a high value may arise when the relationship between the two variables is non-linear). The square root of the is the coefficient of correlation (r) and has the same sign (minus or plus) as the slope. The r value does not have a clear-cut interpretation as the in linear regression. The
and r are simply descriptive measures of the degree of linear association between X and Y in the sample observations.