The proportion of variance in variable Y explained
by variable X is defined as the ratio of the explained variation in Y to
the total variation. The section on partitioning the
variance in prediction shows that:
r² = SSY'/SSY
where r
is the correlation between the two variables, SSY' is the sum of squares
explained, and SSY is the total sum of squares Y. Therefore, r²
is a measure of the proportion of variance explained.
In
multiple regression, the best measure of the proportion of variance
explained by the predictor variables is the adjusted R² (corrected
for shrinkage).