I am trying to do a manual calculation of the proportion of variance explained by one variable, relative to the total explained variance. However, this variable is correlated with another variable that also explains a portion of the total variance. How can I account for this covariance?
For example:
- Total variance explained = 95%, i.e. 5% unexplained
- Variable 1 variance: 0.35
- Variable 2 variance: 0.01
- Covariance var1var2: -0.026
Normally I would compute the variance explained by variable 1 as: 0.35/(0.35+0.05) * 100 = 87.5%. However, I would guess I need to account for the covariance. Is that correct? And how would I do that?
Best Answer
What you want is called the coefficient of partial determination. The coefficient of partial determination (amount of variation explained in Y) between, for example, $Y$ and $X_2$, when $X_1$ is also in the model is
$r^2_{Y2.1} = \frac{SSR(X_2|X_1)}{SSE(X_2)}.$
Likewise, for the general case of more predictors, the coefficient of partial determination between, for example, $Y$ and $X_2$, when $X_1$ and $X_3$ are in the model is
$r^2_{Y2.13} = \frac{SSR(X_2|X_1, X_3)}{SSE(X_1,X_3)}.$
Don't expect to find a lot of information on coefficients of partial determination in a lot of stat books, as it would only exist in better books on regression like Neter, Wasserman, and Kutner, and maybe Draper and Smith. It would probably be online in a lot lectures or software manuals.