I am interested in stating that ___ % of the variance in Y is explained uniquely by X1 and ___ % is explained uniquely by X2.
- Is there some way to obtain this from a multiple regression model, or do I need to obtain adjusted R2 values from a series of residual regressions (sensu Legendre & Legendre et al.)?
- That being asked, is R2 value valid for insignificant OLS regression model?
- Does the value of R2 depend on a statistical test?
Yes, you’re trying to calculate the Extra Sum of Squares. In short you are partitioning the regression sum of squares. Assume we have two X variables, X1 and X2. The SSTO (total sum of squares, made up of the SSR and SSE) is the same regardless of how many X variables we have. Denote the SSR and SSE to indicate which X variables are in the model: e.g.
SSR(X1,X2)=385 and SSE(X1,X2)=110
Now let’s assume we did the regression just on X1 e.g.
SSR(X1)=352 and SSE(X1)=143.
The (marginal) increase in the regression sum of squares in X2 given that X1 is already in the model is:
or equivalently, the extra reduction in the error sum of squares associated with X2 given that X1 is already in the model is:
In the same way we can find:
Of course, this also works for more X variables as well.