Should individual $R^2$ of a predictor always be greater than $\Delta R^2$ when removing that predictor from an expanded model? I'm running some regressions with a set of somewhat correlated predictors. Let's call these predictors $x$, $y$ and $z$, and my dependent variable $d$.
I'm focused on the effect of $x$ on $d$.
I first calculate $R^2_x$ by running a regression using only $x$ as a predictor.
I can also calculate $R^2_{full}$ by running a regression using $x$, $y$ and $z$ as predictors.
I calculate $R^2_{reduced}$ by running a regression using only $y$ and $z$ as predictors.
I calculate $\Delta R^2$ as $R^2_{full} - R^2_{reduced}$.
My question is, shouldn't $R^2_x$ always be greater than $\Delta R^2$? I would think that it would, since in the full model, the predictive power of $x$ has been reduced by the inclusion of the correlated predictors $y$ and $z$. However I am observing many instances of $\Delta R^2 > R^2_x$. Is this to be expected? Thanks!

The short answer is that $x$ and $y$ and $z$ can be correlated with each other in ways that are not correlated with $d$. The delta can be greater than the $R^2(x)$ because adding $X$ to the model can help clean up noise in y and $z,$ helping the model make sense of more of $y$ and $z,$ ultimately increasing the $R^2(\textrm{full}).$ When this concept is taught in classes, they often show neat overlapping circles, but the reality is that the circles are more like amorphous blobs that overlap in weird ways both inside and outside the variance circle for $d$.
