# Definition of partial R-squared?

I'm trying to understand the definition of Partial R-squared values in the context of a regression model. Does anyone have a layman's definition or an intuitive example that might help me better understand what it means?

Thank You!

Let's look at the equation.

$$R^2_{partial} = \dfrac{SSE(reduced) - SSE(full)}{SSE(reduced)}$$

In words, this compares the reduction in SSE from the reduced to the full model to the SSE of the reduced model. (We know the full model will have lower SSE.)

When the reduced model is the intercept-only model, the reduced model always predicts $$\bar{y}$$. This means that $$SSE(reduced) = SSTotal$$. Thus...

$$R^2_{partial} = \dfrac{SSTotal - SSE(full)}{SSTotal}$$

This is the usual $$R^2!$$

Therefore, much as $$R^2$$ describes the variance (proportional to $$SSE$$) of a model, compared to the variance of the intercept-only model (naïvely guessing $$\bar{y}$$, no matter the predictor values), partial $$R^2$$ describes the variance explained by a model, compared to a more complex model than the intercept-only model.

• Thank you for your explanation! This is very helpful and illuminating. – Sharif Amlani Jun 29 at 7:28

• This is consistent with how $R^2$ compares your model fit to the naïve model that always predicts $\bar{y}$ (intercept-only model). – Dave Jun 16 at 19:24