Should partial $R^2$ add up to total $R^2$ in multiple regression?

Question

Following is a model created from mtcars dataset:

> ols(mpg~wt+am+qsec, mtcars)

Linear Regression Model

ols(formula = mpg ~ wt + am + qsec, data = mtcars)

                Model Likelihood     Discrimination    
                   Ratio Test           Indexes        
Obs       32    LR chi2     60.64    R2       0.850    
sigma 2.4588    d.f.            3    R2 adj   0.834    
d.f.      28    Pr(> chi2) 0.0000    g        6.456    

Residuals

    Min      1Q  Median      3Q     Max 
-3.4811 -1.5555 -0.7257  1.4110  4.6610 

          Coef    S.E.   t     Pr(>|t|)
Intercept  9.6178 6.9596  1.38 0.1779  
wt        -3.9165 0.7112 -5.51 <0.0001 
am         2.9358 1.4109  2.08 0.0467  
qsec       1.2259 0.2887  4.25 0.0002  

The model seems good with total $R^2$ of 0.85. However, partial $R^2$ values seen on following plot do not add up to this value. They add up to approx 0.28.

> plot(anova(mod), what='partial R2')

Is there any relation between sum of all partial $R^2$ and total $R^2$ ? The analysis is done with rms package.

Answer 1

No.

One way to understand partial $R^2$ for a given predictor is that it equals the $R^2$ that you would get if you first regress your independent variable on all other predictors, take the residuals, and regress those on the remaining predictor.

So if e.g. all predictors are perfectly identical (collinear), one can have decent $R^2$, but partial $R^2$ for all predictors will be exactly zero, because any single predictor has zero additional explanatory power.

On the other hand, if all predictors together explain the dependent variable perfectly, i.e. $R^2=1$, then partial $R^2$ for each predictor will be $1$ too, because whatever is unexplained by all other predictors can be perfectly explained by the remaining one.

So the sum of all partial $R^2$ can easily be below or above the total $R^2$. They do not have to coincide even if all predictors are orthogonal. Partial $R^2$ is a bit of a weird measure.

See this long thread for many more details: Importance of predictors in multiple regression: Partial $R^2$ vs. standardized coefficients.