Should partial R2R^2 add up to total R2R^2 in multiple regression?

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 R2 of 0.85. However, partial R2 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')

enter image description here

Is there any relation between sum of all partial R2 and total R2 ? The analysis is done with rms package.

Answer

No.

One way to understand partial R2 for a given predictor is that it equals the R2 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 R2, but partial R2 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. R2=1, then partial R2 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 R2 can easily be below or above the total R2. They do not have to coincide even if all predictors are orthogonal. Partial R2 is a bit of a weird measure.

See this long thread for many more details: Importance of predictors in multiple regression: Partial R2 vs. standardized coefficients.

Attribution
Source : Link , Question Author : rnso , Answer Author : Community

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