Solved – R-squared or adjusted R-squared to use when comparing nested models

I have a model with predictor variables x1, x2, and x3.
I have another model with predictor variable x1.

My understanding is that when you have multiple predictors, you use adjusted R-squared, but with only one predictor, you use x1.

So when I am trying to compare the goodness of fit of the two different models I mentioned above, should I be comparing R-squared or adjusted R-squared?

In linear regression, $R^2$ can't decrease when you add a new regressor (another "x" variable), since the fit will be equal or better. It remains the same only if the new "x" is a linear combination of the previous ones you already had. But then is it better to have more variables? $R^2_{adj}$ takes this into account by penalizing models with more variables. In other words, the increase in $R^2$ (i.e. the improvement of the fitting) must be reasonably large for the inclusion of a new variable to cause an increase in $R^2_{adj}$. In conclusion, you should always use $R^2_{adj}$ when comparing models with a different number of regressors. In the case where your models have the same number, it doesn't really matter which one you use. So it is just better to use the adjusted one all the time.

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