interpretation of adjusted $R^2$

Question

the $R^2$ is the amount of variation of the dependend variable $Y$ explained by linear regression. It can take values between 0 and 1.

If I use the adjusted $R^2$ instead, what is the interpretation of this value?

SmallChess · Answer 1 · 2015-06-02T09:25:31.920

0

I think there're many existing posts covering it. The wikipedia has a nice section on it, so I think you should take a look.

When you fit a regression model, you will get a R2 statistic. We all want a model with high R2, but we don't want to overfit our model. Our job is to create a model with variables that can explain the variability significantly. We don't want to pour all variables into the data-set.

Unfortunately, we can always add more variables into the model to achieve a higher R2, regardless of whether those variables are correlated with the dependent variable.

Adjusted R2 is an attempt to take care of it. It can decrease if a new variable isn't useful to explain the variability.

EDITED

Adjusted R2 has no direct and simple interpretation, it is a metric to compare two different models. You can think of it like AIC or BIC. It is a measure of fit.

edited Jun 02 '15 at 09:25

answered Jun 02 '15 at 08:38

SmallChess

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I know how to calculate the adjusted $R^2$ and I know its usefulness, but what does it say? What is its interpretation? Adjusted $R^2$ can take negative values, $R^2$ cannot. – spore234 Jun 02 '15 at 08:56
Check my EDITED answer. – SmallChess Jun 02 '15 at 09:25
1

but what if $n >> p$, then $R^2 \sim$ adj. $R^2$ ? – spore234 Jun 02 '15 at 09:31
Yes, see the comments here: http://stats.stackexchange.com/questions/147688/why-to-report-r-squared/147704#147704 – Christoph Hanck Jun 02 '15 at 09:43

score 0 · Answer 2 · answered Jun 02 '15 at 15:25

The interpretation of $1-R^2_{Adj}$ is the same as that of any other regularized loss function - penalizing a complicated model.

In particular this is similar to $L_0$ norm which also penalizes number of parameters but is not as popular as $ L_1$ and $L_2$ regularization because of their nice properties. See also their logistic counterparts, Elastic Net and Lasso regression.

interpretation of adjusted $R^2$

2 Answers2