# Adjusted R-Squared in terms of variance

Say that I am performing a multiple linear regression with 3 variables. If I want to say that two of these variables account for some percentage of the observed variance in the third variable, should I use my $R ^2$ value, or adjusted $R^2$ value?

I understand that the adjusted R squared value accounts for the fact that I have have more predictors (as compared to a regression of only two variables), but I'm wondering how that translates to my interpretation of the variance in these variables.

• Are you asking about this in the context of assessing multicollinearity? Eg, working up to computing the VIF? – gung May 24 '18 at 12:55
• Possible duplicate of How to split r-squared between predictor variables in multiple regression? – Michael Chernick May 29 '18 at 4:52
• @MichaelChernick it is unclear to me whether the OP want to split $R^2$ or whether it is about the choice between adjusted and unadjusted. Perhaps Matthew can edit to clarify? – mdewey May 29 '18 at 8:45

## 1 Answer

If you want to describe how much of the total variance in $X_1$ is explained by $X_2$ and $X_3$ using a linear model, then use $R^2$ which by definition gives just this number.

Save the adjusted $R^2$ for when you want to assess if it is worthwhile to include yet another variable, say $X_4$, in an attempt to model $X_1$ more closely, since (regular) $R^2$ will always increase when adding more variables.

You might want to read the wiki-page on the subject, which includes a note on the use of adjusted $R^2$.

• My reading of "If I want to say that two of these variables account for some percentage of the observed variance in the third variable" is that the OP is asking about R2 for, say, X1 as a function of X2 & X3, which is what's discussed in this answer. AFAICT, this answer is on point. – gung May 24 '18 at 12:54