# Can you use the coefficient of determination for categorical variables?

## 2006-04-04

[StATS]: Can you use the coefficient of determination for categorical variables? (April 4, 2006)

Dear Professor Mean, How can you compute a coefficient of determination (R squared) for a model that has a dichotomous variable? I thought that you could only compute this in a linear regression model?

That’s an interesting question, because it helps to illustrate the consistency of linear regression and ANOVA models. Your concern, no doubt, is spurred by the classic definition of the coefficient of determination in that it is almost always defined as the square of the correlation coefficient (hence the alternate name, R squared). How can you define a correlation coefficient when your independent variable is categorical?

Well, it turns out that you can come up with something called the point biserial correlation coefficient, but (a) it is a rather obscure quantity, and (b) it doesn’t help you when you have an independent variable with three or more levels.

What you need to do is to use an alternative formula for the coefficient of determination. If you define it as the ratio of SSR/SST (Sum of Squares Due to Regression divided by Sum of Squares Total), then this quantity is well defined for both a linear regression model and for an ANOVA model, since both models allow for calculation of sums of squares.

Surprisingly, I could not find a good example of this explanation on the web, other than my own web page (see below). That may just be a reflection on my lack of good searching skills (or perhaps a measure of my arrogance)!

Most textbooks, however, offer a good explanation of this. For example, in Woolson’s book, the section on ANOVA models has the following quote.

The coefficient of multiple determination, denoted by R^2^, is defined by R^2^ = SSR/SST, analogous to the simple linear regression situation, and again is the proportion of the Y variability accounted for by the fitted model.<U+FFFD> Statistical Methods for the Analysis of Biomedical Data. Robert F. Woolson (1987) New York: John Wiley & Sons. page 299. [BookFinder4U link]

One way to think about this is that the coefficient of determination represents a correlation of the predicted values and the actual values. This correlation always has to be non-negative, of course, and the closer this is to 1.0, the better the model. You can compute predicted values for both a regression model and an ANOVA model.

Most statisticians do not make a distinction between a regression model and an ANOVA model, since they both use the same summary tables and can be fit using the same algorithms. So it is not too surprising that the coefficient of determination is used for both models.