# Goodness of fit (r squared) for circular data

I have started to work with regression models for angular data. In particular with the package circular in R.

I checked previously in this forum if there was something analogous to Pearson R-squared for angles. As it was not completely clear I read the literature about the topic from Sarma and Rao Jammalamadaka (1993) (p. 119, Eq 3.6). Finally I got the book 'Circular statistics in R' in which they confirm (Ch 8.6, p. 169) that the output rho of the function circ.reg, is a measure of the fit analogous to Pearson R-squared, also with values between 0 and 1.

So far so good but when trying to understand the formulas I do not see how this value can be related to the residuals or the variance in the model. Can someone explain?

You are conflating two separate statistical concepts, which is causing confusion.

Explained variance is often indicated by $R^2$, has to do with what proportion of the total variance in the outcome measure can be 'explained' by the predictor. That is, how strong is the relation between the set of predictors and the outcome.

Goodness of fit measures indicate how well the model "fits" the data. That is, can the model generate data that is similar to the observed data, which would indicate that the model we have chosen was chosen well.

The $\rho$ in circular regression is a measure of explained variance, indeed with interpretation analogous to Pearson's $R^2$.

• Probably I was confused because R2 gives also some information about the goodness of fit as is a statistical measure of how well the regression line approximates the real data points. – gis20 Feb 13 '17 at 20:25