Alternative to R$^2$ in linear regression without intercept It has been extensively described in this website the reason why one cannot properly calculate the $R^2$ - neither the Adjusted $R^2$ - in regression models fitted without an intercept. What is a good alternative metrics to compare the goodness-of-fit between regression models with and without intercept? Is doing the Pearson's correlation coefficient between dependent variable and fitted values $r^2$ a good solution?
 A: The reason that $R^2$ is dubious for a model without an intercept is because it compares the model fit (sum of squares errors) to the fit for a model with just an intercept.
However, by expressing interest in $R^2$, you are saying that the sum of squared errors interests you. So compare the sum of squared errors!
What you lost when you do this is the ability to gauge if a model is doing a decent job of predicting. It sounds good to get $R^2=0.95$; the model explains most of the variability. If your sum of squared errors is $17$ that’s fairly meaningless. However, that model has a better fit than a model with a sum of squared errors of $37$.
You can compare the sum of squared errors for any two models on the same data set. It sounds like you’re doing linear models, but it would be perfectly valid to compare to the sum of squared errors of a random forest model (for instance).
Out-of-sample testing might be a topic that interests you for this work.
And as EngrStudent wrote, there could be many other viable metrics, depending on what you value. Sum of squared errors seems to be the default in the absence of an argument for another metric, however.
A: If your goal is to compare two models (one with an intercept, and one without), and you're concerned that the in-sample $R^2$ will be misleading because one model has more parameters than the other, two options come to mind:

*

*See which model produces better out-of-sample predictions by running cross-validation. The simplest version of this will be splitting your dataset in two, estimating both models on one half of the data and making predictions on the other half with the fitted models, then comparing the mean squared errors. A common, more efficient variant of this is K-fold cross validation.


*Compute an in-sample measure of goodness-of-fit which penalizes the model with more parameters, like the AIC.
