# Why is R Squared not a good measure for regressions fit using LASSO?

I have read in several places that R Squared is not an ideal measure when a model is fit using LASSO. However, I'm not clear on exactly why that is.

In addition, could you recommend the best alternative?

The goal of using LASSO is obtaining a sparse representation (of a predicted quantity) in the sense of not having many covariates. Comparing models with $R^2$ tends to favor models with lots of covariates: in fact, adding covariates unrelated to the outcome will never decrease $R^2$ and almost always increases it at least a little bit. The LASSO model will identify the model with the optimal penalized log-likelihood (an unpenalized log-likelihood is monotonically related to the $R^2$). Validation statistics that are more widely used to compare LASSO models to other types of models are, for instance, the BIC or cross-validated $R^2$.
• @Dave I think you have the right idea. The linear regression model is a LASSO with no penalty, and the log-likelihood is just $\log(2\pi)N+1−\log(N)+\log(\sum_{i=1}^n r_i^2)$ whereas the R2 is just $1 - \sum_{i=1}^n r_i^2/\sum_{i=1}^ny_i^2$ . The penalization contributes to error indirectly, it is a price you pay to enforce sparseness. The unpenalized model will always have lower (internal) error. People generally do cross-validation with the same dataset. Testing models in new datasets is a whole other thing(no need for the "cross" part) and it isn't done enough. – AdamO Jun 8 '18 at 18:24