Significance test for coefficients of elastic net

Question

I have a 150x41 dataset, on which I performed variable selection and regression with Elastic Net. The response variable is continuous. I'd like to test the significance of the coefficients that I obtained. Is there a specific significance test for elastic net coefficients? Or something similar?

Answer 1

Three responses:

(1) Given the similarity of the elastic net penalty to the Lasso, you will encounter the same post-selection inference challenges as the Lasso. Thus, you would do well to see the helpful responses to this question:

Inference after using Lasso for variable selection

(2) If you're just looking for a p-value, you could consider a permutation test, but that could be extremely computationally expensive since you would need to re-select for each permutation. Or you could take a Bayesian approach with a continuous shrinkage prior (I am partial to the horseshoe prior) and use the posterior distribution for inference.

(3) All that said, why do you (think you) need a test of significance? The elastic net is intended to prune out the 'zero-valued coefficients' such that you should expect all of the selected coefficients to be significantly different from zero, by virtue of being selected. Can you instead focus on obtaining a properly cross-validated estimate of the improvement in model fit?