# Comparing an elastic net model with a nested linear regression

Suppose I have a linear model $$y_i=\mathbf{x}_i\boldsymbol{\beta}+\mathbf{z}_i\boldsymbol{\gamma}+\epsilon_i$$, where $$\boldsymbol{\gamma}$$ is subject to elastic net regularization. Now I have a nested linear regression $$y_i=\mathbf{x}_i\boldsymbol{\beta}+\epsilon_i$$. Now I want to know whether $$\mathbf{z}_i$$ contributes significantly. i.e., If both models are linear models, we could do a F-test to compare these two models. Or, I can compare their adjusted $$R^2$$. What are the equivalents (F-test and adjusted $$R^2$$) when one of the models involved are under elastic net regularization?

• By nested regression, you mean random effects? Why would you like to do a hypothesis test against regularized model?
– Tim
Commented Apr 2, 2022 at 10:28
• @Tim No. I meant the most basic linear regression. I want to know whether the added variables truly contribute to the model, and in my case, there are so many additional variables that I have to use regularized model.
– wdg
Commented Apr 2, 2022 at 10:41