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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?

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  • $\begingroup$ By nested regression, you mean random effects? Why would you like to do a hypothesis test against regularized model? $\endgroup$
    – Tim
    Commented Apr 2, 2022 at 10:28
  • $\begingroup$ @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. $\endgroup$
    – wdg
    Commented Apr 2, 2022 at 10:41

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