Penalized Poisson Regression Model Using Elastic Net and Least Absolute Shrinkage and Selection Operator (Lasso) Penality

Abstract

Variable selection in count data using Penalized Poisson regression is one of the challenges in applying Poisson regression model when the explanatory variables are correlated. To tackle both estimate the coefficients and perform variable selection simultaneously, Lasso penalty was successfully applied in Poisson regression. However, Lasso has two major limitations. In the p > n case, the lasso selects at most n variables before it saturates, because of the nature of the convex optimization problem. This seems to be a limiting feature for a variable selection method. Moreover, the lasso is not well-defined unless the bound on the L1-norm of the coefficients is smaller than a certain value. If there were a group of variables among which the pairwise correlations are very high, then the lasso tends to select only one variable from the group and does not care which one is selected. To address these issues, we propose the elastic net method between explanatory variables and to provide the consistency of the variable selection simultaneously. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in the model together.