# On the Limit Imbalanced Logistic Regression by Binary Predictors

## TL;DR

This paper proposes a rescaled likelihood approach for imbalanced logistic regression with binary predictors, facilitating regularization and interpretation, especially useful in pharmacovigilance data analysis.

## Contribution

It introduces a novel rescaled likelihood that simplifies regularization and interpretation in imbalanced logistic regression with binary predictors.

## Key findings

- Convergence of maximum likelihood estimates under class imbalance with strong overlap conditions.
- Analytic solutions for lasso regularization paths in binary predictor models.
- An efficient approximate path algorithm based on matrix inversions.

## Abstract

In this work, we introduce a modified (rescaled) likelihood for imbalanced logistic regression. This new approach makes easier the use of exponential priors and the computation of lasso regularization path. Precisely, we study a limiting behavior for which class imbalance is artificially increased by replication of the majority class observations. If some strong overlap conditions are satisfied, the maximum likelihood estimate converges towards a finite value close to the initial one (intercept excluded) as shown by simulations with binary predictors. This solution corresponds to the extremum of a concave function that we refer to as "rescaled" likelihood. In this context, the use of exponential priors has a clear interpretation as a shift on the predictor means for the minority class. Thanks to the simple binary structure, some random designs give analytic path estimators for the lasso regularization problem. An effective approximate path algorithm by piecewise logarithmic functions based on matrix inversions is also presented. This work was motivated by its potential application to spontaneous reports databases in a pharmacovigilance context.