Liu-type Shrinkage Estimators for Mixture of Logistic Regressions: An Osteoporosis Study

TL;DR

This paper introduces Liu-type shrinkage estimators to improve the estimation of mixture logistic regression models, especially under multicollinearity, demonstrated through numerical studies and an osteoporosis case study.

Contribution

It develops and applies Liu-type and ridge shrinkage estimators to address multicollinearity in mixture logistic regression models, enhancing estimation reliability.

Findings

Shrinkage methods outperform traditional estimates in simulations.

Liu-type estimators provide more stable coefficient estimates.

Application to osteoporosis data illustrates practical benefits.

Abstract

The logistic regression model is one of the most powerful statistical methods for the analysis of binary data. The logistic regression allows to use a set of covariates to explain the binary responses. The mixture of logistic regression models is used to fit heterogeneous populations through an unsupervised learning approach. The multicollinearity problem is one of the most common problems in logistics and a mixture of logistic regressions where the covariates are highly correlated. This problem results in unreliable maximum likelihood estimates for the regression coefficients. This research developed shrinkage methods to deal with the multicollinearity in a mixture of logistic regression models. These shrinkage methods include ridge and Liu-type estimators. Through extensive numerical studies, we show that the developed methods provide more reliable results in estimating the coefficients of the mixture. Finally, we applied the shrinkage methods to analyze the bone disorder status of women aged 50 and older.