A Compound Logistic Regression Model for Binary Responses

TL;DR

This paper addresses the limitation of fixed asymptotes in standard logistic regression for binary responses by introducing a compound logistic regression framework that allows covariate-dependent asymptotes and correlated responses. It defines a mean function $\pi_i$ as a composition of multiple logistic components through a compounding function $H$, with instantiations such as the Asymptote COP and Vaccine Efficacy COP, and provides a general inference scheme based on quasi-likelihood/GEEs that accommodates both independent and correlated data. Regularization and cross-validation are integrated to stabilize fitting and perform model selection, and practical fitting strategies are demonstrated through simulations and a MIDUS data example. The approach is implemented in the R package CLmodel, enabling researchers to fit flexible binary response models with interpretable decompositions of exposure, COP threshold, and efficacy components.

Abstract

Logistic regression is the most commonly used method for constructing predictive models for binary responses. One significant drawback to this approach, however, is that the asymptotes of the logistic response function are fixed at 0 and 1, and there are many applications for which this constraint is inappropriate. More flexible models have been proposed for this application, most proceeding by supplementing the logistic response function with additional parameters. In this article we extend these models to allow correlated responses and the inclusion of covariates. This is achieved through the \emph{compound logistic regression model}, for which the mean response is a function of several logistic regression functions. This permits a greater variety of models, while retaining the advantages of logistic regression.

Figures (6)

• Figure 1: Decision tree representation of infection/COP process introduced in Section \ref{['sec.vaccine.efficacy.model']}.
• Figure 2: Response curve for simulated model of Section \ref{['sec.sim']}.
• Figure 3: Histograms of parameter estimate $Z$-scores $({\hat{\theta}}_j - \theta^*_j)/S_{{\hat{\theta}}_j}$, from simulation study of Section \ref{['sec.sim']} (${\lambda} = 0$). Values $|Z| > 5$ are omitted.
• Figure 4: Histograms of parameter estimate $Z$-scores $({\hat{\theta}}_j - \theta^*_j)/S_{{\hat{\theta}}_j}$, from simulation study of Section \ref{['sec.sim']} (${\lambda} = 0.1$). Values $|Z| > 5$ are omitted.
• Figure 5: Fitted response curves for selected COP model, Section \ref{['sec.examples']}