An Extension of Generalized Linear Models to Finite Mixture Outcome Distributions

TL;DR

This paper extends generalized linear models to incorporate finite mixture distributions for outcomes, enabling better modeling of heterogeneity and variability in categorical and count data.

Contribution

It introduces a likelihood-based extension of GLMs that models the response as a finite mixture, linking the mean to the regression, and handles complex mean structures.

Findings

Improved residual plots with extra variation support.

Wider prediction intervals achieved.

Application to success/failure and count data datasets.

Abstract

Finite mixture distributions arise in sampling a heterogeneous population. Data drawn from such a population will exhibit extra variability relative to any single subpopulation. Statistical models based on finite mixtures can assist in the analysis of categorical and count outcomes when standard generalized linear models (GLMs) cannot adequately account for variability observed in the data. We propose an extension of GLM where the response is assumed to follow a finite mixture distribution, while the regression of interest is linked to the mixture's mean. This approach may be preferred over a finite mixture of regressions when the population mean is the quantity of interest; here, only a single regression function must be specified and interpreted in the analysis. A technical challenge is that the mean of a finite mixture is a composite parameter which does not appear explicitly in the density. The proposed model is completely likelihood-based and maintains the link to the regression through a certain random effects structure. We consider typical GLM cases where means are either real-valued, constrained to be positive, or constrained to be on the unit interval. The resulting model is applied to two example datasets through a Bayesian analysis: one with success/failure outcomes and one with count outcomes. Supporting the extra variation is seen to improve residual plots and to appropriately widen prediction intervals.