Finite Mixtures of Generalized Linear Regression Models

Abstract

Summary. Generalized linear models have become a standard technique in the statistical modelling toolbox for investigating relationships between variables. The assumption of homogeneity of regression coefficients over all observations can be relaxed by incorporating generalized linear models into the finite mixture framework.The model class consisting of finite mixtures of generalized linear models is presented. Model identification is discussed given that difficulties might be encountered due to trivial and generic identifiability problems. These problems have already been observed for mixtures of distributions, but the extension to mixtures of regression models introduces additional identifiability problems. Details on model estimation are given and the application is illustrated on several examples.

A note on the identifiability of nonparametric and semiparametric mixtures of GLMs

“…We further established identifiability results for the proposed two models under mild conditions. In particular, our result verifies that the semiparametric and nonparametric mixture models proposed by Grün and Leisch (2008a), Young and Hunter (2010), and Cao and Yao (2012) are identifiable under the conditions given in Theorems 3.1 and 3.2.…”

“…Identifiability results for finite mixtures of ordinary linear models and finite mixtures of GLMs are available, see Hennig (2000) and Grün and Leisch (2008a); but these results are not applicable to models (2.2) and (2.3) as some of parameters are nonparametric functions of covariates.…”

“…These nonparametric and semiparametric extensions of mixtures of GLMs enable more flexible modeling and widen the scope of applications of mixture models. However, these works either lack of identifiability results, such as Grün and Leisch (2008a), Young and Hunter (2010), and Cao and Yao (2012), or only give identifiability results on a case-by-case basis.…”

“…Classical identifiability results given by Teicher (1963) and Yakowitz and Spragins (1968) provide a foundation for finite mixtures of parametric distributions. For the identifiability results of mixtures of linear regression models for continuous responses, and mixtures of generalized linear models (GLMs) for binary or count responses, please see Hennig (2000), Grün and Leisch (2008a), and Grün and Leisch (2008c) and references therein.…”

Erratum to: The Generalized Linear Mixed Cluster-Weighted Model

“…General conditions for identifiability of mixtures of linear models can be found in Hennig (2000); based on such results, Grün and Leisch (2008a) provided results about identifiability for model (8). Follmann and Lambert (1991) and Wang (1994) established identifiability results for mixtures of logistic regression models (the latter paper considered the case with concomitant variables).…”