Frailty mixtures for count regression: separating within- and between-class heterogeneity

Mohieddine Rahmouni; Mohieddine Rahmouni

doi:10.3934/math.2026962

AIMS Mathematics

2026, Volume 11, Issue 8: 23893-23920. doi: 10.3934/math.2026962

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Frailty mixtures for count regression: separating within- and between-class heterogeneity

Mohieddine Rahmouni ^,

Applied College, King Faisal University, Al-Ahsa, Saudi Arabia

Received: 21 April 2026 Revised: 08 July 2026 Accepted: 21 July 2026 Published: 04 August 2026
MSC : 62J12, 62F10, 62P10

We proposed a finite exponential-gamma frailty mixture (FEGFM) regression model for recurrent-event count data observed over a fixed follow-up window. The model targets settings with overdispersion, heavy upper tails, and latent subgroup structure that are not well-captured by a single Poisson or negative binomial regression. Each latent component has its own log-linear mean regression and gamma frailty parameter, while component membership depends on covariates through multinomial logistic gating. After integrating out frailty, each component follows a negative binomial distribution, so the overall model is a covariate-dependent finite mixture of negative binomial regressions. We derived the main distributional properties, identifiability conditions, and an expectation-maximization algorithm for maximum-likelihood estimation. Simulation results showed improved fit and tail calibration relative to Poisson and single-component negative binomial benchmarks, while also indicating weak identification for some parameters in finite samples. An application to physician-visit counts illustrates the practical value of separating within-class frailty from between-class latent heterogeneity.
- recurrent events,
- count regression,
- finite mixtures,
- negative binomial,
- gamma frailty,
- EM algorithm,
- latent heterogeneity,
- overdispersion
Citation: Mohieddine Rahmouni. Frailty mixtures for count regression: separating within- and between-class heterogeneity[J]. AIMS Mathematics, 2026, 11(8): 23893-23920. doi: 10.3934/math.2026962

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Abstract

We proposed a finite exponential-gamma frailty mixture (FEGFM) regression model for recurrent-event count data observed over a fixed follow-up window. The model targets settings with overdispersion, heavy upper tails, and latent subgroup structure that are not well-captured by a single Poisson or negative binomial regression. Each latent component has its own log-linear mean regression and gamma frailty parameter, while component membership depends on covariates through multinomial logistic gating. After integrating out frailty, each component follows a negative binomial distribution, so the overall model is a covariate-dependent finite mixture of negative binomial regressions. We derived the main distributional properties, identifiability conditions, and an expectation-maximization algorithm for maximum-likelihood estimation. Simulation results showed improved fit and tail calibration relative to Poisson and single-component negative binomial benchmarks, while also indicating weak identification for some parameters in finite samples. An application to physician-visit counts illustrates the practical value of separating within-class frailty from between-class latent heterogeneity.

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