A combined gamma frailty and normal random-effects model for repeated,overdispersed time-to-event data

Abstract

This paper presents, extends, and studies a model for repeated, overdispersed time-to-event outcomes, subject to censoring. Building upon work by Molenberghs, Verbeke, and Demétrio (2007) and Molenberghs et al. (2010), gamma and normal random effects are included in a Weibull model, to account for overdispersion and between-subject effects, respectively. Unlike these authors, censoring is allowed for, and two estimation methods are presented. The partial marginalization approach to full maximum likelihood of Molenberghs et al. (2010) is contrasted with pseudo-likelihood estimation. A limited simulation study is conducted to examine the relative merits of these estimation methods. The modeling framework is employed to analyze data on recurrent asthma attacks in children on the one hand and on survival in cancer patients on the other.

Keywords

exponential model generalized Cauchy distribution conjugacy maximum likelihood frailty model pseudo-likelihood strong conjugacy Weibull model

Get full access to this article

View all access options for this article.

References

Duchateau

Janssen

. The frailty model, New York: Springer, 2007.

Hinde

Demétrio

CGB

. Overdispersion: models and estimation. Comput Stat Data Anal 1998; 27: 151–170.

Nelder

Wedderburn

RWM

. Generalized linear models. J R Stat Soc Ser B 1972; 135: 370–384.

McCullagh

Nelder

. Generalized linear models, London: Chapman & Hall/CRC, 1989.

Molenberghs

Verbeke

Demétrio

. A family of generalized linear models for repeated measures with normal and conjugate random effects. Stat Sci 2010; 25: 325–347.

Molenberghs

Verbeke

Demétrio

. An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Anal 2007; 13: 513–531.

Engel

Keen

. A simple approach for the analysis of generalized linear mixed models. Stat Neerland 1994; 48: 1–22.

Breslow

Clayton

. Approximate inference in generalized linear mixed models. J Am Stat Assoc 1993; 88: 9–25.

Wolfinger

O’Connell

. Generalized linear mixed models: a pseudo-likelihood approach. J Stat Comput Simul 1993; 48: 233–243.

10.

Molenberghs

Verbeke

. Models for discrete longitudinal data, New York: Springer, 2005.

11.

Hand

Daly

Lunn

. A handbook of small data sets, 1st ed. London: Chapman & Hall, 1994, pp. 255–255.

12.

Breslow

. Extra-Poisson variation in log-linear models. Appl Stat 1984; 33: 38–44.

13.

Lawless

. Negative binomial and mixed Poisson regression. Can J Stat 1987; 15: 209–225.

14.

Cox

Hinkley

. Theoretical statistics, London: Chapman & Hall/CRC, 1974, pp. 370–370.

15.

Lee

Nelder

Pawitan

. Generalized linear models with random effects: unified analysis via h-likelihood, Boca Raton, FL: Chapman & Hall/CRC, 2006, pp. 178–178.

16.

Thall

Vail

. Some covariance models for longitudinal count data with overdispersion. Biometrics 1990; 46: 657–671.

17.

Dean

Estimating equations for mixed-Poisson models. In: Godambe

(eds). Estimating functions, Oxford: Oxford University Press, 1991, pp. 35–46.

18.

Aerts

Geys

Molenberghs

. Topics in modelling of clustered data, London: Chapman & Hall, 2002.

19.

Liang

K-Y

Zeger

. Longitudinal data analysis using generalized linear models. Biometrika 1986; 73: 13–22.

20.

Arnold

Strauss

. Pseudolikelihood estimation: some examples. Sankhya B 1991; 53: 233–243.

21.

Varin

. On composite marginal likelihoods. Adv Stat Anal 2008; 92: 1–28.

22.

Lindsay

. Composite likelihood methods. Contemp Math 1988; 80: 221–239.

23.

Joe

Lee

. On weighting of bivariate margins in pairwise likelihood. J Multivariate Anal 2008; 100: 670–685.

24.

Geys

Molenberghs

Lipsitz

. A note on the comparison of pseudo-likelihood and generalized estimating equations for marginal odds ratio models. J Stat Comput Simul 1998; 62: 45–72.

25.

Cox

Reid

. A note on pseudolikelihood constructed from marginal densities. Biometrika 2004; 91: 729–737.

26.

Geys

Molenberghs

Ryan

. Pseudo-likelihood modelling of multivariate outcomes in developmental toxicology. J Am Stat Assoc 1999; 94: 734–745.

27.

le Cessie

van Houwelingen

. A goodness-of-fit test for binary regression models, based on smoothing methods. Biometrics 1991; 47: 1267–1282.

28.

Renard

Molenberghs

Geys

. A pairwise likelihood approach to estimation in multilevel probit models. Comput Stat Data Anal 2004; 44: 649–667.

29.

Molenberghs

Verbeke

. On the Weibull-Gamma frailty model, its infinite moments, and its connection to generalized log-logistic, logistic, Cauchy, and extreme-value distributions. J Stat Plan Inf 2011; 141: 861–868.

30.

Litière S. Random-effects misspecification in generalized linear mixed models: a simulation study. PhD Thesis report, Chapter 7, University of Hasselt, Diepenbeek, Belgium, 2007.

31.

van Duijn

MAJ

Gile

Handcock

. Comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graphs. Working paper no. 74, Center for Statistics and the Social Sciences, Seattle: University of Washington, 2007.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.12 MB