zbMATH — the first resource for mathematics

An extended random-effects approach to modeling repeated, overdispersed count data. (English) Zbl 1331.62363
Summary: Non-Gaussian outcomes are often modeled using members of the so-called exponential family. The Poisson model for count data falls within this tradition. The family in general, and the Poisson model in particular, are at the same time convenient since mathematically elegant, but in need of extension since often somewhat restrictive. Two of the main rationales for existing extensions are (1) the occurrence of overdispersion, in the sense that the variability in the data is not adequately captured by the model’s prescribed mean-variance link, and (2) the accommodation of data hierarchies owing to, for example, repeatedly measuring the outcome on the same subject, recording information from various members of the same family, etc. There is a variety of overdispersion models for count data, such as, for example, the negative-binomial model. Hierarchies are often accommodated through the inclusion of subject-specific, random effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these issues may occur simultaneously, models accommodating them at once are less than common. This paper proposes a generalized linear model, accommodating overdispersion and clustering through two separate sets of random effects, of gamma and normal type, respectively. This is in line with the proposal by J. G. Booth et al. [Stat. Model. 3, No. 3, 179–191 (2003; Zbl 1070.62058)]. The model extends both classical overdispersion models for count data [N. E. Breslow, “Extra-Poisson variation in log-linear models”, J. R. Stat. Soc., Ser. C 33, No. 1, 38–44 (1984; doi:10.2307/2347661)], in particular the negative binomial model, as well as the generalized linear mixed model [N. E. Breslow and D. G. Clayton, J. Am. Stat. Assoc. 88, No. 421, 9–25 (1993; Zbl 0775.62195)]. Apart from model formulation, we briefly discuss several estimation options, and then settle for maximum likelihood estimation with both fully analytic integration as well as hybrid between analytic and numerical integration. The latter is implemented in the SAS procedure NLMIXED. The methodology is applied to data from a study in epileptic seizures.

62J12 Generalized linear models (logistic models)
62P10 Applications of statistics to biology and medical sciences; meta analysis
Full Text: DOI
[1] Agresti A (2002) Categorical data analysis, 2nd edn. John Wiley & Sons, New York · Zbl 1018.62002
[2] Aitkin M (1999) A general maximum likelihood analysis of variance components in generalized linear models. Biometrics 55: 117–128 · Zbl 1059.62564 · doi:10.1111/j.0006-341X.1999.00117.x
[3] Alfò M, Aitkin M (2000) Random coefficient models for binary longitudinal responses with attrition. Stat Comput 10: 279–288 · doi:10.1023/A:1008999824193
[4] Booth JG, Casella G, Friedl H, Hobert JP (2003) Negative binomial loglinear mixed models. Stat Model 3: 179–181 · Zbl 1195.62117 · doi:10.1191/1471082X03st058oa
[5] Breslow NE (1984) Extra-Poisson variation in log-linear models. Appl Stat 33: 38–44 · doi:10.2307/2347661
[6] Breslow NE, Clayton DG (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88: 9–25 · Zbl 0775.62195
[7] Breslow NE, Lin X (1995) Bias correction in generalized linear mixed models with a single component of dispersion. Biometrika 82: 81–91 · Zbl 0823.62059 · doi:10.1093/biomet/82.1.81
[8] Dean CB (1991) Estimating equations for mixed-Poisson models. In: Godambe VP(eds) Estimating functions. Oxford University Press, Oxford · Zbl 0850.62273
[9] Engel B, Keen A (1994) A simple approach for the analysis of generalized linear mixed models. Statistica Neerlandica 48: 1–22 · Zbl 0826.62055 · doi:10.1111/j.1467-9574.1994.tb01428.x
[10] Faught E, Wilder BJ, Ramsay RE, Reife RA, Kramer LD, Pledger GW, Karim RM (1996) Topiramate placebo-controlled dose-ranging trial in refractory partial epilepsy using 200-, 400-, and 600-mg daily dosages. Neurology 46: 1684–1690 · doi:10.1212/WNL.46.6.1684
[11] Hinde J, Demétrio CGB (1998) Overdispersion: models and estimation. Comput Stat Data Anal 27: 151–170 · Zbl 1042.62578 · doi:10.1016/S0167-9473(98)00007-3
[12] Hinde J, Demétrio CGB (1998) Overdispersion: models and estimation. XIII Sinape, São Paulo · Zbl 1042.62578
[13] Lawless J (1987) Negative binomial and mixed Poisson regression. Can J Stat 15: 209–225 · Zbl 0632.62060 · doi:10.2307/3314912
[14] Lee Y, Nelder JA (1996) Hierarchical generalized linear models (with discussion). J R Stat Soc Ser B 58: 619–678 · Zbl 0880.62076
[15] Lee Y, Nelder JA (2001) Hierarchical generalized linear models: a synthesis of generalized linear models, random-effect models and structured dispersions. Biometrika 88: 987–1006 · Zbl 0995.62066 · doi:10.1093/biomet/88.4.987
[16] Lee Y, Nelder JA (2003) Extended-REML estimators. J Appl Stat 30: 845–856 · Zbl 1121.62424 · doi:10.1080/0266476032000075930
[17] Liang K-Y, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73: 13–22 · Zbl 0595.62110 · doi:10.1093/biomet/73.1.13
[18] McCullagh P, Nelder JA (1989) Generalized linear models. Chapman & Hall, London · Zbl 0744.62098
[19] Molenberghs G, Verbeke G (2005) Models for discrete longitudinal data. Springer, New York · Zbl 1093.62002
[20] Nelder JA, Wedderburn RWM (1972) Generalized linear models. J R Stat Soc Ser B 135: 370–384 · doi:10.2307/2344614
[21] Thall PF, Vail SC (1990) Some covariance models for longitudinal count data with overdispersion. Biometrics 46: 657–671 · Zbl 0712.62048 · doi:10.2307/2532086
[22] Verbeke G, Molenberghs G (2000) Linear mixed models for longitudinal data. Springer-Verlag, New York · Zbl 0956.62055
[23] Wolfinger R, O’Connell M (1993) Generalized linear mixed models: a pseudo-likelihood approach. J Stat Comput Simul 48: 233–243 · Zbl 0833.62067 · doi:10.1080/00949659308811554
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.