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Aitkin, Murray

A general maximum likelihood analysis of variance components in generalized linear models. (English) Zbl 1059.62564

Biometrics 55, No. 1, 117-128 (1999).
Summary: This paper describes an EM algorithm for nonparametric maximum likelihood (ML) estimation in generalized linear models with variance component structure. The algorithm provides an alternative analysis to approximate MQL and PQL analyses [C. A. McGilchrist and C. W. Aisbett, Biom. J. 33, No. 2, 131–141 (1991; Zbl 0729.62566); N. E. Breslow and D. G. Clayton, J. Am. Stat. Assoc. 88, No. 421, 9–25 (1993; Zbl 0775.62195); C. A. McGilchrist, J. R. Stat. Soc., Ser. B 56, No. 1, 61–69 (1994; Zbl 0800.62433); H. Goldstein, Multilevel Statistical Models (1995)] and to GEE [analyses K.-Y. Liang and S. L. Zeger, Biometrika 73, 13–22 (1986; Zbl 0595.62110)]. The algorithm, first given by Hinde and Wood [Longitudinal data analysis, 110–126 (1987)], is a generalization of that for random effect models for overdispersion in generalized linear models, described in Aitkin [Statistics and Computing 6, 251–262 (1996)]. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully nonparametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters can be sensitive to the specification of a parametric form for the mixing distribution. The nonparametric analysis can be extended straightforwardly to general random parameter models, with full NPML estimation of the joint distribution of the random parameters. This can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters. A simple method is described for obtaining correct standard errors for parameter estimates when using the EM algorithm. Several examples are discussed involving simple variance component and longitudinal models, and small-area estimation.

Cited in 82 Documents

MSC:

62J12 Generalized linear models (logistic models)
62J10 Analysis of variance and covariance (ANOVA)
62G08 Nonparametric regression and quantile regression

Keywords:

EM algorithm; longitudinal data; mixture model; nonparametric maximum likelihood; overdispersion; random effects GLMs; variance components

Citations:

Zbl 0729.62566; Zbl 0775.62195; Zbl 0800.62433; Zbl 0595.62110

Software:

GLIM
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\textit{M. Aitkin}, Biometrics 55, No. 1, 117--128 (1999; Zbl 1059.62564)
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