A Bayesian approach to some overdispersion models. (English) Zbl 0685.62033
Summary: In fitting a generalized linear model, many authors have noticed that data sets can show greater residual variability than predicted under the exponential family. Two main approaches have been used to model this overdispersion. The first approach uses a sampling density which is a conjugate mixture of exponential family distributions. The second uses a quasilikelihood which adds a new scale parameter to the exponential likelihood. The approaches are compared by means of a Bayesian analysis using noninformative priors. In examples, it is indicated that the posterior analysis can be significantly different using the two approaches.
Keywords:
Bayes factors; generalized linear model; residual variability; overdispersion; conjugate mixture of exponential family distributions; quasilikelihood; exponential likelihood; noninformative priors; posterior analysisReferences:
| [1] | Armitage, Studies in the variability of Pock counts, J. Hygiene Camb. 55 pp 564– (1957) |
| [2] | Anderson, Some models for overdispersed binomial data, Austral. J. Statist. 30 pp 125– (1988) · Zbl 0707.62005 |
| [3] | Box, Sampling and Bayes inference in scientific modeling and robustness, J. Roy. Statist. Soc. Ser. A 143 pp 383– (1980) · Zbl 0471.62036 |
| [4] | Breslow, Extra-Poisson variation in log-linear models, Appl. Statist. 33 pp 38– (1984) |
| [5] | Crowder, Beta-binomial Anova for proportions, Appl. Statist. 27 pp 34– (1978) |
| [6] | Diaconis, Conjugate priors for exponential families, Ann. Statist. 7 pp 269– (1979) · Zbl 0405.62011 |
| [7] | Finney, Radiologand assay, Biometrics 32 pp 721– (1976) |
| [8] | Good, A Bayesian significance test for multinomial distributions, J. Roy. Statist. Soc Ser. B 29 pp 399– (1967) |
| [9] | Holford, The estimation of age, period and cohort effects for vital rates, Biometrics 39 pp 311– (1983) |
| [10] | Lawless, Negative binomial and mixed Poisson regression, Canad. J. Statist. 15 pp 209– (1987) · Zbl 0632.62060 |
| [11] | Margolin, Statistical analysis of the Ames Salmonella/microsome test, Proc. Natl. Acad. Sci. U. S. A. 76 pp 3779– (1981) |
| [12] | Morton, A generalised linear model with nested strata of extra-Poisson variation, Biometrika 4 pp 125– (1987) · Zbl 0621.62075 |
| [13] | Morris, Natural exponential families with quadratic variance functions, Ann. Statist. 11 pp 515– (1983) · Zbl 0521.62014 |
| [14] | Stevens, Temporal trends in breast cancer, Amer. J. Epidemiol. 115 pp 759– (1982) |
| [15] | Sweeting, Scale parameters: A Bayesian treatment, J. Roy. Statist. Soc. Ser. B 43 pp 333– (1981) · Zbl 0475.62022 |
| [16] | West, Bayesian Statistics 2 (1985) |
| [17] | Williams, Extra-binomial variation in logistic linear models, Appl. Statist. 31 pp 144– (1982) · Zbl 0488.62055 |
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