Abstract
To clarify the advantage of using the quasilikelihood method, lack of robustness of the maximum likelihood method was demonstrated for the negative-binomial model. Efficiency calculations of the method of moments and the pseudolikelihood method in the estimation of extra-Poisson parameters in a negative-binomial model were carried out. Especially when the overdispersion parameter is small, both methods are relatively highly efficient and the pseudolikelihood estimate is more efficient than the method of moments estimate. Two examples of the quasilikelihood analyses of count data with overdispersion are given. The bootstrap method also is applied to the data to illustrate the advantage of the method of moments or pseudolikelihood method in the estimation of the standard errors of the mean parameter estimates under the negative-binomial model.
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REFERENCES
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, 2nd International Symposium on Information Theory (eds. B. N. Petrov and F. Csaki), 267–281, Akademia Kiado, Budapest.
Breslow, N. E. (1984). Extra-Poisson variation in log-linear models, Appl. Statist., 33, 38–44.
Carroll, R. J. and Ruppert, D. (1982). Robust estimation in heteroscedastic linear models., Ann. Statist., 10, 429–441.
Carroll, R. J. and Ruppert, D. (1988). Transformation and Weighting in Regression, Chapman and Hall, London.
Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion), J. Roy. Statist. Soc. Ser. B, 49, 1–39.
Davidian, M. and Carroll, R. J. (1987). Variance function estimation, J. Amer. Statist. Assoc., 82, 1033–1048.
Davidian, M. and Carroll, R. J. (1988). A note on extended quasi-likelihood, J. Roy. Statist. Soc. Ser. B, 50, 74–82.
Dean, C. B. (1992). Testing for overdispersion in Poisson and binomial regression models, J. Amer. Statist. Assoc., 87, 451–457.
Dean, C. B. (1994). Modified pseudo-likelihood estimator of the overdispersion parameter in Poisson mixture models, J. Appl. Statist., 21, 523–532.
Dean, C. B., Lawless, J. F. and Willmot, G. E. (1989). A mixed Poisson-inverse-Gaussian regression model, Canad. J. Statist., 17, 171–181.
Efron, B., and Tibshirani, R. J. (1993). An Introduction to the Bootstrap, Chapman and Hall, London.
Firth, D. (1987). On the efficiency of quasi-likelihood estimation, Biometrika, 39, 665–674.
Firth, D. (1992). Discussion of the paper: Multivariate regression analysis for categorical data (by Liang, K.-Y., Zeger, S. L. and Qaqish, B.), J. Roy. Statist. Soc. Ser. B, 54, 3–40.
Fisher, R. A. (1949). A biological assay of tuberculin, Biometrics, 5, 300–316.
Frome, E. L. (1983). The analysis of rates using Poisson regression models, Biometrics, 39, 665–674.
Hinde, J. (1982). Compound Poisson regression model, GLIM82: Proc. Internat. Conf. Generalized Linear Models (ed. R. Gilchrist), 109–121, Springer, Berlin.
Holford, T. R. (1983). The estimation of age, period and cohort effects for vital rates, Biometrics, 39, 311–324.
Hurvich, C. M. and Tsai, C.-H. (1995). Model selection for extended quasi-likelihood models in small samples, Biometrics, 51, 1077–1084.
Inagaki, N. (1973). Asymptotic relation between the likelihood estimating function and the maximum likelihood estimator, Ann. Inst. Statist. Math., 25, 1–26.
Lambert, D. and Roeder, K. (1995). Overdispersion diagnostics for generalized linear models, J. Amer. Statist. Assoc., 90, 1225–1236.
Lawless, J. F. (1987). Negative binomial and mixed Poisson regression, Canad. J. Statist., 15, 209–225.
Liang, K.-Y. and McCullagh, P. (1993). Case studies in binary dispersion, Biometrics, 49, 623–630.
Liang, K.-Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models, Biometrika, 73, 13–22.
Manton, K. G., Woodbury, M. A. and Stallard, E. (1981). A variance components approach to categorical data models with heterogeneous populations: Analysis of spatial gradients in lung cancer mortality rates in North Carolina counties, Biometrics, 37, 259–269.
Margolin, B. H., Kaplan, N. and Zeiger, E. (1981). Statistical analysis of the Ames salmonella/microsome test, Proc. Nat. Acad. Sci. U.S.A., 76, 3779–3783.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed., Chapman and Hall, London.
White, H. (1982). Maximum likelihood estimation of misspecified models, Econometrica, 50, 1–25.
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Nakashima, E. Some Methods for Estimation in a Negative-Binomial Model. Annals of the Institute of Statistical Mathematics 49, 101–115 (1997). https://doi.org/10.1023/A:1003114706239
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DOI: https://doi.org/10.1023/A:1003114706239