| Summary: | Thesis (M.Sc.)--Memorial University of Newfoundland, 1995. Mathematics and Statistics Bibliography: leaves 73-77. The application of the Poisson mixed model has been hampered by the difficulty of computation in evaluating the marginal likelihood of the parameters involved. Many approximate approaches have recently been proposed for inference about the generalized linear mixed model which refers to the Poisson mixed model as a special case, for example, the penalized quasi-likelihood (PQL) approach of Breslow and Clayton (1993), and the generalized estimating function (GEF) approach of Waclawiw and Liang (1993). We show in the thesis that both the PQL and GEF produce inconsistent inference for the variance component in the Poisson mixed model. The thesis then proposes a two-step approximate likelihood approach (AL) for the estimation of three types of parameters (fixed effect parameters, random effects and their variance component) in the Poisson mixed model. In the first step, an approximate likelihood function of count data is constructed to estimate the fixed effect parameters and the variance component by applying a conjugate Bayesian theorem. In the second step, the random effects are estimated by minimizing their approximate posterior mean square error. Our estimates are always consistent for both the fixed effect parameters and the variance component. When the actual variance component is near zero, our estimates are almost efficient for both the fixed effect parameters and the variance component, and are almost optimal for the random effects. When the actual variance component is away from zero, our estimates are always asymptotically unbiased for the fixed effect parameters, whereas our estimate is asymptotically negative biased for the variance component. Another desirable merit is that, unlike the existing approaches mentioned above, our estimates for both the fixed effect parameters and the variance component only depend on the distribution of random effects rather than the estimates of random effects. An important finding is that the asymptotic covariance of our estimates for the fixed effect parameters will become smaller in general as the variance component, an index of the intra-cluster association, increases, and can be noticeably reduced by assigning the values of the fixed effect covariates as different as possible among different observations in any cluster. However, if the fixed effect covariate has the same or almost equal values among different observations in any cluster, the asymptotic variance of the estimate for the corresponding fixed effect parameter may increase as the variance component gets larger. This feature may be useful in designing a valid experiment or sampling for the Poisson mixed model. Unless the variance component is small, the fixed effect covariates should be designed to have values as different as possible among different observations in any cluster. It is further shown, through simulation, the proposed approach performs better than the PQL and GEF approaches.
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