| Generalized linear mixed models (GLMM) have received a lot of attention in the past decades or even longer. The allowance of discrete and non-normally distributed responses and the incorporation of random effects have made GLMM a flexible approach to model a transformation of the mean as a function of both fixed and random effects. The application of this kind of models can be addressed to statistical issues, such as heterogeneity, over-dispersion, and intra-cluster correlation. The history of the development for GLMM will be reviewed. The existing parameter estimation methods will also be discussed.;The most common relationships among the random effects can be either nested or crossed. The model diagnostic methods in this dissertation are developed for both of them. For each random effect structure, the dissertation shows step by step from model introduction, parameter estimation, and finally test statistic and its asymptotic property development. The feasibility from computation point of view has also been considered.;The model diagnostic method proposed for GLMM with nested random effects starts from picking minimum chi-square estimate (MCE) as the parameter estimation method. Then, a modified Pearson's chi-square test statistic is defined. Because the random effects are nested, we still have the independence at the subject level, which leads to the application of central limit theorem for independent but not identical observations. The regularity conditions are discussed, so that the requirements for applying this method are specified.;The model diagnostic method proposed for GLMM with crossed random effects starts from using method of simulated moments (MSM) estimate as the parameter estimation method. The test statistic follows the similar formula from the previous chapter. However, we need to consider a different central limit theorem because of the crossed random effects, which lead to the dependence across all the observations. The martingale central limit theorem comes into mind and the conditions needed for using this theorem are proved.;Simulation studies and real data examples will be considered for each method. |
