Second-order Least Squares Estimation in Generalized Linear Mixed Models
Maximum likelihood is an ubiquitous method used in the estimation of generalized linear mixed model (GLMM). However, the method entails computational difficulties and relies on the normality assumption for random effects. We propose a second-order least squares (SLS) estimator based on the first two marginal moments of the response variables. The proposed estimator is computationally feasible and requires less distributional assumptions than the maximum likelihood estimator. To overcome the numerical difficulties of minimizing an objective function that involves multiple integrals, a simulation-based SLS estimator is proposed. We show that the SLS estimators are consistent and asymptotically normally distributed under fairly general conditions in the framework of GLMM. Missing data is almost inevitable in longitudinal studies. Problems arise if the missing data mechanism is related to the response process. This thesis develops the proposed estimators to deal with response data missing at random by either adapting the inverse probability weight method or applying the multiple imputation approach. In practice, some of the covariates are not directly observed but are measured with error. It is well-known that simply substituting a proxy variable for the unobserved covariate in the model will generally lead to biased and inconsistent estimates. We propose the instrumental variable method for the consistent estimation of GLMM with covariate measurement error. The proposed approach does not need any parametric assumption on the distribution of the unknown covariates. This makes the method less restrictive than other methods that rely on either a parametric distribution of the covariates, or to estimate the distribution using some extra information. In the presence of data outliers, it is a concern that the SLS estimators may be vulnerable due to the second-order moments. We investigated the robustness property of the SLS estimators using their influence functions. We showed that the proposed estimators have a bounded influence function and a redescending property so they are robust to outliers. The finite sample performance and property of the SLS estimators are studied and compared with other popular estimators in the literature through simulation studies and real world data examples.