Asymptotic Normality in Generalized Linear Mixed Models
Smith, Paul J.
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Generalized Linear Mixed Models (GLMMs) extend the framework of Generalized Linear Models (GLMs) by including random effects into the linear predictor. This will achieve two main goals of incorporating correlation and allowing broader inference. This thesis investigates estimation of fixed effects as the number of random effects grows large. This model describes cluster analysis with many clusters and also meta-analysis. After reviewing currently available methods, especially the penalized likelihood and conditional likelihood estimators of Jiang (1999), we focus on the random intercept problem. We propose a new estimator of regression coefficient and prove that when m, the number of random effects, grows to infinity at a slower rate than the smallest cluster sample size, the proposed estimator is consistent and given the realization of random effects, is asymptotically normal. We also show how to estimate the standard errors of our estimators. We also study the asymptotic distribution of Jiang(1999) penalized likelihood estimators. In the absence of regression coefficients, the normalized estimated intercept converges to a normal distribution. Difficulties arise in establishing the conditional asymptotic normality of Jiang(1999) penalized likelihood estimator of regression coefficients for fixed effects in a general GLMM. In Chapter 4, we make an extended analysis of the 2 by 2 by m table to show how to verify the general conditions in Chapter 3. We compare our estimator to the Mantel-Haenszel estimator. Simulation studies and real data analysis results validate our theoretical results. In Chapter 5, asymptotic normality of joint fixed effect estimate and scale parameter estimate is proved for the case where m/N does not go to 0. An example was used to verify the general conditions in this case. Simulation studies were performed to validate the theoretical results as well as to investigate conjectures that are not covered in the theoretical proofs. The asymptotic theory for the proposed estimator describes the finite sample behavior of the estimator very accurately. We find that in the case as m/N goes to 0, in the random logistic and Poisson intercept models, consistency and conditional asymptotic normality results appear to hold for the penalized regression coefficient estimates regression coefficients.