Mathematics Ph.D. Dissertations

Title

Generalized Estimating Equations for Mixed Models

Date of Award

2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Hanfeng Chen (Advisor)

Second Advisor

Robert Dyer (Other)

Third Advisor

Wei Ning (Committee Member)

Fourth Advisor

Junfeng Shang (Committee Member)

Abstract

Most statistical approaches of molding the relationship between the explanatory variables and the responses assume subjects are independent. However, in clinical studies the longitudinal data are quite common. In this type of data, each subject is assessed repeatedly over a period of time. Therefore, the independence assumption is unlikely to be valid with longitudinal data due to the correlated observations of each subject. Generalized estimating equations method is a popular choice for longitudinal studies. It is an efficient method since it takes the within-subjects correlation into account by introducing a working correlation matrix. Although the generalized estimating equations’ methodology considers correlation among the repeated observations on the same subject, it ignores the between-subject correlation and assumes subjects are independent. The objective of this dissertation is to provide an extension to the generalized estimating equations to take both within-subject and between-subject correlations into account by incorporating the random effect b to the model. If our interest focuses on the regression coefficients, we regard the correlation parameter as nuisance and estimate the fixed effects " using the estimating equations. If our interest focuses either on both the correlation parameter and the variance of the random effects or on the coefficient parameters and the association structure, then building an additional system of estimating equations analogous to the first estimating equations can serve to estimate either the correlation parameter and coefficients parameter, simultaneously or the variance of the random effects and the coefficient parameter, simultaneously.

This estimating equations method has no closed form solution and can be solved iteratively. For example, Newton-Raphson is a popular iterative method to be used. We illustrate through simulation studies and real data applications the performance of the proposed methods in terms of bias and efficiency. Moreover, we investigate their behaviors compared to those for existing methods such as generalized estimating equations (GEE), generalized linear models (GLM) and generalized linear mixed models (GLMM). For further studying the performance of newly proposed method, the new approach is applied to the epilepsy data that was studied by many others Fitzmaurice, Laird, and Ware (2012).

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