Date of Award

Spring 2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

N. Rao Chaganty

Committee Member

Larry D. Lee

Committee Member

Nak-Kyeong Kim

Committee Member

Tina D. Cunningham

Abstract

Longitudinal or repeated measure data are common in biomedical and clinical trials. These data are often collected on individuals at scheduled times resulting in dependent responses. Inference methods for studying the behavior of responses over time as well as methods to study the association with certain risk factors or covariates taking into account the dependencies are of great importance. In this research we focus our study on the analysis of continuous longitudinal data. To model the dependencies of the responses over time, we consider appropriate correlation structures generated by the stationary and non-stationary time-series models. We develop new estimation procedures depending on the correlation structures considered and compare those procedures with the existing methods.

The first part of this dissertation focuses on the robust correlation structure generated by the first-order autoregressive-moving average (ARMA(1, 1)) stationary time-series model. ARMA(1, 1) correlation structure is characterized by two correlation parameters and this correlation structure reduces to the AR(1), MA(1) and CS structures in special cases. Although standard efficient procedures are preferable to estimate the correlation parameters, there are computational challenges in implementing them. To overcome these challenges we employ an alternative estimation procedure based on pairwise likelihoods. A typical advantage of this approach is that the inference procedure does not involve complex computations and it results in a closed form expressions for the estimators of the correlation parameters. We show that the estimates obtained using the pairwise likelihood method for ARMA(1, 1) correlation structure are highly efficient asymptotically when compared to that of maximum likelihood.

The second part of the dissertation studies correlation structures generated by non-stationary time-series model known as antedependence models of first order. These correlation structures are capable of modeling the non-constant correlations between the same-lagged observations. Note that while this correlation structure has been extensively studied in the case of heterogeneous variance, we model homogenous variance and use a recent and new method known as quasi-least squares to estimate the correlation parameters. A major advantage of the quasi-least squares method is that it yields closed form expressions for the estimators of correlation parameters unlike the maximum likelihood method. We provide the asymptotic and small-sample properties of these estimators and compare their performance using relative efficiencies.

DOI

10.25777/vq67-x035

ISBN

9781303166204

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