Date of Award

Spring 1996

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics & Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Dayanand N. Naik

Committee Member

Narasinga R. Chaganty

Committee Member

John P. Morgan

Committee Member

Ardythe L. Morrow

Abstract

In this dissertation we focus mainly on the analysis of continuous multivariate repeated measurements data based on the assumption of multivariate normality. However certain aspects of the analysis of univariate repeated measures data are also considered. Typically, we have measurements on p variables (possibly correlated) in the form of px1 vectors yijk observed at k = 1,2, ...,tij occasions on j = 1,2, ..., ni individuals from i = 1,2, ..., g groups. We assume a naturally occurring covariance structure Vij ⊗ ∑ among the p variables on the jth individual from ith group made at tij occasions. Here Vij and ∑ are positive definite matrices of order tij x tij and p x p respectively. We develop a general linear model approach to accommodate both balanced and unbalanced repeated measures data.

Our main results are: (1) construction of Rao's score test for a simpler model with p=1 (univariate case) and Vij having a structure as in a mixed effects model, (2) comparison of all the methods for analyzing univariate repeated measures data with time varying covariates, (3) derivation of the maximum likelihood estimates of the covariance matrices V and ∑ in the balanced case, (4) derivation of Satterthwaite type approximation to the distribution of multivariate quadratic forms, (5) estimation of degrees of freedom for these approximations, and (6) derivation of the maximum likelihood estimates of the covariance parameters under certain specific covariance structures for unbalanced case.

Rao's score test is derived in Chapter 2. Analysis of repeated measures in the presence of time varying covariates is a useful but difficult problem. In Chapter 3, we review the existing methods for analyzing repeated measured data with time varying covariates and discuss their computational aspects using SAS software. We also point out that a linear model approach yields a unified tool to analyze these data. In Chapter 4, various results about balanced multivariate repeated measures models are derived. We present the entire scheme of analysis of balanced multivariate data including the computational details. Finally, the analysis of unbalanced multivariate repeated measures is discussed in Chapter 5. In this case we assume two commonly used covariance structures namely equicorrelation and autoregressive structures for Vij and derive the maximum likelihood estimates of the unknown parameters.

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DOI

10.25777/p6wc-j581

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