# Analysis of Growth Curves Under Some Special Covariance Structures

Fall 1995

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematics & Statistics

## Program/Concentration

Computational and Applied Mathematics

D. N. Naik

Larry Lee

Ram Dahiya

Cynthia Jones

## Abstract

In this dissertation we consider the growth curve or generalized MANOVA model in its most general form given by and develop statistical methodology for analyzing data using this model. Here g represents the number of groups, Yij is the observation matrix, ξ is a matrix of unknown parameters, Ai is a known matrix of rank g, and Bij is a matrix of rank k. Further, the rows of the error matrix ∈ij are independent and each distributed as Npij (0, Σij).This model accommodates different kinds of unbalanced data, such as, monotone data, data missing from any occasion, and data observed at unequally spaced time points.

Our main results are: (1) derivation of the formulae for the maximum likelihood estimates (MLEs) of the parameters involved, (2) construction of the tests for testing general linear hypothesis of the form Ho : EqxgξgxkFkxv =0. for known full rank matrices E and F, and (3) derivation of the formulae for prediction of (a) future observations corresponding to an individual, (b) the unobserved portion of a partially observed data for a new individual, and (c) any missing value of an observation vector.

Deriving the maximum likelihood estimates and the prediction formulae for unbalanced data is a challenging problem. We have derived these results by taking two types of covariance structures for Σij. These structures, namely equicorrelation structure and autoregressive structure, are most commonly used in the literature. For the autoregressive structure, the maximum likelihood estimator of the correlation parameter turns out to be a solution of a cubic equation. We prove that this cubic equation has a unique real root in (-1, 1). This proves the uniqueness of the MLE. Further, we notice that the autoregressive structure leads to Markov structure when the data are observed at unequally spaced time intervals. For the model with Markov covariance structure, we derive a formula for estimating a missing value and show that the estimator based on this formula depends on only two neighboring data values. The results for equicorrelation structure are included in Chapter 2 and those for the autoregressive structure (Markov structure as well) are included in Chapter 3.

Finally, in the fourth chapter we point out some draw backs of fitting the linear growth curve models to biological data and suggest fitting nonlinear models to growth data. After reviewing the popular nonlinear models, we show the analysis of nonlinear models with different covariance structures using SAS software.

## Rights

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## DOI

10.25777/rn0v-q848

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