Extensions to profile analysis
Date
1986
Authors
Verbyla, Arunas Petras
Editors
Advisors
Venables, W. N.
Journal Title
Journal ISSN
Volume Title
Type:
Thesis
Citation
Statement of Responsibility
Conference Name
Abstract
The analysis of growth curve data is of wide practical importance. In this thesis we consider aspects of the statistical analysis of profile data of which growth curves form an important special case.
Chapter 1 provides an overview of the area and a more complete description of this work.
Chapter 2 contains a review of the basic model, namely the generalized multivariate analysis of variance model or growth curve model of Potthoff and Roy ( 1964 ). We call this model the profile model to indicate its wider applicability. Some new work is included in this chapter but in the main the results may be found in the literature.
In Chapter 3 we consider the covariance structure which arises from the autoregressive process. A closed form for the inverse covariance matrix is found and the likelihood ratio test for autoregressive covariance matrix presented. A partition of the likelihood ratio test into components which examine aspects of the overall hypothesis is also given. While the results of this chapter are incidental to the main subject of the thesis, these results provide the means for checking the assumption of autoregressive covariance structure for profile data.
The main contributions are given in Chapters 4 and 5.
The analysis of the profile model involves a conditioning argument, and reducing the number of conditioning variables can become an important practical consideration. In Chapter 4 it is shown that reducing the number of conditioning variables depends on the relationship between the inverse covariance matrix and the form of the profile model. Conditions on the profile model are found so that a reduction in the number of
conditioning variables can be achieved when the covariance matrix is of autoregressive form.
The analysis of a model which is the sum of terms of the form of the standard profile model is given in Chapter 5. The model arises naturally in practical situations and may sometimes be used for the analysis of longitudinal data from designed experiments. It is shown that a canonical form for the model is the so - called seemingly unrelated regressions model ( Zellner, 1962 ) provided certain conditions hold. If this canonical reduction is possible, both the examination of identifiability and estimation are handled easily. Tests of hypotheses are also discussed.
We conclude in Chapter 6 by considering non - linear models. It turns out that the linearised form can be written as the sum of profiles model of Chapter 5. This is compared with maximum likelihood and generalized least squares based on a linear approximation of the non - linear model. The problem of finding confidence regions for non - linear parameters in partially non - linear models is considered. The use of non - linear models within a sum of profiles model is also discussed.
School/Discipline
Department of Statistics
Dissertation Note
Thesis (Ph.D.)--Department of Statistics, 1986.