Author ORCID Identifier

https://orcid.org/0000-0002-1123-7228

Year of Publication

2021

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Statistics

First Advisor

Dr. Solomon W. Harrar

Abstract

Multivariate growth curve data naturally arise in various fields, for example, biomedical science, public health, agriculture, social science and so on. For data of this type, the classical approach is to conduct multivariate analysis of variance (MANOVA) based on Wilks' Lambda and other multivariate statistics, which require the assumptions of multivariate normality and homogeneity of within-cell covariance matrices. However, data being analyzed nowadays show marked departure from multivariate normal distribution and homoscedasticity. In this dissertation, we investigate nonparametric testing approaches for multivariate growth curve data from three aspects, i.e., finite-sample, resampling and rank-based methods.

The first project proposes an approximate finite-sample test using modified sums of squares matrices to make them insensitive to the heterogeneity in MANOVA. The modification corrects the associated quadratic forms of the two sums of squares for the effect of heterogeneity. The distribution of the proposed test statistic is invariant to the original data distribution. The proposed approximation method can be used in various experimental designs, for example, factorial design and crossover design. Under various simulation settings, the proposed method outperforms the classical Doubly Multivariate Model and Multivariate Mixed Model, especially for unbalanced sample sizes. The applications of the proposed method are illustrated with ophthalmology data in a factorial design and in a 2 × 2 crossover design.

In the semiparametric situation, parametric and nonparametric bootstraps are known to have satisfactory finite-sample performance in general factorial designs. In this regard, the second project provides resampling-based tests for multivariate growth curve data. Such tests are useful in situations where data are not necessarily exchangeable under the null hypothesis of interest and with small sample sizes. Simulation studies are conducted to evaluate the finite-sample performance of the proposed test procedures under various practical scenarios. Data from an optometry study are used to illustrate the benefits of the nonparametric methods proposed.

For multivariate growth curve data which are measured in ordered categorical scales, the usual mean- and covariance-based inferences are not appropriate anymore. The third project deals with general nonparametric methods for multivariate growth curve data in factorial designs. Treatment effects are characterized in terms of functionals of distribution functions with the sole assumption of nondegenerate marginal distributions. This model accommodates binary, discrete, ordered categorical, and continuous data in a unified manner. Hypotheses are formulated in terms of meaningful nonparametric measures of treatment effects. In this project, the Wald-type statistic is proposed and its asymptotic properties are investigated. In addition, the ANOVA-type statistic and the modified Wilks' Lambda statistic under the nonparametric framework are also presented. The theory can be used to construct confidence intervals for the nonparametric treatment effects. Simulation studies are conducted to show the finite-sample performance of the proposed methods in comparison with other parametric and nonparametric methods. Data from a study of infantile nystagmus syndrome (INS) are analyzed to illustrate the application of the proposed methods.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2021.261

Available for download on Sunday, July 23, 2023

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