HETEROSCEDASTIC UNBALANCED NESTED DESIGNS AND FULLY NONPARAMETRIC ANALYSIS OF COVARIANCE
Open Access
- Author:
- Liao, Shu-Min
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 08, 2009
- Committee Members:
- Michael G Akritas, Dissertation Advisor/Co-Advisor
Michael G Akritas, Committee Chair/Co-Chair
Rana Arnold, Committee Member
Francesca Chiaromonte, Committee Member
Vernon Michael Chinchilli, Committee Member - Keywords:
- nested design
ancova
fully nonparametric model
statistical testing
heteroscedasticity
asymptotic theory - Abstract:
- Analysis of variance is a corner stone of statistical applications. The classical asymptotic results were built either under the normality and homoscedasticity assumptions, or on cases when the numbers of factor levels are all fixed. However, the past decade has witnessed the generation of large data sets which involve a multitude of factor levels while the number of replications per factor combination is very small. The asymptotic theory is considerably more complicated when testing against those high-dimensional alternatives. In the first part of this thesis, we consider the problem of testing for the sub-class effect in the unbalanced two-fold nested models with a large number of sub-classes. It is shown that the classical F-statistic is very sensitive to departures from homoscedasticity, even in balanced designs. We propose new testing procedures to accommodate heteroscedasticity, and the asymptotic distributions of the proposed test statistics, both under the null and local alternative hypotheses, are established. Simulation studies examine the finite sample performance of the proposed statistics and the competing classical F-test. Two real data sets are analyzed and ramifications of these results to the hypothesis of no covariate effect in the analysis of covariance are discussed, which leads to a more sophisticated approach described in the second part of the thesis. Testing for the class effect is also investigated. In the second part of this thesis, we introduce a new approach for testing the covariate effect in the context of the fully nonparametric ANCOVA model which capitalizes on the connection to the testing problems in nested designs. The basic idea behind the proposed method is to think of each distinct covariate value as a level of a sub-class nested in each group/class. A projection-based tool is developed to obtain a new class of quadratic forms, whose asymptotic behavior is then studied to establish the limiting distributions of the proposed test statistic under the null hypothesis and local alternatives. Simulation studies show that this new method, compared with existing alternatives, has better power properties and achieves the nominal level under violations of the classical assumptions. Three data sets are analyzed, and asymptotic results concerning testing for the covariate-adjusted group effect are also included.