Testing in multifactor heteroscedastic ANOVA and repeated measures designs with large number of levels
Open Access
- Author:
- Wang, Haiyan
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- April 29, 2004
- Committee Members:
- Michael G Akritas, Committee Chair/Co-Chair
Rana Arnold, Committee Member
Bing Li, Committee Member
Vernon Michael Chinchilli, Committee Member - Keywords:
- repeated measurements
ANOVA
longitudinal or funcitonal data
nonparametric procedures
large dimentional data
Rank tests - Abstract:
- Testing methods for factorial designs with independent or dependent observations where some of the factors have a large number of levels have received a lot of attention recently. Most results for independent data in the literature have been restricted to procedures using the original observations for the balanced homoscedastic case, which require strong moment assumptions and are sensitive to outliers. The results in the literature for dependent data were extensively studied in parametric, nonparametric and semiparametric, and Bayesian models but all that do inference require large sample sizes or the normality assumption. The first part of my thesis considers the use of rank statistics as robust alternatives for testing hypotheses in balanced and unbalanced, homoscedastic and heteroscedastic one-way and two-way ANOVA models when the number of levels of at least one factor is large. The second part of my thesis deals with various testing problems in possibly unbalanced and heteroscedastic multi-factor designs with arbitrary but fixed number of factors when at least one of the factors have large number of factor levels. Procedures based on both original observations and their (mid-)ranks are presented for the same general setting. The first two parts pertain to independent data. The third part of my thesis is focused on testing hypotheses in functional data, a fully nonparametric method for evaluating the effect of several crossed factors on the curve and their interactions with time. The asymptotics, which rely on the large number of measurements per curve (subject) and not on large group sizes, hold under the general assumption of $alpha$-mixing without specifying the covariance structure, and do not require the measurements to be continuous or homoscedastic. A competing set of (mid-)rank procedures is also developed. The procedures in all three parts can be applied to both continuous and discrete ordinal observations. The rank tests are robust to outliers, invariant under monotone transformations, and do not require any restrictive moment conditions. Simulation studies reveal that the (mid-)rank procedures outperform those based on the original observations in all non-normal situations while they do not lose much power when normality holds. Applications to several data sets are given and potential extensions in several directions are discussed.