(Texas Tech University, 1995-08) Robertson, Stephen D.

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Five different nonparametric approaches, based on ranks, for testing hypothesis in a general linear regression model will be considered, and comparisons will be made between the performances of each of these five statistics, under different hypotheses and conditions. Comparisons will also be made between these nonparametric approaches and the classical parametric F test in order to determine any possible significant differences and similarities between these theoretically different methods. Some versions of these comparisons for some of these nonparametric approaches have been made in Hettmansperger and McKean[l], but this new study considers cases not studied in their study and considers some new ideas that have not yet been presented.

The Bayes Factor is a widely-used summary measure that can be used to test hypotheses in a Bayesian setting. It also performs well in problems of model selection. In this study, Bayes Factors for variance components in the mixed linear model are derived. The formulation used avoids the assumption of a priori independence between the variance components by using a Dirichlet prior on the intraclass correlations. A reference prior, which results in a Bayes Factor that is flexible and easy to use, is suggested. Hypothesis tests using the Bayes Factor avoid difficulties of the classical tests, such as non-uniqueness and invalid asymptotics.
The priors on the nuisance parameters are chosen to be non-informative and the corresponding integrals are carried out analytically. For the parameters of interest, however, numerical methods have to be used. For this purpose, Monte Carlo methods have been investigated. Simple random sampling and Latin hypercube sampling are employed for simulating the prior and a Gibbs sampling scheme has been implemented for simulating the posterior. The resulting estimators are compared on a small data set.