Applications of generalized inference

Abstract

There are many statistical inference problems in mixed linear models for which exact solutions are not available. In these cases, standard statistical software packages such as SAS and S-PLUS either use asymptotic procedures or methods based on the Satterthwaite-type approximations. In this dissertation, it is shown that generalized inference (generalized pivotal quantity, generalized confidence interval, generalized test variable and generalized P-value) may be used as an alternative to asymptotic approximations or other small sample approximations. After presenting the definitions pertaining to generalized inference, we show how to extend the concept of generalized confidence intervals to simultaneous generalized confidence intervals. We also show how to extend generalized tests of hypotheses to multiparameter problems. Specifically, the following problems were addressed. 1. All pairwise comparisons of cell-means in unbalanced heterogeneous one-way ANOVA (Extended Tukey). 2. Pairwise comparisons of treatment means to a control mean in unbalanced heterogeneous one-way ANOVA (Extended Dunnett). 3. All cell-means in balanced two-factor crossed mixed linear model. 4. All pairwise comparisons of cell-means in balanced three-factor nested factorial mixed linear model. Other applications considered in this dissertation include the use of generalized confidence intervals to compare two non-nested linear models when the response and the predictors have jointly a multivariate normal distribution. Generalized tests were developed for the following problems. 1. Testing the equality of the cell means in unbalanced heterogeneous one-way ANOVA. 2. Testing the equality of the cell means in balanced three-factor crossed mixed linear model with interactions. Simulation studies were used to estimate the error rates for each problem considered. Comparisons to other existing procedures were carried out when appropriate.