Nonparametric tests to detect relationship between variables in the presence of heteroscedastic treatment effects

Abstract

Statistical tools to detect nonlinear relationship between variables are commonly needed in various practices. The first part of the dissertation presents a test of independence between a response variable, either discrete or continuous, and a continuous covariate after adjusting for heteroscedastic treatment effects. The method first involves augmenting each pair of the data for all treatments with a fixed number of nearest neighbors as pseudo-replicates. A test statistic is then constructed by taking the difference of two quadratic forms. Using such differences eliminate the need to estimate any nonlinear regression function, reducing the computational time. Although using a fixed number of nearest neighbors poses significant difficulty in the inference compared to when the number of nearest neighbors goes to infinity, the parametric standardizing rate is obtained for the asymptotic distribution of the proposed test statistics. Numerical studies show that the new test procedure maintains the intended type I error rate and has robust power to detect nonlinear dependency in the presence of outliers. The second part of the dissertation discusses the theory and numerical studies for testing the nonparametric effects of no covariate-treatment interaction and no main covariate based on the decomposition of the conditional mean of regression function that is potentially nonlinear. A similar test was discussed in Wang and Akritas (2006) for the effects defined through the decomposition of the conditional distribution function, but with the number of pseudo-replicates going to infinity. Consequently, their test statistics have slow convergence rates and computational speeds. Both test limitations are overcome using new model and tests. The last part of the dissertation develops theory and numerical studies to test for no covariate-treatment interaction, no simple covariate and no main covariate effects for cases when the number of factor levels and the number of covariate values are large.