In many clinical trials, experimenters gather data on different characteristics from individuals. Typically, a researcher is interested in analyzing measurements on one variable, which is called the response. The information from auxiliary variables that are measured with the response, called covariates, is used to construct suitable models for the response variable. A useful statistical tool in constructing models between several variables is multiple regression. Classical multiple regression techniques specify models up to some parameters and estimates the unknown parameters using data. This approach is not suitable in many situations and data analysts have resorted to nonparametric methods to construct multiple regression models. In this project we examine several issues related to nonparametric multiple regression methods. We begin by examining the problem of selecting the optimal number of covariates in a model. Most methods currently available for selecting the most informative variables require assumptions of special model and error structure. We propose to compare and fully develop a few available methods and propose novel selection criteria that are relatively free of restrictions on the model structure. We require that these criteria depend on formal testing methods as opposed to the few existing methods. Second, we study the problem of testing the equality of regression surfaces in a nonparametric setting. We attempt to develop optimal projection methods which avoid high dimensional smoothing. The existing methods would be extended to cases with non-identical covariate design and with unequal sample sizes. New test statistics for comparisons would be examined. As a portion of this research, the estimation of noise variance with multiple covariates is investigated. The results of this project will help users to implement nonparametric multiple regression techniques in an optimal manner. It will provide new tools for developing models and carrying out diagnostics and comparisons without stringent model assumptions.
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