In many clinical trials experimenters gather data on various different characteristics from individuals. Typically a researcher is interested in analyzing measurements on one variable and the information from auxiliary variables, called covariates, that are measured is used to construct suitable models for the variable of interest. One useful statistical tool in constructing models between several variables is multiple regression. Classical multiple regression techniques specify models up to some parameters and one has to estimate the unknown parameters in using such models. This approach is not suitable in many occasions 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 addressing the smoothing parameter selection problems for the existing methods. All methods require some smoothing parameters at one stage or another and the use of improper smoothing parameters can mask many important features of the data. Therefore optimal smoothing parameter selection becomes one of the key problems in implementing the known methods. We examine the empirical as well as theoretical properties of a few criteria that are known to work in general and propose new criteria that will take the individual covariate behavior into account. In the second stage we examine the problem of selecting the optimal number of covariates in a model. The methods currently available for selecting the most informative variables require some special structures and none of them have been studied in depth. We propose to investigate the properties of the available methods and propose selection criteria that are relatively free of restrictions on the model structure. At the same time we address the estimation of the model components in a more robust setting. Finally, we study the impact on testing the equality of regressors and the estimates of the differences between regressors when different models are used to approximate the true regression surface. At the same time, we propose new methods that avoid high dimensional estimation in the construction of test statistics. The results of this project will help a user to implement many nonparametric multiple regression techniques in an optimal manner. It will also provide some new tools for developing models and carrying out diagnostics without stringent model assumptions.
