Tuning Variable Selection Procedures and Score Tests for Dose Effect in the Presence of Non-Responders.

Abstract

There are two topics in this dissertation. The first topic is variable selection in linear regression, and the second topic is hypothesis testing in a regression setting with nonresponders. In the literature, there are many variable selection methods for linear regression whose performance depends critically on the stopping rule. But it appears that many of the rules used in practice do not adequately adapt to each particular data set. Thus we propose a general approach based on adding additional noise to the response variable that allows us to "tune" the stopping rule so that the selection method is not too greedy or too parsimonious and results in choosing a good model. We focus on a forward selection method due to an interest in handling large numbers of explanatory variables. Because the method is analytically intractable, we study it by Monte Carlo methods, compare it with some other methods, and find that it works very well except that it underfits models with large number of active predictors.

For a mixture model where both the logit of the response rate and the response mean are linear functions of the covariate (dose level), we propose new score test statistics for treatment effect. If the linear coefficient for the logit response rate is β and d is the linear coefficient for the mean, then score statistics are derived for H0: β = d = 0 versus H1,β: β≠0, d=0, H0 versus H1,d: d≠0, β=0, and H0 versus H1: β²+d²≠0, respectively. For H0 versus H1 we propose a 2-degree-of-freedom score statistic and also the maximum of the individual score statistics for H0 versus H1,β and H0 versus H1,d, respectively. Permutation critical values are used, and the tests are compared with the simple linear regression method. A simulation study shows that under most of the circumstances considered, the 2-degree-of-freedom statistic gives the best performance, while the simple linear regression is very sensitive to the response rate. The five methods are also applied to several real data sets, and the 2-degree-of-freedom score statistic provides satisfactory results.