Title: Flexible Regression Models for Estimating Interactions between a Treatment and Scalar/Functional Predictors
Description: In this dissertation, we develop regression models for estimating interactions between a treatment variable and a set of baseline predictors in their eect on the outcome in a randomized trial, without restriction to a linear relationship. The proposed semiparametric/nonparametric regression approaches for representing interactions generalize the notion of an interaction between a categorical treatment variable and a set of predictors on the outcome, from a linear model context. In Chapter 2, we develop a model for determining a composite predictor from a set of baseline predictors that can have a nonlinear interaction with the treatment indicator, implying that the treatment efficacy can vary across values of such a predictor without a linearity restriction. We introduce a parsimonious generalization of the single-index models that targets the eect of the interaction between the treatment conditions and the vector of predictors on the outcome. A common approach to interrogate such treatment-by-predictor interaction is to t a regression curve as a function of the predictors separately for each treatment group.

For parsimony and insight, we propose a single-index model with multiple-links that estimates a single linear combination of the predictors (i.e., a single-index), with treatment-specic nonparametrically-dened link functions. The approach emphasizes a focus on the treatment-by-predictors interaction eects on the treatment outcome that are relevant for making optimal treatment decisions. Asymptotic results for estimator are obtained under possible model misspecication. A treatment decision rule based on the derived single-index is dened, and it is compared to other methods for estimating optimal treatment decision rules. An application to a clinical trial for the treatment of depression is presented to illustrate the proposed approach for deriving treatment decision rules. In Chapter 3, we allow the proposed single-index model with multiple-links to have an unspecified main effect of the predictors on the outcome. This extension greatly increases the utility of the proposed regression approach for estimating the treatment-by-predictors interactions.

By obviating the need to model the main eect, the proposed method extends the modied covariate approach of [Tian et al., 2014] into a semiparametric regression framework. Also, the approach extends [Tian et al., 2014] into general K treatment arms. In Chapter 4, we introduce a regularization method to deal with the potential high dimensionality of the predictor space and to simultaneously select relevant treatment effect modiers exhibiting possibly nonlinear associations with the outcome. We present a set of extensive simulations to illustrate the performance of the treatment decision rules estimated from the proposed method. An application to a clinical trial for the treatment of depression is presented to illustrate the proposed approach for deriving treatment decision rules. In Chapter 5, we develop a novel additive regression model for estimating interactions between a treatment and a potentially large number of functional/scalar predictor. If the main effect of baseline predictors is misspecied or high-dimensional (or, innite dimensional), any standard nonparametric or semiparametric approach for estimating the treatment-bypredictors interactions tends to be not satisfactory because it is prone to (possibly severe) inconsistency and poor approximation to the true treatment-by-predictors interaction effect.

To deal with this problem, we impose a constraint on the model space, giving the orthogonality between the main and the interaction effects. This modeling method is particularly appealing in the functional regression context, since a functional predictor, due to its infinite dimensional nature, must go through some sort of dimension reduction, which essentially involves a main effect model misspecication. The main effect and the interaction effect can be estimated separately due to the orthogonality between the two effects, which side-steps the issue of misspecication of the main effect. The proposed approach extends the modied covariate approach of [Tian et al., 2014] into an additive regression model framework. We impose a concave penalty in estimation, and the method simultaneously selects functional/scalar treatment effect modifiers that exhibit possibly nonlinear interaction effects with the treatment indicator. The dissertation concludes in Chapter 6.