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.

DNA methylation plays a crucial role in human health, especially cancer. Traditional DNA methylation analysis aims to identify CpGs/genes with differential methylation (DM) between experimental groups. Differential variability (DV) was recently observed that contributes to cancer heterogeneity and was also shown to be essential in detecting early DNA methylation alterations, notably epigenetic field defects. Moreover, studies have demonstrated that environmental factors may modify the effect of DNA methylation on health outcomes, or vice versa. Therefore, this dissertation seeks to develop new statistical methods for epigenetic data focusing on DV and interactions when efficient analytical tools are lacking. First, as neighboring CpG sites are usually highly correlated, we introduced a new method to detect differentially methylated regions (DMRs) that uses combined DM and DV signals between diseased and non-diseased groups. Next, using both DM and DV signals, we considered the problem of identifying epigenetic field defects, when CpG-site-level DM and DV signals are minimal and hard to be detected by existing methods. We proposed a weighted epigenetic distance-based method that accumulates CpG-site-level DM and DV signals in a gene. Here DV signals were captured by a pseudo-data matrix constructed using centered quadratic methylation measures. CpG-site-level association signal annotations were introduced as weights in distance calculations to up-weight signal CpGs and down-weight noise CpGs to further boost the study power. Lastly, we extended the weighted epigenetic distance-based method to incorporate DNA methylation by environment interactions in the detection of overall association between DNA methylation and health outcomes. A pseudo-data matrix was constructed with cross-product terms between DNA methylation and environmental factors that is able to capture their interactions. The superior performance of the proposed methods were shown through intensive simulation studies and real data applications to multiple DNA methylation data.

Because of the measuring instrument or the biological variability, many studies with survival data involve covariates which are subject to measurement error. In such cases, the naive estimates are usually biased. In this thesis, we propose a bias corrected estimate of the regression parameter for the multinomial probit regression model with covariate measurement error. Our method handles the case when the response variable is subject to interval censoring, a frequent occurrence in many medical and health studies where patients are followed periodically. A sandwich estimator for the variance is also proposed. Our procedure can be generalized to general measurement error distribution as long as the first four moments of the measurement error are known. The results of extensive simulations show that our approach is very effective in eliminating the bias when the measurement error is not too large relative to the error term of the regression model. / Censoring is an intrinsic part in survival analysis. In this thesis, we establish the asymptotic properties of MMLE to general transformation models when data is subject to right or left censoring. We show that MMLE is not only consistent and asymptotically normal, but also asymptotically efficient. Thus our asymptotic results give a definite answer to a long-term argument on the efficiency of the maximum marginal likelihood estimator. The difficulty in establishing these results comes from the fact that the score function derived from the marginal likelihood does not have ordinary independence or martingale structure. We will develop a discretization method in establishing our results. As a special case, our results imply the consistency, asymptotic normality and efficiency for the multinomial probit regression, a popular alternative to the Cox regression model. / General transformation model is an important family of semiparametric models in survival analysis which generalizes the linear transformation model. It not only includes typical Cox regression model, proportional odds model and multinomial probit regression model, but also includes heteroscedastic hazard regression model, general heteroscedastic rank regression model and frailty model. By maximizing the marginal likelihood, a parameter estimation (MMLE) can be obtained with the property that it avoids estimating the baseline survival function and censoring distribution, and such property is enjoyed by the Cox regression model. In this thesis, we study three areas of generalization of general transformation models: main response variable is subject to censoring, covariates are time-varying and covariates are subject to measurement error. / In medical studies, the covariates are not always the same during the whole period of study. Covariates may change at certain time points. For example, at the beginning, n patients accept drug A as treatment. After certain percentage of patients have died, the investigator might add new drug B to the rest of the patients. This corresponds to the case of time-varying covariates. In this thesis, we propose an estimation procedure for the parameters in general transformation model with this type of time-varying covariates. The results of extensive simulations show that our approach works well. / Wu, Yueqin. / Adviser: Ming Gao Gu. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3589. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 74-78). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Tsui Yuen-Yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 53-56). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Parametric Regressions in Survival Analysis --- p.6 / Chapter 2.1 --- Introduction --- p.6 / Chapter 2.2 --- Exponential Regression --- p.7 / Chapter 2.3 --- Weibull Regression --- p.8 / Chapter 2.4 --- Maximum Likelihood Method --- p.9 / Chapter 2.5 --- Diagnostic --- p.10 / Chapter 3 --- Local Influence --- p.13 / Chapter 3.1 --- Introduction --- p.13 / Chapter 3.2 --- Development --- p.14 / Chapter 3.2.1 --- Normal Curvature --- p.14 / Chapter 3.2.2 --- Conformal Normal Curvature --- p.15 / Chapter 3.2.3 --- Q-displacement Function --- p.16 / Chapter 3.3 --- Perturbation Scheme --- p.17 / Chapter 4 --- Examples --- p.21 / Chapter 4.1 --- Halibut Data --- p.21 / Chapter 4.1.1 --- The Data --- p.22 / Chapter 4.1.2 --- Initial Analysis --- p.23 / Chapter 4.1.3 --- Perturbations of σ around 1 --- p.23 / Chapter 4.2 --- Diabetic Data --- p.30 / Chapter 4.2.1 --- The Data --- p.30 / Chapter 4.2.2 --- Initial Anaylsis --- p.31 / Chapter 4.2.3 --- Perturbations of σ around σ --- p.31 / Chapter 5 --- Conclusion Remarks and Further Research Topic --- p.35 / Appendix A --- p.38 / Appendix B --- p.47 / Bibliography --- p.53

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Adelaide, Waite Agricultural Research Institute, 1984. / Includes bibliographical references (leaves 250-264).

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [100]-103).

Thesis (PhD)(Actuarial Science) -- University of Pretoria, 2002. / Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2003. / Title from first page of PDF file. Document formatted into pages; contains xiv, 113 p.; also includes graphics Includes bibliographical references (p. 108-113). Available online via OhioLINK's ETD Center

Thesis (M. Phil.)--University of Hong Kong, 1999. / Includes bibliographical references (leaves 85-88).