Novel applications of and extensions to linear regression methods for the biomedical and materials sciences.

In this work we present three topics, each of which centered on either the application or modification of various linear regression methods. Our work with respect to the “Materials Genome” project while undermined by oversimplification and data integrity issues in its early stages, provides a sound platform from which the project can proceed successfully. Building upon a growing body of knowledge around the use of Weighted Generalized Estimating Equations (WGEE), our second investigation proposes an extension to that framework intended to address the inherent bias present in the analysis of clustered longitudinal data with potentially informative cluster sizes and temporal observation profiles. Having demonstrated the utility of our marginal WGEE’s with respect to mitigating induced bias our final investigation presents a comparison of our marginal WGEE’s to model estimation via Joint Likelihood maximization in certain simulation models. We find, as would be in line with expectation, comparable performance with a loss of efficiency in the marginal WGEE setting

Frailty Probit Models for Clustered Interval-Censored Failure Time Data

Survival analysis is an important branch of statistics that deals with time to event data or survival data. An important feature of such data is that the survival time of interest is usually not completely known but is censored due to the design of the study or an early dropout. In this dissertation we focus on studying clustered interval-censored data, a special type of survival data. Interval-censored data arise in many epidemiological, social science, and medical studies, in which subjects are examined at periodical follow-up visits. The survival (or failure) time of interest is never exactly observed but is known to fall within an interval formed by two examination times with changed status of the event of interest. Clustered intervalcensored data contributes another complication that the failure times within the same cluster are not independent. Chapter 1 of this dissertation provides a detailed description of interval-censored data with several real data examples and reviews existing regression models and approaches for clustered interval-censored data. Chapter 2 proposes a novel frailty Probit model for analyzing clustered intervalcensored data. The proposed model has several appealing properties: (1) the marginal covariate effects are proportional to the conditional effect and (2) the intra-cluster association can be quantified in terms of several nonparametric association measures in closed form. the proposed Bayesian estimation approach is easy to implement because all parameters and latent variables have their full conditionals in standard form. The approach has excellent performance in estimating the regression parameters and the baseline survival function and is also robust to misspecification of the frailty distribution. Chapter 3 extends the frailty Probit model in Chapter 2 to allow modeling both clustered and independent data through the adoption of a mixture distribution for the frailty. The proposed approach provides tests of the existence of intra-cluster association for each cluster via Bayes factors and can identify clusters with strong (weak) correlation. Two different prior structures are considered in our approach, and both lead to good estimation and testing results. Chapter 4 studies a joint modeling of clustered interval-censored failure times and the sizes of the clusters. The cluster size is modeled as an ordinal response using a parametric Probit model, and a separate frailty semiparametric Probit model is used to model the clustered failure times. The two submodels are connected through a shared random effect. The performance of the proposed model is evaluated through a simulation study