Indiana University-Purdue University Indianapolis (IUPUI) / A semi-parametric approach incorporates parametric and nonparametric functions in the model and is very useful in situations when a fully parametric model is inadequate. The objective of this dissertation is to extend statistical methodology employing the semi-parametric modeling approach to analyze data in health science research areas. This dissertation has three parts. The first part discusses the modeling of the dose-response relationship with correlated data by introducing overall drug effects in addition to the deviation of each subject-specific curve from the population average. Here, a penalized spline regression method that allows modeling of the smooth dose-response relationship is applied to data in studies monitoring malaria drug resistance through the ex-vivo assays.The second part of the dissertation extends the SiZer map, which is an exploratory and a powerful visualization tool, to detect underlying significant features (increase, decrease, or no change) of the curve at various smoothing levels. Here, Penalized Spline Significant Zero Crossings of Derivatives (PS-SiZer), using a penalized spline regression, is introduced to investigate significant features in correlated data arising from longitudinal settings. The third part of the dissertation applies the proposed PS-SiZer methodology to analyze HIV data. The durability of significant weight change over a period is explored from the PS-SiZer visualization. PS-SiZer is a graphical tool for exploring structures in curves by mapping areas where rate of change is significantly increasing, decreasing, or does not change. PS-SiZer maps provide information about the significant rate of weigh change that occurs in two ART regimens at various level of smoothing. A penalized spline regression model at an optimum smoothing level is applied to obtain an estimated first-time point where weight no longer increases for different treatment regimens.

In this thesis, Bayesian semiparametric models for the missing data mechanisms of nonignorably missing covariates in logistic regression are developed. In the missing data literature, fully parametric approach is used to model the nonignorable missing data mechanisms. In that approach, a probit or a logit link of the conditional probability of the covariate being missing is modeled as a linear combination of all variables including the missing covariate itself. However, nonignorably missing covariates may not be linearly related with the probit (or logit) of this conditional probability. In our study, the relationship between the probit of the probability of the covariate being missing and the missing covariate itself is modeled by using a penalized spline regression based semiparametric approach. An efficient Markov chain Monte Carlo (MCMC) sampling algorithm to estimate the parameters is established. A WinBUGS code is constructed to sample from the full conditional posterior distributions of the parameters by using Gibbs sampling. Monte Carlo simulation experiments under different true missing data mechanisms are applied to compare the bias and efficiency properties of the resulting estimators with the ones from the fully parametric approach. These simulations show that estimators for logistic regression using semiparametric missing data models maintain better bias and efficiency properties than the ones using fully parametric missing data models when the true relationship between the missingness and the missing covariate has a nonlinear form. They are comparable when this relationship has a linear form.