Shape Detection and Mediation Analysis using Semi-parametric Shape-Restricted Regression Spline with Applications

Linear models are widely used in the field of epidemiology to model the relationship between two continuous variables, such as circulating levels of the placental hormone hCG and infant genital size. When researchers suspect curvilinear relationship exists, some nonparametric techniques can be used to model the relationship. By applying nonparametric techniques, researchers can relax the linearity assumption and capture scientifically meaningful or appropriate shapes. In the first part of the dissertation, a shape detection method based on regression splines is developed. The proposed method can help researchers select the most suitable shape to describe their data among increasing, decreasing, convex and concave shapes. Specifically, we develop a technique based on mixed effects regression spline to analyze hormonal data, but the method is general enough to be applied to other similar problems. Analyzing the association between two variables is usually the first step of some research project. Researchers also want to explore the causal relationship between an exposure and a potential outcome caused by the exposure. In many cases, the exposure may not directly lead to the outcome, but instead, it induces the outcome through a process. Mediation analysis is designed to explain the causal relationship between the exposure and the outcome by examining the intermediate stage, which helps researchers understand the pathway whereby the exposure affects the outcome. In the second part of the dissertation, we develop a method to analytically estimate the direct and indirect effects when we have some prior knowledge on the relationship between the mediator and the outcome (increasing, decreasing, convex or concave). In order to make suitable inferences, the asymptotic confidence intervals of those effects are obtained via delta method. Public health significance: The shape detection technique can help researchers make judgements on the potential relationship between the exposure and the outcome while controlling for confounders. With such judgements, researchers can avoid the bias caused by model misspecification when building models. The regression-based mediation analysis within the shape-restricted framework offers researchers a flexible and efficient approach to perform the causal inference. The method helps researchers estimate causal effects using reasonable models