Bayesian order restricted methods with biomedical applications

This dissertation focuses on Bayesian order restricted inference, with interest in applying new methodology to biomedical examples. The first section considers samples of curves restricted to follow a particular shape. For example, progesterone levels in healthy women increase during the menstrual cycle to a random peak with decreases thereafter. Reproductive epidemiologists are interested in studying the distribution of the peak and the trajectory for women in different groups. Motivated by this application, we propose a simple approach for restricting each woman's mean trajectory to follow an umbrella shape. An unconstrained hierarchical Bayesian model is used to characterize the data, and draws from the posterior distribution obtained using a Gibbs sampler are then mapped to the constrained space. Inferences are based on the resulting posterior distribution for the peak and individual woman trajectories. Methods are applied to a study comparing progesterone trajectories for conception and non-conception cycles.; The second section addresses studies that collect event time data in which it is often appropriate to assume non-decreasing hazards across dose groups, though dose effects may vary with time. Motivated by this application, we propose a Bayesian approach for order restricted inference using a non-proportional hazards model with time-varying coefficients. In order to make inferences on equalities versus increases in hazard functions, a prior is chosen for the time-varying coefficients that assigns positive probability to no dose effect while restricting coefficients to be non-negative. By using a high dimensional piecewise constant model and smoothing functions by coupling Markov beta and gamma processes, we obtain a flexible and computationally tractable approach for identifying sets of dose and age values at which hazards increase. This approach can also be used to estimate dose response and survival curves. The methods are illustrated through application to data from a toxicology study.