Bayesian Noninferiority Testing and Nonparametric Survival Regression using Bernstein Polynomials

In this dissertation we study two distinctly important topics in biostatistics. In both, Bernstein polynomials are used as building blocks for the statistical models. The first part involves the development of a new Bayesian procedure for hypothesis testing of noninferiority. A semiparametric testing approach based on Bayes factor is developed for non-inferiority trials with binary endpoints. The proposed method is shown to work for a broad class of hypotheses by accommodating a variety of dissimilarity measures between two binomial parameters. Two of the unique features of the proposed testing procedure include: (i) construction of a flexible class of conjugate priors using a mixture of Beta densities to maintain approximate equality of prior probabilities of the competing hypotheses; and (ii) automatic determination of the cut-off value of the Bayes factor to facilitate the decision making process. In contrast to the use of Jeffreys' rule of thumb, two forms of total weighted average error criteria are used to determine the cut-off value. The second part of the dissertation focuses on nonparametric regression models for right-censored data. We present a new nonparametric regression model for the conditional hazard rate using a suitable sieve of Bernstein polynomials. The proposed nonparametric methodology has three key features: (i) the smooth estimator of the conditional hazard rate is shown to be a unique solution of a strictly convex optimization problem for a wide range of applications; making it computationally attractive, (ii) the model is shown to nest the popular Cox proportional hazard model, and (iii) large sample properties including consistency and convergence rates are established under a set of mild regularity conditions. Finally, we proposed a Bayesian nonparametric model for the conditional hazard function based on Bernstein polynomials with varying degrees. The most important feature of the proposed Bayesian method is its capability of modeling the uncertainty about the order of Bernstein polynomials using reversible jump Markov Chain Monte Carlo (RJ-MCMC). This has been a great challenge for the proposed method from a frequentist perspective because of the lack of cross-validation procedures for censored data in regression settings.