Shape Constrained Single Index Models for Biomedical Studies

• Other Authors: Dhara, Kumaresh (author)
• Format: Others
• Language: English; English
• Published: Florida State University
• Subjects: Statistics
• Online Access: http://purl.flvc.org/fsu/fd/2018_Su_Dhara_fsu_0071E_14739

For many biomedical, environmental and economic studies with an unknown non-linear relationship between the response and its multiple predictors, a single index model provides practical dimension reduction and good physical interpretation. However widespread uses of existing Bayesian analysis for such models are lacking in biostatistics due to some major impediments including slow mixing of the Markov Chain Monte Carlo (MCMC), inability to deal with missing covariates and a lack of theoretical justification of the rate of convergence. We present a new Bayesian single index model with associated MCMC algorithm that incorporates an efficient Metropolis Hastings (MH) step for the conditional distribution of the index vector. Our method leads to a model with good biological interpretation and prediction, implementable Bayesian inference, fast convergence of the MCMC, and a first time extension to accommodate missing covariates. We also obtain for the first time, the set of sufficient conditions for obtaining the optimal rate of convergence of the overall regression function. We illustrate the practical advantages of our method and computational tool via re-analysis of an environmental study. I have proposed a frequentist and a Bayesian methods for a monotone single-index models using the Bernstein polynomial basis to represent the link function. The monotonicity of the unknown link function creates a clinically interpretable index, along with the relative importance of the covariates on the index. We develop a computationally-simple, iterative, profile likelihood-based method for the frequentist analysis. To ease the computational complexity of the Bayesian analysis, we also develop a novel and efficient Metropolis-Hastings step to sample from the conditional posterior distribution of the index parameters. These methodologies and their advantages over existing methods are illustrated via simulation studies. These methods are also used to analyze depression based measures among adolescent girls. === A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. === Summer Semester 2018. === July 13, 2018. === Adolescent Depression, Gaussian Processes, Markov Chain Monte Carlo, Mode Aligned Proposal Density, Monotone Single Index Models, Single Index Models === Includes bibliographical references. === Debajyoti Sinha, Professor Co-Directing Dissertation; Debdeep Pati, Professor Co-Directing Dissertation; Greg Hajcak, University Representative; Elizabeth Slate, Committee Member; Eric Chicken, Committee Member.