Date of Award

Summer 7-8-2014

Document Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Vanja M. Dukic

Second Advisor

David M. Bortz

Third Advisor

Jem N. Corcoran

Abstract

The multiresolution estimator, originally a wavelet-based method for density estimation, was recently extended for estimation of hazard functions. The multiresolution hazard (MRH) method's main advantage is its multiscale property, making simultaneous modeling and inference at multiple time scales possible. Additional advantages, stemming from its Bayesian foundation, are its simple computational implementation, estimation and inference procedures, and ability to easily quantify the uncertainty in hazard function estimates (via point-wise or curve-wise credible bands) adjusted for uncertainty in other model parameters, such as covariate effects. In this dissertation, we further extend the MRH methodology to accommodate the case of varying smoothness in the hazard function over time. The proposed pruned multiresolution hazard (PMRH) performs data-driven "fusing" of adjacent hazard intervals, increasing computational efficiency and reducing uncertainty in hazard rate estimation over regions with low event counts. We apply the PMRH method to examine patterns of failure after treatment for prostate cancer, using data from a large-scale randomized clinical trial.

Additionally, one of the main goals of survival analysis centers around how predictors affect the hazard function, and today, more and more datasets have time-varying predictors and biomarkers, which are functions of time. We extend the MRH methodology to handle time-varying covariates. We study several missingness scenarios, and conclude that when there is no missing data our MRH models perform well and efficiently with time-varying covariates as well. When the amount of missing time-varying covariates increases, our results show how increasing L2 norm of the predictor function minus its mean within an interval is related to the bias and variance in the MRH model parameter estimators.

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