Spatially dependent polya tree modeling for survival data

Abstract

With the proliferation of spatially oriented time-to-event data, spatial modeling in the survival context has received increased recent attention. A traditional way to capture a spatial pattern is to introduce frailty terms in the linear predictor of a semiparametric model, such as proportional hazards or accelerated failure time. We propose a new methodology to capture the spatial pattern by assuming a prior based on a mixture of spatially dependent Polya trees for the baseline survival in the proportional hazards model. Thanks to modern Markov chain Monte Carlo (MCMC) methods, this approach remains computationally feasible in a fully hierarchical Bayesian framework. We compare the spatially dependent mixture of Polya trees (MPT) approach to the traditional spatial frailty approach, and illustrate the usefulness of this method with an analysis of Iowan breast cancer survival data from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute. Our method provides better goodness of fit over the traditional alternatives as measured by log pseudo marginal likelihood (LPML), the deviance information criterion (DIC), and full sample score (FSS) statistics.

Figure 1

Estimated 10th percentiles of survival (months) for women with mean diagnosis age and local stage of disease, Models 1, 1A, 2 and 3.

Figure 2

Posterior mean and standard deviation of 10th and 40th percentiles of survival (months) for women with mean diagnosis age and local stage of disease, Model 1.

Figure 3

Fitted predictive hazard of women with mean diagnosis age and local stage of breast cancer from Models 1, 1A, and 2.

Figure 4

True and estimated densities. Columns 1, 2, 3 are the simulations and rows 1 – 7 are the counties. Thick solid is the truth, thin solid CAR MPT, dashed HBF-frailty, and dots are CAR B-spline.