Modeling Spatial Survival Data Using Semiparametric Frailty Models

Abstract

We propose a new class of semiparametric frailty models for spatially correlated survival data. Specifically, we extend the ordinary frailty models by allowing random effects accommodating spatial correlations to enter into the baseline hazard function multiplicatively. We prove identifiability of the models and give sufficient regularity conditions. We propose drawing inference based on a marginal rank likelihood. No parametric forms of the baseline hazard need to be assumed in this semiparametric approach. Monte Carlo simulations and the Laplace approach are used to tackle the intractable integral in the likelihood function. Different spatial covariance structures are explored in simulations and the proposed methods are applied to the East Boston Asthma Study to detect prognostic factors leading to childhood asthma.

Modelling geographically referenced survival data with a cure fraction

“…When multiple parameter sets (such as frailties in cure fractions and the Weibull link) need to be modelled jointly, as opposed to independently, we resort to multivariate CAR models originally proposed by Mardia 34 ; see also the works of Carlin and Banerjee35 and Gelfand and Vounatsou.36 Let be a vector of p variables associated with the ith region. Collecting these effects into , the joint distribution can be written down as (5) where B R is an np × np matrix with block elements (B R ) ij = R i B ij and 0 as diagonals, R i and B ij are p × p matrices, and Γ is an np × np block diagonal matrix with block elements Γ i , i = 1,…, n. Although the Kronecker structure offers computational and interpretational simplicity, there has been much research on developing more general spatial covariances, notably by Kim et al, 37 Carlin and Banerjee 35 and Jin et al 20 The last work offers a Generalized Multivariate Conditionally Auto Regressive (GMCAR) model with emphasis on computational simplicity.…”

Semiparametric Models: A Review of Progress since BKRW (1993)

“…Efficiency of the proposed inference procedures is not necessarily a goal here; e.g. Li and Ryan (2002). Murphy (1995) provided key asymptotic theory for maximum likelihood estimation methods in a basic gamma-frailty model, and her methods were extended to more complicated frailty models by Parner (1998).…”

spatsurv: An R Package for Bayesian Inference with Spatial Survival Models

“…In Henderson, Shimakura, and Gorst (2002), the authors use a proportional hazards (PH) framework based on Gamma frailties and the Breslow estimate of the hazard function (Breslow 1974) to model the survival times of individuals diagnosed with leukemia in the north west of England. In Li and Ryan (2002), the authors propose the use of log-Gaussian frailties in a semi-parametric framework. In Diva, Dey, and Banerjee (2008), the authors consider both PH and proportional odds (PO) models for the SEER (surveillance, epidemiology and end results) cancer data (Howlader, Noone, Krapcho, Garshell, Neyman, Altekruse, Kosary, Yu, Ruhl, Tatalovich, Cho, Mariotto, DR, and EJ 2013); the baseline hazard function of their parametric PH model is based on the Weibull survival model.…”

“…In each of the above cases, the authors used a Markov chain Monte Carlo (MCMC) algorithm to produce samples from a posterior, enabling them to perform Bayesian inference for each class of models. Li and Ryan (2002) also investigated the Laplace approximation as a means of approximate posterior inference.…”