Applied Bayesian Modelling - Free

384 SURVIVAL AND EVENT HISTORY MODELS9.5 ACCOUNTING FOR FRAILTY IN EVENT HISTORY AND SURVIVALMODELSWhether the event history or survival analysis is in discrete or continuous time, unobserveddifferences between subjects may be confounded with the estimated survivalcurve and the estimated impacts of observed covariates. While there is considerabledebate regarding sensitivity of inferences to the specification of unobserved heterogeneity,this is an important aspect to consider, especially in complex models with timevaryingeffects of predictors or clustering of subjects.Thus, frailty differences, whether modelled by parametric random effects or by nonparametricmethods, provide a way to account for within-cluster correlations in eventhistory outcomes (Guo and Rodriguez, 1992), or for multivariate survival times wherean underlying common influence is present (Keiding et al., 1997). Suppose subjects(patients, children) are arranged within aggregate units or clusters (hospitals, families)and event times are affected by cluster characteristics, known and unknown, as well asby the characteristics of individuals. Thus, for survival after surgery, patients areclustered within hospitals, while for age at pre-marital maternity, adolescent femalesare clustered according to family of origin (Powers and Xie, 2000). In these examples,random effects at cluster level are intended to account for unmeasured differencesbetween clusters that may affect the outcome at the subject level.Example 9.11 Unmeasured heterogeneity and proportional hazards in discrete timeregression McCall (1994) discusses methods for assessing the proportional hazardsassumption in discrete time regression in a single level example (with no clustering), inthe context of data on joblessness durations in months. He considers tests for timevarying coefficients b j in Equation (9.15), which are equivalent to testing the proportionalhazards assumption in the underlying continuous time model (e.g. Kay, 1977;Cox, 1972). In particular, he considers the need to allow for unmeasured heterogeneitywhen applying such tests.McCall uses simulated data based on the real joblessness example, with a sample ofn ˆ 500 persons observed for a maximum of 60 intervals. An underlying continuoushazard is assumed with l 0 (t) ˆ 0:07, and an individual level gamma frailty u i modifyingthe corresponding discrete time event probabilities, and distributed with mean andvariance of 1, namely u i G(1, 1). Thus, for subject i at interval j, the most generalmodel involves time varying predictor x and z and time varying regression coefficientsb 1 and b 2 :h ij ˆ [1 exp ( u i exp {g j b 1j x ij b 2j z ij })]andS ij ˆ exp!X j1u i exp (g k b 1k x ik b 2k z ik )kˆ1There is a concurrent 0.01 probability of being right-censored in any interval (e.g. viaemigration or death in the joblessness example).Here we consider only one of the scenarios of McCall, involving fixed predictors xand z, with fixed coefficients b 1 and b 2 , and no trend in the g j . For n ˆ 100, thecovariates are generated via