survival times

The Cox proportional hazards model is the most commonly used method when analyzing the impact of covariates on continuous survival times. In its classical form, the Cox model was introduced in the setting of right-censored observations. However, in practice other sampling schemes are frequently encountered and there-fore extensions allowing for interval and left censoring or left truncation are clearly desired. Furthermore, many applications require a more flexible modeling of covari-ate information than the usual linear predictor. For example, effects of continuous covariates are likely to be of nonlinear form or spatial information is to be included appropriately. Further extensions should allow for time-varying effects of covariates or covariates that are themselves time-varying. Such models relax the assumption of proportional hazards. We propose a regression model for the hazard rate that com-bines and extends the above-mentioned features on the basis of a unifying Bayesian model formulation. Nonlinear and time-varying effects as well as the baseline haz-ard rate are modeled by penalized splines. Spatial effects can be included based on either Markov random fields or stationary Gaussian random fields. The model allows for arbitrary combinations of left, right and interval censoring as well as left truncation. Estimation is based on a reparameterisation of the model as a vari-ance components mixed model. The variance parameters corresponding to inverse smoothing parameters can then be estimated based on an approximate marginal likelihood approach. As an application we present an analysis on childhood mortal-ity in Nigeria, where the interval censoring framework also allows to deal with the problem of heaped survival times caused by memory effects. In a simulation study we investigate the effect of ignoring the impact of interval censored observations. Key words: extended Cox model, interval censoring, left truncation, marginal likelihood, mixed models, geoadditive hazard regression, time-varying covariate