A mixed model approach for structured hazard regression

The classical Cox proportional hazards model is a benchmark approach to analyze continuous survival times in the presence of covariate information. In a number of applications, there is a need to relax one or more of its inherent assumptions, such as linearity of the predictor or the proportional hazards property. Also, one is often interested in jointly estimating the baseline hazard together with covariate effects or one may wish to add a spatial component for spatially correlated survival data. We propose an extended Cox model, where the (log-)baseline hazard is weakly parameterized using penalized splines and the usual linear predictor is replaced by a structured additive predictor incorporating nonlinear effects of continuous covariates and further time scales, spatial effects, frailty components, and more complex interactions. Inclusion of time-varying coefficients leads to models that relax the proportional hazards assumption. Nonlinear and time-varying effects are modelled through penalized splines, and spatial components are treated as correlated random effects following either a Markov random field or a stationary Gaussian random field. All model components, including smoothing parameters, are specified within a unified framework and are estimated simultaneously based on mixed model methodology. The estimation procedure for such general mixed hazard regression models is derived using penalized likelihood for regression coefficients and (approximate) marginal likelihood for smoothing parameters. Performance of the proposed method is studied through simulation and an application to leukemia survival data in Northwest England

Parametrization and penalties in spline models with an application to survival analysis

In this paper we show how a simple parametrization, built from the definition of cubic splines, can aid in the implementation and interpretation of penalized spline models, whatever configuration of knots we choose to use. We call this parametrization value-first derivative parametrization. We perform Bayesian inference by exploring the natural link between quadratic penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some penalty functionals. Alternatives include empirical Bayes methods involving model selection type criteria. The proposed methodology is illustrated by an application to survival analysis where the usual Cox model is extended to allow for time-varying regression coefficients

Geoadditive hazard regression for interval censored 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 therefore extensions allowing for interval and left censoring or left truncation are clearly desired. Furthermore, many applications require a more flexible modeling of covariate 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 combines 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 hazard 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 variance 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 mortality 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

Semiparametric Bayesian Time-Space Analysis of Unemployment Duration

In this paper, we analyze unemployment duration in Germany with official data from the German Federal Employment Office for the years 1980-1995. Conventional hazard rate models for leaving unemployment cannot cope with simultaneous and flexible fitting of duration dependence, nonlinear covariate effects, trend and seasonal calendar time components and a large number of regional effects. We apply a semiparametric hierarchical Bayesian modelling approach that is suitable for time-space analysis of unemployment duration by simultaneously including and estimating effects of several time scales, regional variation and further covariates. Inference is fully Bayesian and uses recent Markov chain Monte Carlo techniques

Variable Selection and Model Choice in Structured Survival Models

In many situations, medical applications ask for flexible survival models that allow to extend the classical Cox-model via the inclusion of time-varying and nonparametric effects. These structured survival models are very flexible but additional difficulties arise when model choice and variable selection is desired. In particular, it has to be decided which covariates should be assigned time-varying effects or whether parametric modeling is sufficient for a given covariate. Component-wise boosting provides a means of likelihood-based model fitting that enables simultaneous variable selection and model choice. We introduce a component-wise likelihood-based boosting algorithm for survival data that permits the inclusion of both parametric and nonparametric time-varying effects as well as nonparametric effects of continuous covariates utilizing penalized splines as the main modeling technique. Its properties and performance are investigated in simulation studies. The new modeling approach is used to build a flexible survival model for intensive care patients suffering from severe sepsis. A software implementation is available to the interested reader

Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set-up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterisation thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models allowing the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.Comment: Accepted for publication by the Scandinavian Journal of Statistics 2017-06-1