Bayesian semiparametric multi-state models

Multi-state models provide a unified framework for the description of the evolution of discrete phenomena in continuous time. One particular example is Markov processes which can be characterised by a set of time-constant transition intensities between the states. In this paper, we will extend such parametric approaches to semiparametric models with flexible transition intensities based on Bayesian versions of penalised splines. The transition intensities will be modelled as smooth functions of time and can further be related to parametric as well as nonparametric covariate effects. Covariates with time-varying effects and frailty terms can be included in addition. Inference will be conducted either fully Bayesian (using Markov chain Monte Carlo simulation techniques) or empirically Bayesian (based on a mixed model representation). A counting process representation of semiparametric multi-state models provides the likelihood formula and also forms the basis for model validation via martingale residual processes. As an application, we will consider human sleep data with a discrete set of sleep states such as REM and non-REM phases. In this case, simple parametric approaches are inappropriate since the dynamics underlying human sleep are strongly varying throughout the night and individual specific variation has to be accounted for using covariate information and frailty terms

A Bayesian geoadditive relative survival analysis of registry data on breast cancer mortality

In this paper we develop a so called relative survival analysis, that is used to model the excess risk of a certain subpopulation relative to the natural mortality risk, i.e. the base risk that is present in the whole population. Such models are typically used in the area of clinical studies, that aim at identifying prognostic factors for disease specific mortality with data on specific causes of death being not available. Our work has been motivated by continuous-time spatially referenced survival data on breast cancer where causes of death are not known. This paper forms an extension of the analyses presented in Sauleau et al. (2007), where those data are analysed via a geoadditive, semiparametric approach, however without allowance to incorporate natural mortality. The usefulness of this relative survival approach is supported by means of a simulated data set

Nonparametric Bayesian hazard rate models based on penalized splines

Extensions of the traditional Cox proportional hazard model, concerning the following features are often desirable in applications: Simultaneous nonparametric estimation of baseline hazard and usual fixed covariate effects, modelling and detection of time-varying covariate effects and nonlinear functional forms of metrical covariates, and inclusion of frailty components. In this paper, we develop Bayesian multiplicative hazard rate models for survival and event history data that can deal with these issues in a flexible and unified framework. Some simpler models, such as piecewise exponential models with a smoothed baseline hazard, are covered as special cases. Embedded in the counting process approach, nonparametric estimation of unknown nonlinear functional effects of time or covariates is based on Bayesian penalized splines. Inference is fully Bayesian and uses recent MCMC sampling schemes. Smoothing parameters are an integral part of the model and are estimated automatically. We investigate performance of our approach through simulation studies, and illustrate it with a real data application

Bayesian Semiparametric Multi-State Models

Multi-state models provide a unified framework for the description of the evolution of discrete phenomena in continuous time. One particular example are Markov processes which can be characterised by a set of time-constant transition intensities between the states. In this paper, we will extend such parametric approaches to semiparametric models with flexible transition intensities based on Bayesian versions of penalised splines. The transition intensities will be modelled as smooth functions of time and can further be related to parametric as well as nonparametric covariate effects. Covariates with time-varying effects and frailty terms can be included in addition. Inference will be conducted either fully Bayesian using Markov chain Monte Carlo simulation techniques or empirically Bayesian based on a mixed model representation. A counting process representation of semiparametric multi-state models provides the likelihood formula and also forms the basis for model validation via martingale residual processes. As an application, we will consider human sleep data with a discrete set of sleep states such as REM and Non-REM phases. In this case, simple parametric approaches are inappropriate since the dynamics underlying human sleep are strongly varying throughout the night and individual-specific variation has to be accounted for using covariate information and frailty terms

Adaptive Gaussian Markov Random Fields with Applications in Human Brain Mapping

Functional magnetic resonance imaging (fMRI) has become the standard technology in human brain mapping. Analyses of the massive spatio-temporal fMRI data sets often focus on parametric or nonparametric modeling of the temporal component, while spatial smoothing is based on Gaussian kernels or random fields. A weakness of Gaussian spatial smoothing is underestimation of activation peaks or blurring of high-curvature transitions between activated and non-activated brain regions. In this paper, we introduce a class of inhomogeneous Markov random fields (MRF) with spatially adaptive interaction weights in a space-varying coefficient model for fMRI data. For given weights, the random field is conditionally Gaussian, but marginally it is non-Gaussian. Fully Bayesian inference, including estimation of weights and variance parameters, is carried out through efficient MCMC simulation. An application to fMRI data from a visual stimulation experiment demonstrates the performance of our approach in comparison to Gaussian and robustified non-Gaussian Markov random field models

Bayesian mapping of brain regions using compound Markov random field priors

Human brain mapping, i.e. the detection of functional regions and their connections, has experienced enormous progress through the use of functional magnetic resonance imaging (fMRI). The massive spatio-temporal data sets generated by this imaging technique impose challenging problems for statistical analysis. Many approaches focus on adequate modeling of the temporal component. Spatial aspects are often considered only in a separate postprocessing step, if at all, or modeling is based on Gaussian random fields. A weakness of Gaussian spatial smoothing is possible underestimation of activation peaks or blurring of sharp transitions between activated and non-activated regions. In this paper we suggest Bayesian spatio-temporal models, where spatial adaptivity is improved through inhomogeneous or compound Markov random field priors. Inference is based on an approximate MCMC technique. Performance of our approach is investigated through a simulation study, including a comparison to models based on Gaussian as well as more robust spatial priors in terms of pixelwise and global MSEs. Finally we demonstrate its use by an application to fMRI data from a visual stimulation experiment for assessing activation in visual cortical areas

Geoadditive Survival Models: A Supplement

This technical report supplements the paper Geoadditive Survival Models (Hennerfeind, Brezger and Fahrmeir, 2005, Revised for JASA). In particular, we describe the simulation study of this paper in greater detail, present additional results for the application, and provide a complete proof of Theorem 1, Corollary 1, as well as the lemmata and corollaries in the appendix

Geoadditive survival models

Survival data often contain small-area geographical or spatial information, such as the residence of individuals. In many cases the impact of such spatial effects on hazard rates is of considerable substantive interest. Therefore, extensions of known survival or hazard rate models to spatial models have been suggested recently. Mostly, a spatial component is added to the usual linear predictor of the Cox model. We propose flexible continuous-time geoadditive models, extending the Cox model with respect to several aspects often needed in applications: The common linear predictor is generalized to an additive predictor, including nonparametric components for the log-baseline hazard, time-varying effects and possibly nonlinear effects of continuous covariates or further time scales, and a spatial component for geographical effects. In addition, uncorrelated frailty effects or nonlinear two-way interactions can be incorporated. Inference is developed within a unified fully Bayesian framework. We prefer to use penalized regression splines and Markov random fields as basic building blocks, but geostatistical (kriging) models are also considered. Posterior analysis uses computationally efficient MCMC sampling schemes. Smoothing parameters are an integral part of the model and are estimated automatically. Propriety of posteriors is shown under fairly general conditions, and practical performance is investigated through simulation studies. We apply our approach to data from a case study in London and Essex that aims to estimate the effect of area of residence and further covariates on waiting times to coronary artery bypass graft (CABG). Results provide clear evidence of nonlinear time-varying effects, and considerable spatial variability of waiting times to bypass graft

Geoadditive survival models

Survival data often contain geographical or spatial information, such as the residence of individuals. We propose geoadditive survival models for analyzing spatial effects jointly with possibly nonlinear effects of other covariates. Within a unified Bayesian framework, our approach extends the classical Cox model to a more general multiplicative hazard rate model, augmenting the common linear predictor with a spatial component and nonparametric terms for nonlinear effects of time and metrical covariates. Markov random fields and penalized regression splines are used as basic building blocks. Inference is fully Bayesian and uses computationally efficient MCMC sampling schemes. Smoothing parameters are an integral part of the model and are estimated automatically. Perfomance is investigated through simulation studies. We apply our approach to data from a case study in London and Essex that aims to estimate the effect of area of residence and further covariates on waiting times to coronary artery bypass graft (CABG)