Simultaneous probability statements for Bayesian P-splines

P-splines are a popular approach for fitting nonlinear effects of continuous covariates in semiparametric regression models. Recently, a Bayesian version for P-splines has been developed on the basis of Markov chain Monte Carlo simulation techniques for inference. In this work we adopt and generalize the concept of Bayesian contour probabilities to Bayesian P-splines within a generalized additive models framework. More specifically, we aim at computing the maximum credible level (sometimes called Bayesian p-value) for which a particular parameter vector of interest lies within the corresponding highest posterior density (HPD) region. We are particularly interested in parameter vectors that correspond to a constant, linear or more generally a polynomial fit. As an alternative to HPD regions simultaneous credible intervals could be used to define pseudo contour probabilities. Efficient algorithms for computing contour and pseudo contour probabilities are developed. The performance of the approach is assessed through simulation studies and applications to data for the Munich rental guide and on undernutrition in Zambia and Tanzania

Generalized structured additive regression based on Bayesian P-splines

Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX

Monotonic regression based on Bayesian P-splines: an application to estimating price response functions from store-level scanner data

Generalized additive models have become a widely used instrument for flexible regression analysis. In many practical situations, however, it is desirable to restrict the flexibility of nonparametric estimation in order to accommodate a presumed monotonic relationship between a covariate and the response variable. For example, consumers usually will buy less of a brand if its price increases, and therefore one expects a brand's unit sales to be a decreasing function in own price. We follow a Bayesian approach using penalized B-splines and incorporate the assumption of monotonicity in a natural way by an appropriate specification of the respective prior distributions. We illustrate the methodology in an empirical application modeling demand for a brand of orange juice and show that imposing monotonicity constraints for own- and cross-item price effects improves the predictive validity of the estimated sales response function considerably

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

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

BayesX: Analysing Bayesian structured additive regression models

There has been much recent interest in Bayesian inference for generalized additive and related models. The increasing popularity of Bayesian methods for these and other model classes is mainly caused by the introduction of Markov chain Monte Carlo (MCMC) simulation techniques which allow the estimation of very complex and realistic models. This paper describes the capabilities of the public domain software BayesX for estimating complex regression models with structured additive predictor. The program extends the capabilities of existing software for semiparametric regression. Many model classes well known from the literature are special cases of the models supported by BayesX. Examples are Generalized Additive (Mixed) Models, Dynamic Models, Varying Coefficient Models, Geoadditive Models, Geographically Weighted Regression and models for space-time regression. BayesX supports the most common distributions for the response variable. For univariate responses these are Gaussian, Binomial, Poisson, Gamma and negative Binomial. For multicategorical responses, both multinomial logit and probit models for unordered categories of the response as well as cumulative threshold models for ordered categories may be estimated. Moreover, BayesX allows the estimation of complex continuous time survival and hazardrate models

BayesX: Analyzing Bayesian Structural Additive Regression Models

There has been much recent interest in Bayesian inference for generalized additive and related models. The increasing popularity of Bayesian methods for these and other model classes is mainly caused by the introduction of Markov chain Monte Carlo (MCMC) simulation techniques which allow realistic modeling of complex problems. This paper describes the capabilities of the free software package BayesX for estimating regression models with structured additive predictor based on MCMC inference. The program extends the capabilities of existing software for semiparametric regression included in S-PLUS, SAS, R or Stata. Many model classes well known from the literature are special cases of the models supported by BayesX. Examples are generalized additive (mixed) models, dynamic models, varying coefficient models, geoadditive models, geographically weighted regression and models for space-time regression. BayesX supports the most common distributions for the response variable. For univariate responses these are Gaussian, Binomial, Poisson, Gamma, negative Binomial, zero inflated Poisson and zero inflated negative binomial. For multicategorical responses, both multinomial logit and probit models for unordered categories of the response as well as cumulative threshold models for ordered categories can be estimated. Moreover, BayesX allows the estimation of complex continuous time survival and hazard rate models.

Simultaneous probability statements for Bayesian P-splines

P-splines are a popular approach for fitting nonlinear effects of continuous covariates in semiparametric regression models. Recently, a Bayesian version for P-splines has been developed on the basis of Markov chain Monte Carlo simulation techniques for inference. In this work we adopt and generalize the concept of Bayesian contour probabilities to additive models with Gaussian or multicategorical responses. More specifically, we aim at computing the maximum credible level (sometimes called Bayesian p-value) for which a particular parameter vector of interest lies within the corresponding highest posterior density (HPD) region. We are particularly interested in parameter vectors that correspond to a constant, linear or more generally a polynomial fit. As an alternative to HPD regions simultaneous credible intervals could be used to define pseudo contour probabilities. Efficient algorithms for computing contour and pseudo contour probabilities are developed. The performance of the approach is assessed through simulation studies. Two applications on the determinants of undernutrition in developing countries and the health status of trees show how contour probabilities may be used in practice to assist the analyst in the model building process

BayesX - Software for Bayesian Inference based on Markov Chain Monte Carlo simulation techniques

BayesX is a Software tool for Bayesian inference based on Markov Chain Monte Carlo (MCMC) inference techniques. The main feature of BayesX so far, is a very powerful regression tool for Bayesian semiparametric regression within the Generalized linear models framework. BayesX is able to estimate nonlinear effects of metrical covariates, trends and flexible seasonal patterns of time scales, structured and/or unstructured random effects of spatial covariates (geographical data) and unstructured random effects of unordered group indicators. Moreover, BayesX is able to estimate varying coefficients models with metrical and even spatial covariates as effectmodifiers. The distribution of the response can be either Gaussian, binomial or Poisson. In addition, BayesX has some useful functions for handling and manipulating datasets and geographical maps

Bayesian P-Splines

P-splines are an attractive approach for modelling nonlinear smooth effects of covariates within the generalized additive and varying coefficient models framework. In this paper we propose a Bayesian version for P-splines and generalize the approach for one dimensional curves to two dimensional surface fitting for modelling interactions between metrical covariates. A Bayesian approach to P-splines has the advantage of allowing for simultaneous estimation of smooth functions and smoothing parameters. Moreover, it can easily be extended to more complex formulations, for example to mixed models with random effects for serially or spatially correlated response. Additionally, the assumption of constant smoothing parameters can be replaced by allowing the smoothing parameters to be locally adaptive. This is particularly useful in situations with changing curvature of the underlying smooth function or where the function is highly oscillating. Inference is fully Bayesian and uses recent MCMC techniques for drawing random samples from the posterior. In a couple of simulation studies the performance of Bayesian P-splines is studied and compared to other approaches in the literature. We illustrate the approach by a complex application on rents for flats in Munich