Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

A general approach to heteroscedastic linear regression

Our article presents a general treatment of the linear regression model, in which the error distribution is modelled nonparametrically and the error variances may be heteroscedastic, thus eliminating the need to transform the dependent variable in many data sets. The mean and variance components of the model may be either parametric or nonparametric, with parsimony achieved through variable selection and model averaging. A Bayesian approach is used for inference with priors that are data-based so that estimation can be carried out automatically with minimal input by the user. A Dirichlet process mixture prior is used to model the error distribution nonparametrically; when there are no regressors in the model, the method reduces to Bayesian density estimation, and we show that in this case the estimator compares favourably with a well-regarded plug-in density estimator. We also consider a method for checking the fit of the full model. The methodology is applied to a number of simulated and real examples and is shown to work well.     

Asymptotic distributions of error density and distribution function estimators in nonparametric regression

This paper considers the problem of estimating the error density and distribution functions in nonparametric regression models. The asymptotic distribution of a suitably standardized density estimator at a fixed point is shown to be normal while that of the maximum of a suitably normalized deviation of the density estimator from the true density function is the same as in the case of the one sample set up. Finally, the standardized residual empirical process is shown to be uniformly close to the similarly standardized empirical process of the errors. This paper thus generalizes some of the well known results about the residual density estimators and the empirical process in parametric regression models to nonparametric regression models, thereby enhancing the domain of their applications.     

Bayesian nonparametric modelling of the link function in the single-index model using a Bernstein–Dirichlet process prior

This paper proposes the use of the Bernstein–Dirichlet process prior for a new nonparametric approach to estimating the link function in the single-index model (SIM). The Bernstein–Dirichlet process prior has so far mainly been used for nonparametric density estimation. Here we modify this approach to allow for an approximation of the unknown link function. Instead of the usual Gaussian distribution, the error term is assumed to be asymmetric Laplace distributed which increases the flexibility and robustness of the SIM. To automatically identify truly active predictors, spike-and-slab priors are used for Bayesian variable selection. Posterior computations are performed via a Metropolis-Hastings-within-Gibbs sampler using a truncation-based algorithm for stick-breaking priors. We compare the efficiency of the proposed approach with well-established techniques in an extensive simulation study and illustrate its practical performance by an application to nonparametric modelling of the power consumption in a sewage treatment plant.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Slice sampling mixture models

We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (Commun. Stat., Simul. Comput. 36:45–54, 2007). This new sampler allows for the fitting of infinite mixture models with a wide-range of prior specifications. To illustrate this flexibility we consider priors defined through infinite sequences of independent positive random variables. Two applications are considered: density estimation using mixture models and hazard function estimation. In each case we show how the slice efficient sampler can be applied to make inference in the models. In the mixture case, two submodels are studied in detail. The first one assumes that the positive random variables are Gamma distributed and the second assumes that they are inverse-Gaussian distributed. Both priors have two hyperparameters and we consider their effect on the prior distribution of the number of occupied clusters in a sample. Extensive computational comparisons with alternative “conditional” simulation techniques for mixture models using the standard Dirichlet process prior and our new priors are made. The properties of the new priors are illustrated on a density estimation problem.     

Estimation,prediction and interpretation of NGG random effects models: an application to Kevlar fibre failure times

We propose a class of Bayesian semiparametric mixed-effects models; its distinctive feature is the randomness of the grouping of observations, which can be inferred from the data. The model can be viewed under a more natural perspective, as a Bayesian semiparametric regression model on the log-scale; hence, in the original scale, the error is a mixture of Weibull densities mixed on both parameters by a normalized generalized gamma random measure, encompassing the Dirichlet process. As an estimate of the posterior distribution of the clustering of the random-effects parameters, we consider the partition minimizing the posterior expectation of a suitable class of loss functions. As a merely illustrative application of our model we consider a Kevlar fibre lifetime dataset (with censoring). We implement an MCMC scheme, obtaining posterior credibility intervals for the predictive distributions and for the quantiles of the failure times under different stress levels. Compared to a previous parametric Bayesian analysis, we obtain narrower credibility intervals and a better fit to the data. We found that there are three main clusters among the random-effects parameters, in accordance with previous frequentist analysis.     

A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

Restricted Estimation in Partially Linear Errors-in-Variables Models

As a compromise between parametric regression and nonparametric regression, partially linear models are frequently used in statistical modelling. This article considers statistical inference for this semiparametric model when the linear covariate is measured with additive error and some additional linear restrictions on the parametric component are assumed to hold. We propose a restricted corrected profile least-squares estimator for the parametric component, and study the asymptotic normality of the estimator. To test hypothesis on the parametric component, we construct a Wald test statistic and obtain its limiting distribution. Some simulation studies are conducted to illustrate our approaches.     

A mixture of beta–Dirichlet processes prior for Bayesian analysis of event history data

In this paper, we propose a mixture of beta–Dirichlet processes as a nonparametric prior for the cumulative intensity functions of a Markov process. This family of priors is a natural extension of a mixture of Dirichlet processes or a mixture of beta processes which are devised to compromise advantages of parametric and nonparametric approaches. They give most of their prior mass to the small neighborhood of a specific parametric model. We show that a mixture of beta–Dirichlet processes prior is conjugate with Markov processes. Formulas for computing the posterior distribution are derived. Finally, results of analyzing credit history data are given.     

Modelling stochastic order in the analysis of receiver operating characteristic data: Bayesian non-parametric approaches

Summary.  The evaluation of the performance of a continuous diagnostic measure is a commonly encountered task in medical research. We develop Bayesian non-parametric models that use Dirichlet process mixtures and mixtures of Polya trees for the analysis of continuous serologic data. The modelling approach differs from traditional approaches to the analysis of receiver operating characteristic curve data in that it incorporates a stochastic ordering constraint for the distributions of serologic values for the infected and non-infected populations. Biologically such a constraint is virtually always feasible because serologic values from infected individuals tend to be higher than those for non-infected individuals. The models proposed provide data-driven inferences for the infected and non-infected population distributions, and for the receiver operating characteristic curve and corresponding area under the curve. We illustrate and compare the predictive performance of the Dirichlet process mixture and mixture of Polya trees approaches by using serologic data for Johne's disease in dairy cattle.     

Bayesian Semiparametric Regression for Median Residual Life

Abstract.  With survival data there is often interest not only in the survival time distribution but also in the residual survival time distribution. In fact, regression models to explain residual survival time might be desired. Building upon recent work of Kottas & Gelfand [ J. Amer. Statist. Assoc. 96 (2001) 1458], we formulate a semiparametric median residual life regression model induced by a semiparametric accelerated failure time regression model. We utilize a Bayesian approach which allows full and exact inference. Classical work essentially ignores covariates and is either based upon parametric assumptions or is limited to asymptotic inference in non-parametric settings. No regression modelling of median residual life appears to exist. The Bayesian modelling is developed through Dirichlet process mixing. The models are fitted using Gibbs sampling. Residual life inference is implemented extending the approach of Gelfand & Kottas [ J. Comput. Graph. Statist. 11 (2002) 289]. Finally, we present a fairly detailed analysis of a set of survival times with moderate censoring for patients with small cell lung cancer.     

Bayesian Nonparametric Instrumental Variables Regression Based on Penalized Splines and Dirichlet Process Mixtures

We propose a Bayesian nonparametric instrumental variable approach under additive separability that allows us to correct for endogeneity bias in regression models where the covariate effects enter with unknown functional form. Bias correction relies on a simultaneous equations specification with flexible modeling of the joint error distribution implemented via a Dirichlet process mixture prior. Both the structural and instrumental variable equation are specified in terms of additive predictors comprising penalized splines for nonlinear effects of continuous covariates. Inference is fully Bayesian, employing efficient Markov chain Monte Carlo simulation techniques. The resulting posterior samples do not only provide us with point estimates, but allow us to construct simultaneous credible bands for the nonparametric effects, including data-driven smoothing parameter selection. In addition, improved robustness properties are achieved due to the flexible error distribution specification. Both these features are challenging in the classical framework, making the Bayesian one advantageous. In simulations, we investigate small sample properties and an investigation of the effect of class size on student performance in Israel provides an illustration of the proposed approach which is implemented in an R package bayesIV. Supplementary materials for this article are available online.     

INFERENCE IN DIRICHLET PROCESS MIXED GENERALIZED LINEAR MODELS BY USING MONTE CARLO EM

Generalized linear mixed models are widely used for describing overdispersed and correlated data. Such data arise frequently in studies involving clustered and hierarchical designs. A more flexible class of models has been developed here through the Dirichlet process mixture. An additional advantage of using such mixture models is that the observations can be grouped together on the basis of the overdispersion present in the data. This paper proposes a partial empirical Bayes method for estimating all the model parameters by adopting a version of the EM algorithm. An augmented model that helps to implement an efficient Gibbs sampling scheme, under the non‐conjugate Dirichlet process generalized linear model, generates observations from the conditional predictive distribution of unobserved random effects and provides an estimate of the average number of mixing components in the Dirichlet process mixture. A simulation study has been carried out to demonstrate the consistency of the proposed method. The approach is also applied to a study on outdoor bacteria concentration in the air and to data from 14 retrospective lung‐cancer studies.     

Bayesian nonparametric survival analysis using mixture of Burr XII distributions

ABSTRACT

Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.     

Bayesian Kernel Mixtures for Counts

Although Bayesian nonparametric mixture models for continuous data are well developed, there is a limited literature on related approaches for count data. A common strategy is to use a mixture of Poissons, which unfortunately is quite restrictive in not accounting for distributions having variance less than the mean. Other approaches include mixing multinomials, which requires finite support, and using a Dirichlet process prior with a Poisson base measure, which does not allow smooth deviations from the Poisson. As a broad class of alternative models, we propose to use nonparametric mixtures of rounded continuous kernels. An efficient Gibbs sampler is developed for posterior computation, and a simulation study is performed to assess performance. Focusing on the rounded Gaussian case, we generalize the modeling framework to account for multivariate count data, joint modeling with continuous and categorical variables, and other complications. The methods are illustrated through applications to a developmental toxicity study and marketing data. This article has supplementary material online.     

Prior Distributions on Measure Space

A measure is the formal representation of the non-negative additive functions that abound in science. We review and develop the art of assigning Bayesian priors to measures. Where necessary, spatial correlation is delegated to correlating kernels imposed on otherwise uncorrelated priors. The latter must be infinitely divisible (ID) and hence described by the Lévy–Khinchin representation. Thus the fundamental object is the Lévy measure, the choice of which corresponds to different ID process priors. The general case of a Lévy measure comprising a mixture of assigned base measures leads to a prior process comprising a convolution of corresponding processes. Examples involving a single base measure are the gamma process, the Dirichlet process (for the normalized case) and the Poisson process. We also discuss processes that we call the supergamma and super-Dirichlet processes, which are double base measure generalizations of the gamma and Dirichlet processes. Examples of multiple and continuum base measures are also discussed. We conclude with numerical examples of density estimation.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.