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.

     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

A Bayesian approach to non-parametric monotone function estimation

Summary.  The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.     

Nonparametric Bayesian modeling for monotonicity in catch ratio

This article proposes nonparametric Bayesian approaches to monotone function estimation. This approach uses a hierarchical Bayes framework and a characterization of stick-breaking process that allows unconstrained estimation of the monotone function. In order to avoid the limitation of parametric modeling, a general class of prior distributions, called stick-breaking priors, is considered. It accommodates much more flexible forms and can easily deal with skewness, multimodality, etc., of the dependent variable response. The proposed approach is incorporated to model the catch ratio based on automatic weather station (AWS) data.     

Bayesian local influence for survival models

The aim of this paper is to develop a Bayesian local influence method (Zhu et al. 2009, submitted) for assessing minor perturbations to the prior, the sampling distribution, and individual observations in survival analysis. We introduce a perturbation model to characterize simultaneous (or individual) perturbations to the data, the prior distribution, and the sampling distribution. We construct a Bayesian perturbation manifold to the perturbation model and calculate its associated geometric quantities including the metric tensor to characterize the intrinsic structure of the perturbation model (or perturbation scheme). We develop local influence measures based on several objective functions to quantify the degree of various perturbations to statistical models. We carry out several simulation studies and analyze two real data sets to illustrate our Bayesian local influence method in detecting influential observations, and for characterizing the sensitivity to the prior distribution and hazard function.     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

Bayesian Semiparametric Cure Rate Model with an Unknown Threshold

Abstract.  We propose a Bayesian semiparametric model for survival data with a cure fraction. We explicitly consider a finite cure time in the model, which allows us to separate the cured and the uncured populations. We take a mixture prior of a Markov gamma process and a point mass at zero to model the baseline hazard rate function of the entire population. We focus on estimating the cure threshold after which subjects are considered cured. We can incorporate covariates through a structure similar to the proportional hazards model and allow the cure threshold also to depend on the covariates. For illustration, we undertake simulation studies and a full Bayesian analysis of a bone marrow transplant data set.     

A Bayesian approach to Weibull survival models—Application to a cancer clinical trial

In this paper we outline a class of fully parametric proportional hazards models, in which the baseline hazard is assumed to be a power transform of the time scale, corresponding to assuming that survival times follow a Weibull distribution. Such a class of models allows for the possibility of time varying hazard rates, but assumes a constant hazard ratio. We outline how Bayesian inference proceeds for such a class of models using asymptotic approximations which require only the ability to maximize the joint log posterior density. We apply these models to a clinical trial to assess the efficacy of neutron therapy compared to conventional treatment for patients with tumors of the pelvic region. In this trial there was prior information about the log hazard ratio both in terms of elicited clinical beliefs and the results of previous studies. Finally, we consider a number of extensions to this class of models, in particular the use of alternative baseline functions, and the extension to multi-state data.     

The Kumaraswamy generalized gamma distribution with application in survival analysis

We introduce and study the so-called Kumaraswamy generalized gamma distribution that is capable of modeling bathtub-shaped hazard rate functions. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a large number of well-known lifetime special sub-models such as the exponentiated generalized gamma, exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma, generalized Rayleigh, among others. Some structural properties of the new distribution are studied. We obtain two infinite sum representations for the moments and an expansion for the generating function. We calculate the density function of the order statistics and an expansion for their moments. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The usefulness of the new distribution is illustrated in two real data sets.     

Bayesian partial likelihood approach for tied observations

The Bayesian analysis based on the partial likelihood for Cox's proportional hazards model is frequently used because of its simplicity. The Bayesian partial likelihood approach is often justified by showing that it approximates the full Bayesian posterior of the regression coefficients with a diffuse prior on the baseline hazard function. This, however, may not be appropriate when ties exist among uncensored observations. In that case, the full Bayesian and Bayesian partial likelihood posteriors can be much different. In this paper, we propose a new Bayesian partial likelihood approach for many tied observations and justify its use.     

Bayesian variable selection for proportional hazards models

The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semi-parametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regression coefficients. For the baseline hazard, they use a discrete gamma process prior, and for the regression coefficients and the model space, they propose a semi-automatic parametric informative prior specification that focuses on the observables rather than the parameters. To implement the methodology, they propose a Markov chain Monte Carlo method to compute the posterior model probabilities. Examples using simulated and real data are given to demonstrate the methodology.     

Propriety of posteriors in structured additive regression models: Theory and empirical evidence

Structured additive regression comprises many semiparametric regression models such as generalized additive (mixed) models, geoadditive models, and hazard regression models within a unified framework. In a Bayesian formulation, non-parametric functions, spatial effects and further model components are specified in terms of multivariate Gaussian priors for high-dimensional vectors of regression coefficients. For several model terms, such as penalized splines or Markov random fields, these Gaussian prior distributions involve rank-deficient precision matrices, yielding partially improper priors. Moreover, hyperpriors for the variances (corresponding to inverse smoothing parameters) may also be specified as improper, e.g. corresponding to Jeffreys prior or a flat prior for the standard deviation. Hence, propriety of the joint posterior is a crucial issue for full Bayesian inference in particular if based on Markov chain Monte Carlo simulations. We establish theoretical results providing sufficient (and sometimes necessary) conditions for propriety and provide empirical evidence through several accompanying simulation studies.     

On selection of a suitable prior for the posterior analysis of censored mixture of Topp Leone distribution

The paper aims to select a suitable prior for the Bayesian analysis of the two-component mixture of the Topp Leone model under doubly censored samples and left censored samples for the first component and right censored samples for the second component. The posterior analysis has been carried out under the assumption of a class of informative and noninformative priors using a couple of loss functions. The comparison among the different Bayes estimators has been made under a simulation study and a real life example. The model comparison criterion has been used to select a suitable prior for the Bayesian analysis. The hazard rate of the Topp Leone mixture model has been compared for a range of parametric values.     

A Bayesian semiparametric method for analyzing length-biased data

Survival data obtained from prevalent cohort study designs are often subject to length-biased sampling. Frequentist methods including estimating equation approaches, as well as full likelihood methods, are available for assessing covariate effects on survival from such data. Bayesian methods allow a perspective of probability interpretation for the parameters of interest, and may easily provide the predictive distribution for future observations while incorporating weak prior knowledge on the baseline hazard function. There is lack of Bayesian methods for analyzing length-biased data. In this paper, we propose Bayesian methods for analyzing length-biased data under a proportional hazards model. The prior distribution for the cumulative hazard function is specified semiparametrically using I-Splines. Bayesian conditional and full likelihood approaches are developed for analyzing simulated and real data.     

Bayesian generalized varying coefficient models for longitudinal proportional data with errors-in-covariates

This paper is motivated from a neurophysiological study of muscle fatigue, in which biomedical researchers are interested in understanding the time-dependent relationships of handgrip force and electromyography measures. A varying coefficient model is appealing here to investigate the dynamic pattern in the longitudinal data. The response variable in the study is continuous but bounded on the standard unit interval (0, 1) over time, while the longitudinal covariates are contaminated with measurement errors. We propose a generalization of varying coefficient models for the longitudinal proportional data with errors-in-covariates. We describe two estimation methods with penalized splines, which are formalized under a Bayesian inferential perspective. The first method is an adaptation of the popular regression calibration approach. The second method is based on a joint likelihood under the hierarchical Bayesian model. A simulation study is conducted to evaluate the efficacy of the proposed methods under different scenarios. The analysis of the neurophysiological data is presented to demonstrate the use of the methods.     

Bayesian Survival Analysis in Proportional Hazard Models with Logistic Relative Risk

Abstract.  The traditional Cox proportional hazards regression model uses an exponential relative risk function. We argue that under various plausible scenarios, the relative risk part of the model should be bounded, suggesting also that the traditional model often might overdramatize the hazard rate assessment for individuals with unusual covariates. This motivates our working with proportional hazards models where the relative risk function takes a logistic form. We provide frequentist methods, based on the partial likelihood, and then go on to semiparametric Bayesian constructions. These involve a Beta process for the cumulative baseline hazard function and any prior with a density, for example that dictated by a Jeffreys-type argument, for the regression coefficients. The posterior is derived using machinery for Lévy processes, and a simulation recipe is devised for sampling from the posterior distribution of any quantity. Our methods are illustrated on real data. A Bernshtĕn–von Mises theorem is reached for our class of semiparametric priors, guaranteeing asymptotic normality of the posterior processes.     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.     

Generation of pseudo random numbers and estimation under Cox models with time-dependent covariates

ABSTRACT

We study the method for generating pseudo random numbers under various special cases of the Cox model with time-dependent covariates when the baseline hazard function may not be constant and the random variable may equal infinity with a positive probability. During our simulation studies in computing the partial likelihood estimates, in between 3% and 20% of the time with a moderate sample size, it happens that the partial likelihood estimate of the regression coefficient is ∞ for the data from the Cox model. We propose a semi-parametric estimator as a modification for such a case. We present simulation results on the asymptotic properties of the semi-parametric estimator.     

Monte Carlo Methods for Bayesian Inference on the Linear Hazard Rate Distribution

The Bayesian estimation and prediction problems for the linear hazard rate distribution under general progressively Type-II censored samples are considered in this article. The conventional Bayesian framework as well as the Markov Chain Monte Carlo (MCMC) method to generate the Bayesian conditional probabilities of interest are discussed. Sensitivity of the prior for the model is also examined. The flood data on Fox River, Wisconsin, from 1918 to 1950, are used to illustrate all the methods of inference discussed in this article.