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A computational approach for full nonparametric Bayesian inference under Dirichlet process mixture models
 Journal of Computational and Graphical Statistics
, 2002
"... Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or ksample problems,many classical nonparametricapproachesare limited ..."
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Cited by 27 (7 self)
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Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or ksample problems,many classical nonparametricapproachesare limited to hypothesistesting. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric speci � cations are often most appealing for smaller sample sizes. Bayesian nonparametricapproachesavoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulationbasedmodel � tting approaches which yield inference that is con � ned to posterior moments of linear functionals of the population distribution.This article provides a computationalapproach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a ksample problem, and comparison of survival times from different populations under fairly heavy censoring.
HIGHER ORDER SEMIPARAMETRIC FREQUENTIST INFERENCE WITH THE PROFILE SAMPLER
 SUBMITTED TO THE ANNALS OF STATISTICS
, 2006
"... We consider higher order frequentist inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution. The first order validity of this procedure established by Lee, Kosorok and Fine (2005) is extended to second order validity in the setting ..."
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Cited by 7 (5 self)
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We consider higher order frequentist inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution. The first order validity of this procedure established by Lee, Kosorok and Fine (2005) is extended to second order validity in the setting where the infinite dimensional nuisance parameter achieves the parametric rate. Specifically, we obtain higher order estimates of the maximum profile likelihood estimator and of the efficient Fisher information. Moreover, we prove that an exact frequentist confidence interval for the parametric component at level alpha can be estimated by the alpha level credible set from the profile sampler with an error of order OP (n −1). As far as we are aware, these results are the first higher order frequentist results obtained for semiparametric estimation. A fully Bayesian interpretation is established under a certain data dependent prior. The theory is verified for three specific examples.
Nonparametric Bayesian Data Analysis
"... We review the current state of nonparametric Bayesian inference. The discussion follows a list of important statistical inference problems, including density estimation, regression, survival analysis, hierarchical models and model validation. For each inference problem we review relevant nonparametr ..."
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Cited by 3 (0 self)
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We review the current state of nonparametric Bayesian inference. The discussion follows a list of important statistical inference problems, including density estimation, regression, survival analysis, hierarchical models and model validation. For each inference problem we review relevant nonparametric Bayesian models and approaches including Dirichlet process (DP) models and variations, Polya trees, wavelet based models, neural network models, spline regression, CART, dependent DP models, and model validation with DP and Polya tree extensions of parametric models. 1
A Bayesian Semiparametric Transformation Model Incorporating Frailties
"... This paper is concerned with the Bayesian analysis of failure time data in the presence of covariate information. In this paper, the focus will be on the semiparametric Bayesian model. The semiparametric nature of the models allows considerable generality and applicability but enough structure for u ..."
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Cited by 2 (0 self)
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This paper is concerned with the Bayesian analysis of failure time data in the presence of covariate information. In this paper, the focus will be on the semiparametric Bayesian model. The semiparametric nature of the models allows considerable generality and applicability but enough structure for useful physical interpretation and understanding particular applications in the medical research. The popularity of semiparametric approches for analyzing univariate survival data begins with the seminal paper of Cox(1972) on the proportional hazards model. This model assumes a constant relative risk compared to the baseline hazard function for all values of the failure time given the covariate values. When the assumption is violated, the proportional odds model provides an alternative. In this model the logodds of the failure time distribution depends linearly on the covariates. There is a temptaion to use proportional hazards model to analyse failure time observations without any formal model checking, even when the model doesnot fit the data well due to its large sample inference properties (Andersen ans Gill, 1982) and easy access to statistical software for this model. In this paper we introduce a general class of models which contains both proportional hazards and odds model. The semiparametric nature of this model creates the flexibility to fit the data well. Also we will propose a criterion to choose between proportional hazards and proportional odds model. Furthermore, a simple extension to the model allows us to include frailties (Clayton and Cuzick, 1985) for multivariate survival data. For nice discussion of Bayesian semiparametric models see Gelfand, 1996 and for Bayesian semiparametric analysis of survival data see Sinha and Dey, 1998. Let T 1 , . . . , T n be f...
Monte Carlo Methods for Bayesian Analysis of Survival Data Using Mixtures of Dirichlet Process Priors
, 1998
"... Consider the model in which the data consist of possibly censored lifetimes, and one puts a mixture of Dirichlet process priors on the common survival distribution. The exact computation of the posterior distribution of the survival function is in general impossible to obtain. This paper develops ..."
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Cited by 1 (1 self)
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Consider the model in which the data consist of possibly censored lifetimes, and one puts a mixture of Dirichlet process priors on the common survival distribution. The exact computation of the posterior distribution of the survival function is in general impossible to obtain. This paper develops and compares the performance of several simulation techniques, based on Markov chain Monte Carlo and sequential importance sampling, for approximating this posterior distribution. One scheme, whose derivation is based on sequential importance sampling, gives an exactly iid sample from the posterior for the case of right censored data. A second contribution of this paper is a battery of programs that implement the various schemes discussed in this paper. The programs and methods are illustrated on a data set of intervalcensored times arising from two treatments for breast cancer.
Two Level Proportional Hazards Models
"... This paper will discuss a two level proportional hazards model that incorporates random variability in the baseline risk and random coefficients for an individual level covariate. Connections between this model and other multilevel models for survival times will be discussed. Next, we derive the lik ..."
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This paper will discuss a two level proportional hazards model that incorporates random variability in the baseline risk and random coefficients for an individual level covariate. Connections between this model and other multilevel models for survival times will be discussed. Next, we derive the likelihood and elucidate the assumptions behind the likelihood. A simulation study will follow to empirically demonstrate properties of the model. The CVFS data will be used to demonstrate the applicability of this two level hazard model. Lastly, advantages, disadvantages and areas in need of further research will be discussed.
Mixture Models in Econometric Duration Analysis
, 2002
"... Econometric duration analysis has become an important part of methodology in econometrics, bringing forth a plenty of applications. The probability distribution of the duration of a time span is modeled through its conditional hazard rate given the covariates. When some of the covariates are unobser ..."
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Econometric duration analysis has become an important part of methodology in econometrics, bringing forth a plenty of applications. The probability distribution of the duration of a time span is modeled through its conditional hazard rate given the covariates. When some of the covariates are unobservable, the duration, given the observable covariates, has a mixture distribution. The paper surveys and discusses...
Modeling Multilevel Sleep Transitional Data Via Poisson LogLinear Multilevel Models
, 2009
"... This paper proposes Poisson loglinear multilevel models to investigate population variability in sleep state transition rates. We specifically propose a Bayesian Poisson regression model that is more flexible, scalable to larger studies, and easily fit than other attempts in the literature. We furt ..."
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This paper proposes Poisson loglinear multilevel models to investigate population variability in sleep state transition rates. We specifically propose a Bayesian Poisson regression model that is more flexible, scalable to larger studies, and easily fit than other attempts in the literature. We further use hierarchical random effects to account for pairings of individuals and repeated measures within those individuals, as comparing diseased to nondiseased subjects while minimizing bias is of epidemiologic importance. We estimate essentially nonparametric piecewise constant hazards and smooth them, and allow for time varying covariates and segment of the night comparisons. The Bayesian Poisson regression is justified through a rederivation of a classical algebraic likelihood equivalence of Poisson regression with a log(time) offset and survival regression assuming piecewise constant hazards. This relationship allows us to synthesize two methods currently used to analyze sleep transition phenomena: stratified multistate proportional hazards models and loglinear models with GEE for transition counts. An example data set from the Sleep Heart Health Study is analyzed.
Bayesian Analysis of Survival Models with
"... This paper develops a non/semiparametric Bayesian analysis of bathtub shaped hazard rates given different types of censored data. Computations are rendered straightforward based on a new stochastic substitution method. This method, which we term Auxiliary Random Functions (ARF), is particularly sui ..."
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This paper develops a non/semiparametric Bayesian analysis of bathtub shaped hazard rates given different types of censored data. Computations are rendered straightforward based on a new stochastic substitution method. This method, which we term Auxiliary Random Functions (ARF), is particularly suited to simulate from the posterior conditional distributions in a Gibbs sampler (Smith and Roberts, 1993) in the present context
Mixtures of Polya Trees for Flexible Spatial Frailty Survival Modeling
, 2007
"... Mixtures of Polya trees offer a very flexible, nonparametric approach for modeling timetoevent data. Many such settings also feature spatial association that requires further sophistication, either at a point (geostatistical) or areal (lattice) level. In this paper we combine these two aspects with ..."
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Mixtures of Polya trees offer a very flexible, nonparametric approach for modeling timetoevent data. Many such settings also feature spatial association that requires further sophistication, either at a point (geostatistical) or areal (lattice) level. In this paper we combine these two aspects within three competing survival models, obtaining a data analytic approach that remains computationally feasible in a fully hierarchical Bayesian framework thanks to modern Markov chain Monte Carlo (MCMC) methods. We illustrate the usefulness of our proposed methods with an analysis of spatially oriented breast cancer survival data from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute. Our results indicate appreciable advantages for our approach over previous, competing methods that impose unrealistic parametric assumptions, ignore spatial association, or both.