Bayesian Estimation for the Stable Distributions in the Presence of Covariates with Applications in Clinical Issues (original) (raw)

A Bayesian Estimation of Stable Distributions

Journal of Statistical and Econometric Methods, 2012

Stable distributions are a rich class of probability distributions that are widely used to model leptokurtic data. Since the probability density and distribution functions are not known in closed form, stable distributions are often specified by their characteristic functions. This paper reviews both the techniques used to compute the density functions and the methods used to estimate parameters of the stable distributions. A new Bayesian approach using Metropolis random walk chain and direct numerical integration is proposed. The performance of the method is examined by a simulation study.

An application of bayesian analysis to medical follow-up data

Statistics in Medicine, 1985

Posterior distributions can provide effective summaries of the main conclusions of medical follow-up studies, In this article, we use Bayesian methods for the analysis of survival data. We describe posterior distributions for various parameters of clinical interest in the presence of arbitrary right censorship. Non-informative reference priors result from transformation of a two-parameter Weibull model into a location-scale family. We suggest an approach for checking adequacy. For illustration, we apply the methods to a well-known acute leukemia data set.

Bayesian Statistics from Methods to Models and Applications

Springer Proceedings in Mathematics & Statistics, 2015

Bayesian nonparametric marginal methods are very popular since they lead to fairly easy implementation due to the formal marginalization of the infinitedimensional parameter of the model. However, the straightforwardness of these methods also entails some limitations: they typically yield point estimates in the form of posterior expectations, but cannot be used to estimate non-linear functionals of the posterior distribution, such as median, mode or credible intervals. This is particularly relevant in survival analysis where non-linear functionals such as e.g. the median survival time, play a central role for clinicians and practitioners. The main goal of this paper is to summarize the methodology introduced in Arbel et al (2015) for hazard mixture models in order to draw approximate Bayesian inference on survival functions that is not limited to the posterior mean. In addition, we propose a practical implementation of an R package called momentify designed for moment-based density approximation, and, by means of an extensive simulation study, we thoroughly compare the introduced methodology with standard marginal methods and empirical estimation.

Flexible objective Bayesian linear regression with applications in survival analysis

Journal of Applied Statistics, 2016

We study objective Bayesian inference for linear regression models with residual errors distributed according to the class of two-piece scale mixtures of normal distributions. These models allow for capturing departures from the usual assumption of normality of the errors in terms of heavy tails, asymmetry, and certain types of heteroscedasticity. We propose a general noninformative, scale-invariant, prior structure and provide sufficient conditions for the propriety of the posterior distribution of the model parameters, which cover cases when the response variables are censored. These results allow us to apply the proposed models in the context of survival analysis. This paper represents an extension to the Bayesian framework of the models proposed in [19]. We present a simulation study that shows good frequentist properties of the posterior credible intervals as well as point estimators associated to the proposed priors. We illustrate the performance of these models with real data in the context of survival analysis of cancer patients.

Modeling heavy-tailed distributions in healthcare utilization by parametric and Bayesian methods

2012

Distributions of healthcare utilization such as hospital length of stay and inpatient cost are generally right skewed. The extremes represent legitimate observations on patients who, because of the severity of their illness and need for medical intervention, have long in-stays and incur large costs. In this context we demonstrate the application of several parametric models for fitting heavy tailed data. Both maximum likelihood and Bayesian methods are used for estimation in certain Coxian phase-type models, mixtures of exponential distributions, and for comparison, the lognormal, loglogistic, Weibull, generalized gamma and generalized Pareto —including the standard Pareto and Burr distributions. We focus on the mean and percentiles of the response, and illustrate our methods with an empirical example on fitting models to hospital stays for acute myocardial infarction in the Nationwide Inpatient Sample of the Healthcare Utilization Project. A suite of SAS procedures is used in all ...

A Bayesian Approach for Stable Distributions: Some Computational Aspects

Open Journal of Statistics, 2013

In this work, we study some computational aspects for the Bayesian analysis involving stable distributions. It is well known that, in general, there is no closed form for the probability density function of stable distributions. However, the use of a latent or auxiliary random variable facilitates to obtain any posterior distribution when being related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to two examples: one is related to daily price returns of Abbey National shares, considered in [1], and the other is the length distribution analysis of coding and non-coding regions in a Homo sapiens chromosome DNA sequence, considered in [2]. Posterior summaries of interest are obtained using the OpenBUGS software.

A Flexible Bayesian Parametric Proportional Hazard Model: Simulation and Applications to Right-Censored Healthcare Data

Journal of Healthcare Engineering, 2022

Survival analysis is a collection of statistical techniques which examine the time it takes for an event to occur, and it is one of the most important fields in biomedical sciences and other variety of scientific disciplines. Furthermore, the computational rapid advancements in recent decades have advocated the application of Bayesian techniques in this field, giving a powerful and flexible alternative to the classical inference. e aim of this study is to consider the Bayesian inference for the generalized log-logistic proportional hazard model with applications to right-censored healthcare data sets. We assume an independent gamma prior for the baseline hazard parameters and a normal prior is placed on the regression coefficients. We then obtain the exact form of the joint posterior distribution of the regression coefficients and distributional parameters. e Bayesian estimates of the parameters of the proposed model are obtained using the Markov chain Monte Carlo (McMC) simulation technique. All computations are performed in Bayesian analysis using Gibbs sampling (BUGS) syntax that can be run with Just Another Gibbs Sampling (JAGS) from the R software. A detailed simulation study was used to assess the performance of the proposed parametric proportional hazard model. Two real-survival data problems in the healthcare are analyzed for illustration of the proposed model and for model comparison. Furthermore, the convergence diagnostic tests are presented and analyzed. Finally, our research found that the proposed parametric proportional hazard model performs well and could be beneficial in analyzing various types of survival data.

Análisis bayesiano en presencia de covariables para datos de sobrevivencia multivariados: un ejemplo de aplicación

2011

En este artículo, se introduce un análisis bayesiano para datos multivariados de sobrevivencia en presencia de un vector de covariables y observaciones censuradas. Diferentes “fragilidades” o variables latentes son consideradas para capturar la correlación entre los tiempos de sobrevivencia para un mismo individuo. Asumimos distribuciones Weibull o Gamma generalizadas considerando datos de tiempo de vida a derecha. Desarrollamos el análisis bayesiano usando métodos Markov Chain Monte Carlo (MCMC).In this paper, we introduce a Bayesian analysis for survival multivariate data in the presence of a covariate vector and censored observations. Different “frailties” or latent variables are considered to capture the correlation among the survival times for the same individual. We assumeWeibull or generalized Gamma distributions considering right censored lifetime data. We develop the Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods

DPpackage: Bayesian Semi- and Nonparametric Modeling inR

Journal of Statistical Software, 2011

Data analysis sometimes requires the relaxation of parametric assumptions in order to gain modeling flexibility and robustness against mis-specification of the probability model. In the Bayesian context, this is accomplished by placing a prior distribution on a function space, such as the space of all probability distributions or the space of all regression functions. Unfortunately, posterior distributions ranging over function spaces are highly complex and hence sampling methods play a key role. This paper provides an introduction to a simple, yet comprehensive, set of programs for the implementation of some Bayesian nonparametric and semiparametric models in R, DPpackage. Currently, DPpackage includes models for marginal and conditional density estimation, receiver operating characteristic curve analysis, interval-censored data, binary regression data, item response data, longitudinal and clustered data using generalized linear mixed models, and regression data using generalized additive models. The package also contains functions to compute pseudo-Bayes factors for model comparison and for eliciting the precision parameter of the Dirichlet process prior, and a general purpose Metropolis sampling algorithm. To maximize computational efficiency, the actual sampling for each model is carried out using compiled C, C++ or Fortran code.

Robust Bayesian inference for multivariate longitudinal data by using normal/independent distributions

Statistics in medicine, 2013

Many randomized clinical trials collect multivariate longitudinal measurements in different scales, for example, binary, ordinal, and continuous. Multilevel item response models are used to evaluate the global treatment effects across multiple outcomes while accounting for all sources of correlation. Continuous measurements are often assumed to be normally distributed. But the model inference is not robust when the normality assumption is violated because of heavy tails and outliers. In this article, we develop a Bayesian method for multilevel item response models replacing the normal distributions with symmetric heavy-tailed normal/independent distributions. The inference is conducted using a Bayesian framework via Markov Chain Monte Carlo simulation implemented in BUGS language. Our proposed method is evaluated by simulation studies and is applied to Earlier versus Later Levodopa Therapy in Parkinson's Disease study, a motivating clinical trial assessing the effect of Levodopa...