Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

A Bayesian approach to mixture cure models with spatial frailties for population‐based cancer relative survival data

As the treatments of cancer progress, a certain number of cancers are curable if diagnosed early. In population‐based cancer survival studies, cure is said to occur when mortality rate of the cancer patients returns to the same level as that expected for the general cancer‐free population. The estimates of cure fraction are of interest to both cancer patients and health policy makers. Mixture cure models have been widely used because the model is easy to interpret by separating the patients into two distinct groups. Usually parametric models are assumed for the latent distribution for the uncured patients. The estimation of cure fraction from the mixture cure model may be sensitive to misspecification of latent distribution. We propose a Bayesian approach to mixture cure model for population‐based cancer survival data, which can be extended to county‐level cancer survival data. Instead of modeling the latent distribution by a fixed parametric distribution, we use a finite mixture of the union of the lognormal, loglogistic, and Weibull distributions. The parameters are estimated using the Markov chain Monte Carlo method. Simulation study shows that the Bayesian method using a finite mixture latent distribution provides robust inference of parameter estimates. The proposed Bayesian method is applied to relative survival data for colon cancer patients from the Surveillance, Epidemiology, and End Results (SEER) Program to estimate the cure fractions. The Canadian Journal of Statistics 40: 40–54; 2012 © 2012 Statistical Society of Canada     

A NONPARAMETRIC MIXED‐EFFECTS MODEL FOR CANCER MORTALITY

There are several ways to handle within‐subject correlations with a longitudinal discrete outcome, such as mortality. The most frequently used models are either marginal or random‐effects types. This paper deals with a random‐effects‐based approach. We propose a nonparametric regression model having time‐varying mixed effects for longitudinal cancer mortality data. The time‐varying mixed effects in the proposed model are estimated by combining kernel‐smoothing techniques and a growth‐curve model. As an illustration based on real data, we apply the proposed method to a set of prefecture‐specific data on mortality from large‐bowel cancer in Japan.     

On the clustering term in ecological analysis: how do different prior specifications affect results?

We study how different prior assumptions on the spatially structured heterogeneity term of the convolution hierarchical Bayesian model for spatial disease data could affect the results of an ecological analysis when response and exposure exhibit a strong spatial pattern. We show that in this case the estimate of the regression parameter could be strongly biased, both by analyzing the association between lung cancer mortality and education level on a real dataset and by a simulation experiment. The analysis is based on a hierarchical Bayesian model with a time dependent covariate in which we allow for a latency period between exposure and mortality, with time and space random terms and misaligned exposure-disease data.     

Multivariate time series prediction using a hybridization of VARMA models and Bayesian networks

In this paper, a new hybrid model of vector autoregressive moving average (VARMA) models and Bayesian networks is proposed to improve the forecasting performance of multivariate time series. In the proposed model, the VARMA model, which is a popular linear model in time series forecasting, is specified to capture the linear characteristics. Then the errors of the VARMA model are clustered into some trends by K-means algorithm with Krzanowski–Lai cluster validity index determining the number of trends, and a Bayesian network is built to learn the relationship between the data and the trend of its corresponding VARMA error. Finally, the estimated values of the VARMA model are compensated by the probabilities of their corresponding VARMA errors belonging to each trend, which are obtained from the Bayesian network. Compared with VARMA models, the experimental results with a simulation study and two multivariate real-world data sets indicate that the proposed model can effectively improve the prediction performance.     

Efficient estimation for the conditional autoregressive model

Recently, spatial regression models have been attracting a great deal of attention in areas ranging from effect of traffic congestion on accident rates to the analysis of trends in gastric cancer mortality. In this paper, we propose efficient estimators for the regression coefficients of the spatial conditional autoregressive model, when uncertain auxiliary information is available about these coefficients. We provide efficiency comparisons of the proposed estimators based on asymptotic risk analysis and Monte Carlo simulations. We apply the proposed methods to real data on Boston housing prices and illustrate how a bootstrapping approach can be employed to compute prediction errors of the estimators.     

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.

     

A generalized Bayesian nonlinear mixed‐effects regression model for zero‐inflated longitudinal count data in tuberculosis trials

In this paper, we investigate Bayesian generalized nonlinear mixed‐effects (NLME) regression models for zero‐inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis (TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero‐inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log‐transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti‐TB drugs. We therefore recommend that zero‐inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.     

The Impact of Ovarian Cancer on Life Expectancy in Japan

The purpose of this study was to determine how life expectancy is modified by ovarian cancer from 1950-2000. The contributions of ovarian cancer to life expectancy were estimated. The age characteristics of ovarian cancer were detected using the Gompertz relational mortality model. The patterns between years of potential life lost (YPLL) and mortality were obtained by fitting a linear regression equation to the natural logarithm of their ratios. YPLLs are substantially higher in Ireland than in Japan. However, the rates of change were much higher in Japan than in Ireland. YPLLs changed from 0.02 year in 1950 to 0.12 year in 2000. In Japan, there was a sixfold increase in the proportion of YPLLs for death from ovarian cancer relative to those for death from gynaecological cancers during the last half century. The impact of ovarian cancer on life expectancy clearly increased and the age-specific mortality tend to ageing.     

Partial logistic relevance vector machines in survival analysis

The use of relevance vector machines to flexibly model hazard rate functions is explored. This technique is adapted to survival analysis problems through the partial logistic approach. The method exploits the Bayesian automatic relevance determination procedure to obtain sparse solutions and it incorporates the flexibility of kernel-based models. Example results are presented on literature data from a head-and-neck cancer survival study using Gaussian and spline kernels. Sensitivity analysis is conducted to assess the influence of hyperprior distribution parameters. The proposed method is then contrasted with other flexible hazard regression methods, in particular the HARE model proposed by Kooperberg et al. [16]. A simulation study is conducted to carry out the comparison. The model developed in this paper exhibited good performance in the prediction of hazard rate. The application of this sparse Bayesian technique to a real cancer data set demonstrated that the proposed method can potentially reveal characteristics of the hazards, associated with the dynamics of the studied diseases, which may be missed by existing modeling approaches based on different perspectives on the bias vs. variance balance.     

Bayesian Elastic Net Tobit Quantile Regression

In this article, the problem of parameter estimation and variable selection in the Tobit quantile regression model is considered. A Tobit quantile regression with the elastic net penalty from a Bayesian perspective is proposed. Independent gamma priors are put on the l₁ norm penalty parameters. A novel aspect of the Bayesian elastic net Tobit quantile regression is to treat the hyperparameters of the gamma priors as unknowns and let the data estimate them along with other parameters. A Bayesian Tobit quantile regression with the adaptive elastic net penalty is also proposed. The Gibbs sampling computational technique is adapted to simulate the parameters from the posterior distributions. The proposed methods are demonstrated by both simulated and real data examples.     

Efficient Bayesian inference for COM-Poisson regression models

COM-Poisson regression is an increasingly popular model for count data. Its main advantage is that it permits to model separately the mean and the variance of the counts, thus allowing the same covariate to affect in different ways the average level and the variability of the response variable. A key limiting factor to the use of the COM-Poisson distribution is the calculation of the normalisation constant: its accurate evaluation can be time-consuming and is not always feasible. We circumvent this problem, in the context of estimating a Bayesian COM-Poisson regression, by resorting to the exchange algorithm, an MCMC method applicable to situations where the sampling model (likelihood) can only be computed up to a normalisation constant. The algorithm requires to draw from the sampling model, which in the case of the COM-Poisson distribution can be done efficiently using rejection sampling. We illustrate the method and the benefits of using a Bayesian COM-Poisson regression model, through a simulation and two real-world data sets with different levels of dispersion.     

Bayesian spatial regression models with closed skew normal correlated errors and missing observations

This paper is concerned with Bayesian estimation of a spatial regression model with skew non-Gaussian errors. The regression parameters are estimated by using a closed skew normal (CSN) distribution, which is closed under conditioning and linear combination. The proposed model captures skewness in the response variable. Sometimes, we may encounter missing observations in the response variable, accordingly we model and predict the missing observations by a Bayesian approach using Gibbs sampling methods. Next, a simulation study is performed to asses our model validity. Also, the proposed model in this work is applied to CO data from Tehran, the capital city of Iran. Then, the accuracy of the CSN and Gaussian models is compared by cross validation criterion.     

Bayesian spatial quantile regression for areal count data,with application on substitute care placements in Texas

Quantile regression (QR) allows one to model the effect of covariates across the entire response distribution, rather than only at the mean, but QR methods have been almost exclusively applied to continuous response variables and without considering spatial effects. Of the few studies that have performed QR on count data, none have included random spatial effects, which is an integral facet of the Bayesian spatial QR model for areal counts that we propose. Additionally, we introduce a simplifying alternative to the response variable transformation currently employed in the QR for counts literature. The efficacy of the proposed model is demonstrated via simulation study and on a real data application from the Texas Department of Family and Protective Services (TDFPS). Our model outperforms a comparable non-spatial model in both instances, as evidenced by the deviance information criterion (DIC) and coverage probabilities. With the TDFPS data, we identify one of four covariates, along with the intercept, as having a nonconstant effect across the response distribution.     

Bayesian adaptive Lasso for quantile regression models with nonignorably missing response data

Abstract

Handling data with the nonignorably missing mechanism is still a challenging problem in statistics. In this paper, we develop a fully Bayesian adaptive Lasso approach for quantile regression models with nonignorably missing response data, where the nonignorable missingness mechanism is specified by a logistic regression model. The proposed method extends the Bayesian Lasso by allowing different penalization parameters for different regression coefficients. Furthermore, a hybrid algorithm that combined the Gibbs sampler and Metropolis-Hastings algorithm is implemented to simulate the parameters from posterior distributions, mainly including regression coefficients, shrinkage coefficients, parameters in the non-ignorable missing models. Finally, some simulation studies and a real example are used to illustrate the proposed methodology.     

Bayesian neural tree models for nonparametric regression

Frequentist and Bayesian methods differ in many aspects but share some basic optimal properties. In real-life prediction problems, situations exist in which a model based on one of the above paradigms is preferable depending on some subjective criteria. Nonparametric classification and regression techniques, such as decision trees and neural networks, have both frequentist (classification and regression trees (CARTs) and artificial neural networks) as well as Bayesian counterparts (Bayesian CART and Bayesian neural networks) to learning from data. In this paper, we present two hybrid models combining the Bayesian and frequentist versions of CART and neural networks, which we call the Bayesian neural tree (BNT) models. BNT models can simultaneously perform feature selection and prediction, are highly flexible, and generalise well in settings with limited training observations. We study the statistical consistency of the proposed approaches and derive the optimal value of a vital model parameter. The excellent performance of the newly proposed BNT models is shown using simulation studies. We also provide some illustrative examples using a wide variety of standard regression datasets from a public available machine learning repository to show the superiority of the proposed models in comparison to popularly used Bayesian CART and Bayesian neural network models.     

Bayesian goodness-of-fit test for censored data

We propose a Bayesian computation and inference method for the Pearson-type chi-squared goodness-of-fit test with right-censored survival data. Our test statistic is derived from the classical Pearson chi-squared test using the differences between the observed and expected counts in the partitioned bins. In the Bayesian paradigm, we generate posterior samples of the model parameter using the Markov chain Monte Carlo procedure. By replacing the maximum likelihood estimator in the quadratic form with a random observation from the posterior distribution of the model parameter, we can easily construct a chi-squared test statistic. The degrees of freedom of the test equal the number of bins and thus is independent of the dimensionality of the underlying parameter vector. The test statistic recovers the conventional Pearson-type chi-squared structure. Moreover, the proposed algorithm circumvents the burden of evaluating the Fisher information matrix, its inverse and the rank of the variance–covariance matrix. We examine the proposed model diagnostic method using simulation studies and illustrate it with a real data set from a prostate cancer study.     

Models for papilloma multiplicity and regression: applications to transgenic mouse studies

In cancer studies that use transgenic or knockout mice, skin tumour counts are recorded over time to measure tumorigenicity. In these studies cancer biologists are interested in the effect of endogenous and/or exogenous factors on papilloma onset, multiplicity and regression. In this paper an analysis of data from a study conducted by the National Institute of Environmental Health Sciences on the effect of genetic factors on skin tumorigenesis is presented. Papilloma multiplicity and regression are modelled by using Bernoulli, Poisson and binomial latent variables, each of which can depend on covariates and previous outcomes. An EM algorithm is proposed for parameter estimation, and generalized estimating equations adjust for extra dependence between outcomes within individual animals. A Cox proportional hazards model is used to describe covariate effects on the onset of tumours.     

Bayesian correlated factor analysis of socio-demographic indicators

Recent changes in European family dynamics are often linked to common latent trends of economic and ideational change. Using Bayesian factor analysis, we extract three latent variables from eight socio-demographic indicators related to family formation, dissolution, and gender system and collected on 19 European countries within four periods (1970, 1980, 1990, 1998). The flexibility of the Bayesian approach allows us to introduce an innovative temporal factor model, adding the temporal dimension to the traditional factorial analysis. The underlying structure of the Bayesian factor model proposed reflects our idea of an autoregressive pattern in the latent variables relative to adjacent time periods. The results we obtain are consistent with current interpretations in European demographic trends.