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.     

Fully Bayesian logistic regression with hyper-LASSO priors for high-dimensional feature selection

Feature selection arises in many areas of modern science. For example, in genomic research, we want to find the genes that can be used to separate tissues of different classes (e.g. cancer and normal). One approach is to fit regression/classification models with certain penalization. In the past decade, hyper-LASSO penalization (priors) have received increasing attention in the literature. However, fully Bayesian methods that use Markov chain Monte Carlo (MCMC) for regression/classification with hyper-LASSO priors are still in lack of development. In this paper, we introduce an MCMC method for learning multinomial logistic regression with hyper-LASSO priors. Our MCMC algorithm uses Hamiltonian Monte Carlo in a restricted Gibbs sampling framework. We have used simulation studies and real data to demonstrate the superior performance of hyper-LASSO priors compared to LASSO, and to investigate the issues of choosing heaviness and scale of hyper-LASSO priors.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

Bayesian variable selection and estimation in maximum entropy quantile regression

Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis.     

Bayesian variable selection in logistic regression: predicting company earnings direction

This paper presents a Bayesian technique for the estimation of a logistic regression model including variable selection. As in Ou & Penman (1989), the model is used to predict the direction of company earnings, one year ahead, from a large set of accounting variables from financial statements. To estimate the model, the paper presents a Markov chain Monte Carlo sampling scheme that includes the variable selection technique of Smith & Kohn (1996) and the non-Gaussian estimation method of Mira & Tierney (2001). The technique is applied to data for companies in the United States and Australia. The results obtained compare favourably to the technique used by Ou & Penman (1989) for both regions.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Bayesian variable selection for multioutcome models through shared shrinkage

Variable selection over a potentially large set of covariates in a linear model is quite popular. In the Bayesian context, common prior choices can lead to a posterior expectation of the regression coefficients that is a sparse (or nearly sparse) vector with a few nonzero components, those covariates that are most important. This article extends the “global‐local” shrinkage idea to a scenario where one wishes to model multiple response variables simultaneously. Here, we have developed a variable selection method for a K‐outcome model (multivariate regression) that identifies the most important covariates across all outcomes. The prior for all regression coefficients is a mean zero normal with coefficient‐specific variance term that consists of a predictor‐specific factor (shared local shrinkage parameter) and a model‐specific factor (global shrinkage term) that differs in each model. The performance of our modeling approach is evaluated through simulation studies and a data example.     

A simulation based method for assessing the statistical significance of logistic regression models after common variable selection procedures

Classification models can demonstrate apparent prediction accuracy even when there is no underlying relationship between the predictors and the response. Variable selection procedures can lead to false positive variable selections and overestimation of true model performance. A simulation study was conducted using logistic regression with forward stepwise, best subsets, and LASSO variable selection methods with varying total sample sizes (20, 50, 100, 200) and numbers of random noise predictor variables (3, 5, 10, 15, 20, 50). Using our critical values can help reduce needless follow-up on variables having no true association with the outcome.     

Gibbs sampling methods for Bayesian quantile regression

This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Variable selection in quantile regression via Gibbs sampling

Due to computational challenges and non-availability of conjugate prior distributions, Bayesian variable selection in quantile regression models is often a difficult task. In this paper, we address these two issues for quantile regression models. In particular, we develop an informative stochastic search variable selection (ISSVS) for quantile regression models that introduces an informative prior distribution. We adopt prior structures which incorporate historical data into the current data by quantifying them with a suitable prior distribution on the model parameters. This allows ISSVS to search more efficiently in the model space and choose the more likely models. In addition, a Gibbs sampler is derived to facilitate the computation of the posterior probabilities. A major advantage of ISSVS is that it avoids instability in the posterior estimates for the Gibbs sampler as well as convergence problems that may arise from choosing vague priors. Finally, the proposed methods are illustrated with both simulation and real data.     

Graphics for studying logistic regression models

In this article we focus on logistic regression models for binary responses. An existing result shows that the log-odds can be modelled depending on the log of the ratio between the conditional densities of the predictors given the response variable. This suggests that relevant statistical information could be extracted investigating the inverse problem. Thus, we present different methods for studying the log-density ratio through graphs, which allow us to select which predictors are needed, and how they should be included in a logistic regression model. We also discuss data analysis examples based on real datasets available in literature in order to provide further insights into the methodology proposed.     

Prior distribution elicitation for generalized linear and piecewise-linear models

An elicitation method is proposed for quantifying subjective opinion about the regression coefficients of a generalized linear model. Opinion between a continuous predictor variable and the dependent variable is modelled by a piecewise-linear function, giving a flexible model that can represent a wide variety of opinion. To quantify his or her opinions, the expert uses an interactive computer program, performing assessment tasks that involve drawing graphs and bar-charts to specify medians and other quantiles. Opinion about the regression coefficients is represented by a multivariate normal distribution whose parameters are determined from the assessments. It is practical to use the procedure with models containing a large number of parameters. This is illustrated through practical examples and the benefit from using prior knowledge is examined through cross-validation.     

Bayesian elastic net single index quantile regression

Single index model conditional quantile regression is proposed in order to overcome the dimensionality problem in nonparametric quantile regression. In the proposed method, the Bayesian elastic net is suggested for single index quantile regression for estimation and variables selection. The Gaussian process prior is considered for unknown link function and a Gibbs sampler algorithm is adopted for posterior inference. The results of the simulation studies and numerical example indicate that our propose method, BENSIQReg, offers substantial improvements over two existing methods, SIQReg and BSIQReg. The BENSIQReg has consistently show a good convergent property, has the least value of median of mean absolute deviations and smallest standard deviations, compared to the other two methods.     

The extreme residuals in logistic regression models

Goodness-of-fit tests for logistic regression models using extreme residuals are considered. Approximations to the moments of the Pearson residuals are given for model fits made by maximum likelihood, minimum chi-square and weighted least squares and used to define modified residuals. Approximations to the critical values of the extreme statistics based on the ordinary and modified Pearson residuals are developed and assessed for the case of a single explanatory variable.     

Bayesian quantile regression for longitudinal data models

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained.     

A non-iterative Bayesian sampling algorithm for censored Student-t linear regression models

ABSTRACT

In this paper, we consider an effective Bayesian inference for censored Student-t linear regression model, which is a robust alternative to the usual censored Normal linear regression model. Based on the mixture representation of the Student-t distribution, we propose a non-iterative Bayesian sampling procedure to obtain independently and identically distributed samples approximately from the observed posterior distributions, which is different from the iterative Markov Chain Monte Carlo algorithm. We conduct model selection and influential analysis using the posterior samples to choose the best fitted model and to detect latent outliers. We illustrate the performance of the procedure through simulation studies, and finally, we apply the procedure to two real data sets, one is the insulation life data with right censoring and the other is the wage rates data with left censoring, and we get some interesting results.     

Bayesian analysis of two overdispersed poisson regression models

Shookri and Consul (1989) and Scollnik (1995) have previously considered the Bayesian analysis of an overdispersed generalized Poisson model. Scollnik (1995) also considered the Bayesian analysis of an ordinary Poisson and over-dispersed generalized Poisson mixture model. In this paper, we discuss the Bayesian analysis of these models when they are utilised in a regression context. Markov chain Monte Carlo methods are utilised, and an illustrative analysis is provided.     

Confidence bands for the logistic and probit regression models over intervals

This article presents methods for the construction of two-sided and one-sided simultaneous hyperbolic bands for the logistic and probit regression models when the predictor variable is restricted to a given interval. The bands are constructed based on the asymptotic properties of the maximum likelihood estimators. Past articles have considered building two-sided asymptotic confidence bands for the logistic model, such as Piegorsch and Casella (1988 Piegorsch, W.W., Casella, G. (1988). Confidence bands for logistic regression with restricted predictor variables. Biometrics 44:739–750.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). However, the confidence bands given by Piegorsch and Casella are conservative under a single interval restriction, and it is shown in this article that their bands can be sharpened using the methods proposed here. Furthermore, no method has yet appeared in the literature for constructing one-sided confidence bands for the logistic model, and no work has been done for building confidence bands for the probit model, over a limited range of the predictor variable. This article provides methods for computing critical points in these areas.