Bayesian beta nonlinear models with constrained parameters to describe ruminal degradation kinetics

The models used to describe the kinetics of ruminal degradation are usually nonlinear models where the dependent variable is the proportion of degraded food. The method of least squares is the standard approach used to estimate the unknown parameters but this method can lead to unacceptable predictions. To solve this issue, a beta nonlinear model and the Bayesian perspective is proposed in this article. The application of standard methodologies to obtain prior distributions, such as the Jeffreys prior or the reference priors, involves serious difficulties here because this model is a nonlinear non-normal regression model, and the constrained parameters appear in the log-likelihood function through the Gamma function. This paper proposes an objective method to obtain the prior distribution, which can be applied to other models with similar complexity, can be easily implemented in OpenBUGS, and solves the problem of unacceptable predictions. The model is generalized to a larger class of models. The methodology was applied to real data with three models that were compared using the Deviance Information Criterion and the root mean square prediction error. A simulation study was performed to evaluate the coverage of the credible intervals.     

A beta-inflated mean regression model with mixed effects for fractional response variables

Abstract

In this article we propose a new mixed-effects regression model for fractional bounded response variables. Our model allows us to incorporate covariates directly to the expected value, so we can quantify exactly the influence of these covariates in the mean of the variable of interest rather than to the conditional mean. Estimation is carried out from a Bayesian perspective. Due to the complexity of the augmented posterior distribution, we use a Hamiltonian Monte Carlo algorithm, the No-U-Turn sampler, implemented using the Stan software. A simulation study was performed showing that our model has a better performance than other traditional longitudinal models for bounded variables. Finally, we applied our beta-inflated mean mixed-effects regression model to real data which consists of utilization of credit lines in the peruvian financial system.     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

Augmented mixed beta regression models with skew-normal independent distributions: Bayesian analysis of labor force data

Abstract

Augmented mixed beta regression models are suitable choices for modeling continuous response variables on the closed interval [0, 1]. The random eeceeects in these models are typically assumed to be normally distributed, but this assumption is frequently violated in some applied studies. In this paper, an augmented mixed beta regression model with skew-normal independent distribution for random effects are used. Next, we adopt a Bayesian approach for parameter estimation using the MCMC algorithm. The methods are then evaluated using some intensive simulation studies. Finally, the proposed models have applied to analyze a dataset from an Iranian Labor Force Survey.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. 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 the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

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.     

A Comparison of Frailty and Other Models for Bivariate Survival Data

Multivariate survival data arise when eachstudy subject may experience multiple events or when study subjectsare clustered into groups. Statistical analyses of such dataneed to account for the intra-cluster dependence through appropriatemodeling. Frailty models are the most popular for such failuretime data. However, there are other approaches which model thedependence structure directly. In this article, we compare thefrailty models for bivariate data with the models based on bivariateexponential and Weibull distributions. Bayesian methods providea convenient paradigm for comparing the two sets of models weconsider. Our techniques are illustrated using two examples.One simulated example demonstrates model choice methods developedin this paper and the other example, based on a practical dataset of onset of blindness among patients with diabetic Retinopathy,considers Bayesian inference using different models.     

Bayesian modelling of the mean and covariance matrix in normal nonlinear models

An important problem in statistics is the study of longitudinal data taking into account the effect of other explanatory variables such as treatments and time. In this paper, a new Bayesian approach for analysing longitudinal data is proposed. This innovative approach takes into account the possibility of having nonlinear regression structures on the mean and linear regression structures on the variance–covariance matrix of normal observations, and it is based on the modelling strategy suggested by Pourahmadi [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterizations, Biometrika, 87 (1999), pp. 667–690.]. We initially extend the classical methodology to accommodate the fitting of nonlinear mean models then we propose our Bayesian approach based on a generalization of the Metropolis–Hastings algorithm of Cepeda [E.C. Cepeda, Variability modeling in generalized linear models, Unpublished Ph.D. Thesis, Mathematics Institute, Universidade Federal do Rio de Janeiro, 2001]. Finally, we illustrate the proposed methodology by analysing one example, the cattle data set, that is used to study cattle growth.     

A class of beta regression models with multiplicative log-normal measurement errors

In this article, we propose a beta regression model with multiplicative log-normal measurement errors. Three estimation methods are presented, namely, naive, calibration regression, and pseudo likelihood. The nuisance parameters are estimated from a system of estimation equations using replicated data and these estimates are used to propose a pseudo likelihood function. A simulation study was performed to assess some properties of the proposed methods. Results from an example with a real dataset, including diagnostic tools, are also reported.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.     

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.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

The distribution of the first j digits of beta related random variables

This paper gives the discrete distribution of the first j significant digits of two random variables: (1) a beta variable with integer parameter n and the other parameter m > 0, and (2) the reciprocal of (1). As a special case for n=1, we obtain the distribution of the first j significant digits of the pwoers of uniformly distributed random variables. These generalize the results of Kennard and Reith (1981) and Friedberg (1984), who considered only uniformly distributed random variables.     

Bivariate degradation analysis of products based on Wiener processes and copulas

Modern highly reliable products usually have complex structure and many functions. This means that they may have two or more performance characteristics. All the performance characteristics can reflect the product's performance degradation over time, and they may be independent or dependent. If the performance characteristics are independent, they can be modelled separately. But if they are not independent, it is very important to find the joint distribution function of the performance characteristics for estimating the reliability of the product as accurately as possible. Here, we suppose that a product has two performance characteristics and the degradation paths of these two performance characteristics can be governed by a Wiener process with a time-scale transformation, and that the dependency of the performance characteristics can be described by a copula function. The parameters of the two performance characteristics and the copula function can be estimated jointly. The model in such a situation is very complicated and analytically intractable and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum-likelihood estimates of the parameters to be determined in an efficient manner. For an illustration of the proposed model, a numerical example about fatigue cracks is presented.     

Bivariate location–scale models for regression analysis, with applications to lifetime data

Summary.  The literature on multivariate linear regression includes multivariate normal models, models that are used in survival analysis and a variety of models that are used in other areas such as econometrics. The paper considers the class of location–scale models, which includes a large proportion of the preceding models. It is shown that, for complete data, the maximum likelihood estimators for regression coefficients in a linear location–scale framework are consistent even when the joint distribution is misspecified. In addition, gains in efficiency arising from the use of a bivariate model, as opposed to separate univariate models, are studied. A major area of application for multivariate regression models is to clustered, 'parallel' lifetime data, so we also study the case of censored responses. Estimators of regression coefficients are no longer consistent under model misspecification, but we give simulation results that show that the bias is small in many practical situations. Gains in efficiency from bivariate models are also examined in the censored data setting. The methodology in the paper is illustrated by using lifetime data from the Diabetic Retinopathy Study.     

Bivariate binary models of efficacy and toxicity in dose-ranging trials

Methods for the simultaneous analysis of the relationships of binary variables for efficacy and toxicity to dosage of an experimental drug are developed. Properties of two models of ‘within-dose’ dependence of efficacy and toxicity in parallel designs - one a bivariate analogue of the familiar univariate logistic model, and the other an adaptation of a general model developed by D.R. Cox– are explored. The cell probabilities predicted by these models are often quite similar to those predicted by a model of independence of efficacy and toxicity, but large discrepancies can occur when there is approximate equality of the median effective and median toxic doses. Asymptotic variances of estimates of parameters involved in assessing correlation are large when there is little or no dependence in the data, but parameters can be estimated with good precision in at least some cases of moderate to strong dependence between efficacy and toxicity.     

Estimating the parameters of mixed linear models with modal estimators

This paper presents a procedure to estimate the variance components and fixed effects of mixed linear models. The mode of the joint posterior distribution of all the parameters is obtained by an iterative technique.

The proposed method is illustrated with one-way and two-fold nested random models. Two numerical examples demonstrate the iterative solution.     

Estimation of the Association for Bivariate Interval-censored Failure Time Data

Abstract.  Multivariate failure time data frequently occur in medical studies and the dependence or association among survival variables is often of interest ( Biometrics , 51 , 1995, 1384; Stat. Med. , 18 , 1999, 3101; Biometrika , 87 , 2000, 879; J. Roy. Statist. Soc. Ser. B , 65 , 2003, 257). We study the problem of estimating the association between two related survival variables when they follow a copula model and only bivariate interval-censored failure time data are available. For the problem, a two-stage estimation procedure is proposed and the asymptotic properties of the proposed estimator are established. Simulation studies are conducted to assess the finite sample properties of the presented estimate and the results suggest that the method works well for practical situations. An example from an acquired immunodeficiency syndrome clinical trial is discussed.