Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.     

Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models

The ordinal probit, univariate or multivariate, is a generalized linear model (GLM) structure that arises frequently in such disparate areas of statistical applications as medicine and econometrics. Despite the straightforwardness of its implementation using the Gibbs sampler, the ordinal probit may present challenges in obtaining satisfactory convergence.We present a multivariate Hastings-within-Gibbs update step for generating latent data and bin boundary parameters jointly, instead of individually from their respective full conditionals. When the latent data are parameters of interest, this algorithm substantially improves Gibbs sampler convergence for large datasets. We also discuss Monte Carlo Markov chain (MCMC) implementation of cumulative logit (proportional odds) and cumulative complementary log-log (proportional hazards) models with latent data.     

Bayesian latent variable models for clustered mixed outcomes

A general framework is proposed for modelling clustered mixed outcomes. A mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables, and an arbitrary function relates the underlying variables to be observed outcomes. The model accommodates multilevel data structures, general covariate effects and distinct link functions and error distributions for each underlying variable. Within the framework proposed, novel models are developed for clustered multiple binary, unordered categorical and joint discrete and continuous outcomes. A Markov chain Monte Carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables. Because of the flexibility of the modelling framework and estimation procedure, extensions to ordered categorical outcomes and more complex data structures are straightforward. The methods are illustrated by using data from a reproductive toxicity study.     

Exact and Monte Carlo calculations of integrated likelihoods for the latent class model

The latent class model or multivariate multinomial mixture is a powerful approach for clustering categorical data. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. In this paper, we exploit the fact that a fully Bayesian analysis with Jeffreys non-informative prior distributions does not involve technical difficulty to propose an exact expression of the integrated complete-data likelihood, which is known as being a meaningful model selection criterion in a clustering perspective. Similarly, a Monte Carlo approximation of the integrated observed-data likelihood can be obtained in two steps: an exact integration over the parameters is followed by an approximation of the sum over all possible partitions through an importance sampling strategy. Then, the exact and the approximate criteria experimentally compete, respectively, with their standard asymptotic BIC approximations for choosing the number of mixture components. Numerical experiments on simulated data and a biological example highlight that asymptotic criteria are usually dramatically more conservative than the non-asymptotic presented criteria, not only for moderate sample sizes as expected but also for quite large sample sizes. This research highlights that asymptotic standard criteria could often fail to select some interesting structures present in the data.     

Latent variable models with mixed continuous and polytomous data

Owing to the nature of the problems and the design of questionnaires, discrete polytomous data are very common in behavioural, medical and social research. Analysing the relationships between the manifest and the latent variables based on mixed polytomous and continuous data has proven to be difficult. A general structural equation model is investigated for these mixed outcomes. Maximum likelihood (ML) estimates of the unknown thresholds and the structural parameters in the covariance structure are obtained. A Monte Carlo–EM algorithm is implemented to produce the ML estimates. It is shown that closed form solutions can be obtained for the M-step, and estimates of the latent variables are produced as a by-product of the analysis. The method is illustrated with a real example.     

Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution

In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of a modified Weibull distribution based on a complete sample. While maximum-likelihood estimation (MLE) is the most used method for parameter estimation, MCMC has recently emerged as a good alternative. When applied to parameter estimation, MCMC methods have been shown to be easy to implement computationally, the estimates always exist and are statistically consistent, and their probability intervals are convenient to construct. Details of applying MCMC to parameter estimation for the modified Weibull model are elaborated and a numerical example is presented to illustrate the methods of inference discussed in this paper. To compare MCMC with MLE, a simulation study is provided, and the differences between the estimates obtained by the two algorithms are examined.     

Reversible jump Markov chain Monte Carlo algorithms for Bayesian variable selection in logistic mixed models

In this article, to reduce computational load in performing Bayesian variable selection, we used a variant of reversible jump Markov chain Monte Carlo methods, and the Holmes and Held (HH) algorithm, to sample model index variables in logistic mixed models involving a large number of explanatory variables. Furthermore, we proposed a simple proposal distribution for model index variables, and used a simulation study and real example to compare the performance of the HH algorithm with our proposed and existing proposal distributions. The results show that the HH algorithm with our proposed proposal distribution is a computationally efficient and reliable selection method.     

General location multivariate latent variable models for mixed correlated bounded continuous,ordinal, and nominal responses with non-ignorable missing data

Using a multivariate latent variable approach, this article proposes some new general models to analyze the correlated bounded continuous and categorical (nominal or/and ordinal) responses with and without non-ignorable missing values. First, we discuss regression methods for jointly analyzing continuous, nominal, and ordinal responses that we motivated by analyzing data from studies of toxicity development. Second, using the beta and Dirichlet distributions, we extend the models so that some bounded continuous responses are replaced for continuous responses. The joint distribution of the bounded continuous, nominal and ordinal variables is decomposed into a marginal multinomial distribution for the nominal variable and a conditional multivariate joint distribution for the bounded continuous and ordinal variables given the nominal variable. We estimate the regression parameters under the new general location models using the maximum-likelihood method. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms of the model on the maximal normal curvature. The proposed models are applied to two data sets: BMI, Steatosis and Osteoporosis data and Tehran household expenditure budgets.     

Bias‐reduced marginal Akaike information criteria based on a Monte Carlo method for linear mixed‐effects models

In linear mixed‐effects (LME) models, if a fitted model has more random‐effect terms than the true model, a regularity condition required in the asymptotic theory may not hold. In such cases, the marginal Akaike information criterion (AIC) is positively biased for (?2) times the expected log‐likelihood. The asymptotic bias of the maximum log‐likelihood as an estimator of the expected log‐likelihood is evaluated for LME models with balanced design in the context of parameter‐constrained models. Moreover, bias‐reduced marginal AICs for LME models based on a Monte Carlo method are proposed. The performance of the proposed criteria is compared with existing criteria by using example data and by a simulation study. It was found that the bias of the proposed criteria was smaller than that of the existing marginal AIC when a larger model was fitted and that the probability of choosing a smaller model incorrectly was decreased.     

Marginal likelihood estimation from the Metropolis output: tips and tricks for efficient implementation in generalized linear latent variable models

The marginal likelihood can be notoriously difficult to compute, and particularly so in high-dimensional problems. Chib and Jeliazkov employed the local reversibility of the Metropolis–Hastings algorithm to construct an estimator in models where full conditional densities are not available analytically. The estimator is free of distributional assumptions and is directly linked to the simulation algorithm. However, it generally requires a sequence of reduced Markov chain Monte Carlo runs which makes the method computationally demanding especially in cases when the parameter space is large. In this article, we study the implementation of this estimator on latent variable models which embed independence of the responses to the observables given the latent variables (conditional or local independence). This property is employed in the construction of a multi-block Metropolis-within-Gibbs algorithm that allows to compute the estimator in a single run, regardless of the dimensionality of the parameter space. The counterpart one-block algorithm is also considered here, by pointing out the difference between the two approaches. The paper closes with the illustration of the estimator in simulated and real-life data sets.     

Fuzzy clustering algorithm for latent class model

The expectation maximization (EM) algorithm is a widely used parameter approach for estimating the parameters of multivariate multinomial mixtures in a latent class model. However, this approach has unsatisfactory computing efficiency. This study proposes a fuzzy clustering algorithm (FCA) based on both the maximum penalized likelihood (MPL) for the latent class model and the modified penalty fuzzy c-means (PFCM) for normal mixtures. Numerical examples confirm that the FCA-MPL algorithm is more efficient (that is, requires fewer iterations) and more computationally effective (measured by the approximate relative ratio of accurate classification) than the EM algorithm.     

Variable selection in partially linear hazard regression for multivariate failure time data

The aim of this paper is to explore variable selection approaches in the partially linear proportional hazards model for multivariate failure time data. A new penalised pseudo-partial likelihood method is proposed to select important covariates. Under certain regularity conditions, we establish the rate of convergence and asymptotic normality of the resulting estimates. We further show that the proposed procedure can correctly select the true submodel, as if it was known in advance. Both simulated and real data examples are presented to illustrate the proposed methodology.     

A Bayesian Approach for Zero-Inflated Count Regression Models by Using the Reversible Jump Markov Chain Monte Carlo Method and an Application

In this study, estimation of the parameters of the zero-inflated count regression models and computations of posterior model probabilities of the log-linear models defined for each zero-inflated count regression models are investigated from the Bayesian point of view. In addition, determinations of the most suitable log-linear and regression models are investigated. It is known that zero-inflated count regression models cover zero-inflated Poisson, zero-inflated negative binomial, and zero-inflated generalized Poisson regression models. The classical approach has some problematic points but the Bayesian approach does not have similar flaws. This work points out the reasons for using the Bayesian approach. It also lists advantages and disadvantages of the classical and Bayesian approaches. As an application, a zoological data set, including structural and sampling zeros, is used in the presence of extra zeros. In this work, it is observed that fitting a zero-inflated negative binomial regression model creates no problems at all, even though it is known that fitting a zero-inflated negative binomial regression model is the most problematic procedure in the classical approach. Additionally, it is found that the best fitting model is the log-linear model under the negative binomial regression model, which does not include three-way interactions of factors.     

The characterization of Monte Carlo errors for the quantification of the value of forensic evidence

Recent developments in forensic science have lead to a proliferation of methods for quantifying the probative value of evidence by constructing a Bayes Factor that allows a decision-maker to select between the prosecution and defense models. Unfortunately, the analytical form of a Bayes Factor is often computationally intractable. A typical approach in statistics uses Monte Carlo integration to numerically approximate the marginal likelihoods composing the Bayes Factor. This article focuses on developing a generally applicable method for characterizing the numerical error associated with Monte Carlo integration techniques used in constructing the Bayes Factor. The derivation of an asymptotic Monte Carlo standard error (MCSE) for the Bayes Factor will be presented and its applicability to quantifying the value of evidence will be explored using a simulation-based example involving a benchmark data set. The simulation will also explore the effect of prior choice on the Bayes Factor approximations and corresponding MCSEs.     

The effects of different choices of order for autoregressive approximation on the Gaussian likelihood estimates for ARMA models

This paper investigates, by means of Monte Carlo simulation, the effects of different choices of order for autoregressive approximation on the fully efficient parameter estimates for autoregressive moving average models. Four order selection criteria, AIC, BIC, HQ and PKK, were compared and different model structures with varying sample sizes were used to contrast the performance of the criteria. Some asymptotic results which provide a useful guide for assessing the performance of these criteria are presented. The results of this comparison show that there are marked differences in the accuracy implied using these alternative criteria in small sample situations and that it is preferable to apply BIC criterion, which leads to greater precision of Gaussian likelihood estimates, in such cases. Implications of the findings of this study for the estimation of time series models are highlighted.