Composite Likelihood Estimation for Multivariate Probit Latent Traits Models

Inference in generalized linear mixed models with multivariate random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. This article presents an inferential methodology based on the marginal composite likelihood approach for the probit latent traits models. This method belonging to the broad class of pseudo-likelihood involves marginal pairs probabilities of the responses which has analytical expression. The different results are illustrated with a simulation study and with an analysis of real data from health related quality of life.     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.     

Another look at fisher'S solution to the problem of the weighted mean

The marginal likelihood function of the common mean of two normal populations is considered. Transformed versions of the marginal likelihood function are plotted to illustrate the difficulties of the point estimate approach. Conditions for bimodality and asymmetry are also discussed     

Pointwise and functional approximations in Monte Carlo maximum likelihood estimation

We consider the use of Monte Carlo methods to obtain maximum likelihood estimates for random effects models and distinguish between the pointwise and functional approaches. We explore the relationship between the two approaches and compare them with the EM algorithm. The functional approach is more ambitious but the approximation is local in nature which we demonstrate graphically using two simple examples. A remedy is to obtain successively better approximations of the relative likelihood function near the true maximum likelihood estimate. To save computing time, we use only one Newton iteration to approximate the maximiser of each Monte Carlo likelihood and show that this is equivalent to the pointwise approach. The procedure is applied to fit a latent process model to a set of polio incidence data. The paper ends by a comparison between the marginal likelihood and the recently proposed hierarchical likelihood which avoids integration altogether.     

Structural inference of the parameters of the heteroscedastic simultaneous equation model

The structural approach of inference for the parameters of a simultaneous equation model with heteroscedastic error variance is investigated in this paper. The joint and the marginal structural distributions for the coefficients of the exogenous variables and the scale parameters of the error variables, and the marginal likelihood function of the coefficients of the endogenous variables have been derived. The estimates are directly obtainable from the structural distribution and the marginal likelihood function of the parameters. The marginal distribution of a subset of coefficients of exogenous variables provides the basis for making inference for a particular subset of parameter of interest.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Bayesian marginal inference via candidate's formula

Computing marginal probabilities is an important and fundamental issue in Bayesian inference. We present a simple method which arises from a likelihood identity for computation. The likelihood identity, called Candidate's formula, sets the marginal probability as a ratio of the prior likelihood to the posterior density. Based on Markov chain Monte Carlo output simulated from the posterior distribution, a nonparametric kernel estimate is used to estimate the posterior density contained in that ratio. This derived nonparametric Candidate's estimate requires only one evaluation of the posterior density estimate at a point. The optimal point for such evaluation can be chosen to minimize the expected mean square relative error. The results show that the best point is not necessarily the posterior mode, but rather a point compromising between high density and low Hessian. For high dimensional problems, we introduce a variance reduction approach to ease the tension caused by data sparseness. A simulation study is presented.     

Exact dimensionality selection for Bayesian PCA

We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high-dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a normal-gamma prior distribution. In this context, we exhibit a closed-form expression of the marginal likelihood which allows to infer an optimal number of components. We also propose a heuristic based on the expected shape of the marginal likelihood curve in order to choose the hyperparameters. In nonasymptotic frameworks, we show on simulated data that this exact dimensionality selection approach is competitive with both Bayesian and frequentist state-of-the-art methods.     

Profile likelihood approaches for semiparametric copula and frailty models for clustered survival data

ABSTRACT

In clustered survival data, the dependence among individual survival times within a cluster has usually been described using copula models and frailty models. In this paper we propose a profile likelihood approach for semiparametric copula models with different cluster sizes. We also propose a likelihood ratio method based on profile likelihood for testing the absence of association parameter (i.e. test of independence) under the copula models, leading to the boundary problem of the parameter space. For this purpose, we show via simulation study that the proposed likelihood ratio method using an asymptotic chi-square mixture distribution performs well as sample size increases. We compare the behaviors of the two models using the profile likelihood approach under a semiparametric setting. The proposed method is demonstrated using two well-known data sets.     

A SEMIPARAMETRIC MIXTURE MODEL FOR THE ANALYSIS OF COMPETING RISKS DATA

This paper deals with the regression analysis of failure time data when there are censoring and multiple types of failures. We propose a semiparametric generalization of a parametric mixture model of Larson & Dinse (1985), for which the marginal probabilities of the various failure types are logistic functions of the covariates. Given the type of failure, the conditional distribution of the time to failure follows a proportional hazards model. A marginal like lihood approach to estimating regression parameters is suggested, whereby the baseline hazard functions are eliminated as nuisance parameters. The Monte Carlo method is used to approximate the marginal likelihood; the resulting function is maximized easily using existing software. Some guidelines for choosing the number of Monte Carlo replications are given. Fixing the regression parameters at their estimated values, the full likelihood is maximized via an EM algorithm to estimate the baseline survivor functions. The methods suggested are illustrated using the Stanford heart transplant data.     

Composite Marginal Likelihoods to the Normal Bradley-Terry Model

Inference in generalized linear mixed models with crossed random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. This article presents two inferential approaches based on the marginal composite likelihood for the normal Bradley-Terry model. The two approaches are illustrated by a simulation study to evaluate their performance. Thereafter, the asymptotic variances of the estimated variance component are compared.     

Generalized Estimating Equations to Binary Probit Model

Inference in generalized linear mixed models with multivariate random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. This article presents an inferential methodology based on the generalized estimating equations for the probit latent traits models. This method belonging to the broad class of semi parametric approaches involves marginal joint moments of order 1 and 2, which has analytical expression. The different results are illustrated with a simulation study.     

A multivariate logit model with marginal canonical association

In this work, a generalization of the Goodman Association Model to the case of q, q > 2, categorical variables which is based on the idea of marginal modelling discussed by Gloneck–McCullagh is introduced; the difference between the proposed generalization and two models, previously introduced by Becker and Colombi, is discussed. The Becker generalization is not a marginal model because it does not imply Logit Models for the marginal probabilities, and because it is based on the conditional approach of modelling the association. The Colombi model is only partially a marginal model because it uses simple logit models for the univariate marginal probabilities but is based on the conditional approach of modelling the association. It is also shown that the maximum likelihood estimation of the parameters of the new model is feasible and, to compute the maximum likelihood estimates, an algorithm is proposed, which is a numerically convenient compromise between the constrained optimization approach of Lang and the straightforward use of the Fisher Scoring Algorithm suggested by Glonek–McCullagh.Finally, the proposed model is used to analyze a data set concerning work accidents which occurred to workers at some Italian firms during the years 1994–1996.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

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.     

Likelihood-Based Inference for the Shape Parameter of a Two Parameter Weibull Distribution

Based on a Type 2 censored sample, we use the likelihood-based approach to draw likelihood inference on the shape parameter gamma of a two-parameter Weibull distribution. In particular, we derive the profile, conditional and marginal likelihoods of gamma. Numerical results along with some concluding remarks regarding the use of likelihood-based methods for inference are provided.     

On marginal likelihood inference for the intra-class correlation coefficient

A p-component set of responses have been constructed by a location-scale transformation to a p-component set of error variables, the covariance matrix of the set of error variables being of intra-class covariance structure:all variances being unity, and covariance being equal [IML0001]. A sample of size n has been described as a conditional structural model, conditional on the value of the intra-class correlation coefficient ρ. The conditional technique of structural inference provides the marginal likelihood function of ρ based on the standardized residuals. For the normal case, the marginal likelihood function of ρ is seen to be dependent on the standardized residuals through the sample intra-class correlation coefficient. By the likelihood modulation technique, the nonnull distribution of the sample intra-class correlation coefficient has also been obtained.     

Statistical Inference for the Multidimensional Mixed Rasch Model

Inference in generalized linear mixed models with multivariate random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. This article presents an inferential methodology based on the GEE approach. This method involves the approximations of the marginal likelihood and joint moments of the variables. It is also proposed an approximate Akaike and Bayesian information criterions based on the approximate marginal likelihood using the estimation of the parameters by the GEE approach. The different results are illustrated with a simulation study and with an analysis of real data from health-related quality of life.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Inference about regression parameters using highly stratified survey count data with over-dispersion and repeated measurements

We study methods to estimate regression and variance parameters for over-dispersed and correlated count data from highly stratified surveys. Our application involves counts of fish catches from stratified research surveys and we propose a novel model in fisheries science to address changes in survey protocols. A challenge with this model is the large number of nuisance parameters which leads to computational issues and biased statistical inferences. We use a computationally efficient profile generalized estimating equation method and compare it to marginal maximum likelihood (MLE) and restricted MLE (REML) methods. We use REML to address bias and inaccurate confidence intervals because of many nuisance parameters. The marginal MLE and REML approaches involve intractable integrals and we used a new R package that is designed for estimating complex nonlinear models that may include random effects. We conclude from simulation analyses that the REML method provides more reliable statistical inferences among the three methods we investigated.