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.     

Pairwise- and marginal-likelihood estimation for the mixed Rasch model with binary data

A marginal–pairwise-likelihood estimation approach is examined in the mixed Rasch model with the binary response and logit link. This method belonging to the broad class of composite likelihood provides estimators with desirable asymptotic properties such as consistency and asymptotic normality. We study the performance of the proposed methodology when the random effect distribution is misspecified. A simulation study was conducted to compare this approach with the maximum marginal likelihood. The different results are also illustrated with an analysis of the real data set from a quality-of-life study.     

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.     

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.     

A Composite Likelihood Approach to Multivariate Survival Data

This paper is about the statistical analysis of multivariate survival data. We discuss the additive and multiplicative frailty models which have been the most popular models for multivariate survival data. As an alternative to the additive and multiplicative frailty models, we propose basing inference on a composite likelihood function that only requires modelling of the marginal distribution of pairs of failure times. Each marginal distribution of a pair of failure times is here assumed to follow a shared frailty model. The method is illustrated with a real-life example.     

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.     

General partially linear varying-coefficient transformation model with right censored data

In this paper, a unified maximum marginal likelihood estimation procedure is proposed for the analysis of right censored data using general partially linear varying-coefficient transformation models (GPLVCTM), which are flexible enough to include many survival models as its special cases. Unknown functional coefficients in the models are approximated by cubic B-spline polynomial. We estimate B-spline coefficients and regression parameters by maximizing marginal likelihood function. One advantage of this procedure is that it is free of both baseline and censoring distribution. Through simulation studies and a real data application (VA data from the Veteran's Administration Lung Cancer Study Clinical Trial), we illustrate that the proposed estimation procedure is accurate, stable and practical.     

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.     

Bias Reduction of Likelihood Estimators in Semiparametric Frailty Models

Abstract. Frailty models with a non‐parametric baseline hazard are widely used for the analysis of survival data. However, their maximum likelihood estimators can be substantially biased in finite samples, because the number of nuisance parameters associated with the baseline hazard increases with the sample size. The penalized partial likelihood based on a first‐order Laplace approximation still has non‐negligible bias. However, the second‐order Laplace approximation to a modified marginal likelihood for a bias reduction is infeasible because of the presence of too many complicated terms. In this article, we find adequate modifications of these likelihood‐based methods by using the hierarchical likelihood.     

Fitting via alternative random-effect models

When there are two alternative random-effect models leading to the same marginal model, inferences from one model can be used for the other model. We illustrate how a likelihood method for fitting models with independent random effects can be applied to seemingly very different models with correlated random effects. We also discuss some merits of using these alternative models.     

Properties and Applications of the Generalized Likelihood as a Summary Function for Prediction Problems

The generalized likelihood plays an important role in parametric inference for prediction and empirical Bayesian models. This paper emphasizes the utility of the generalized likelihood as a summarization procedure in general prediction models. Properties of the generalized likelihood when used in this setting, and examples of its use as a data analytic tool are given in a series of numerical examples.     

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.     

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.     

Forecast Combination and Model Averaging Using Predictive Measures

We extend the standard approach to Bayesian forecast combination by forming the weights for the model averaged forecast from the predictive likelihood rather than the standard marginal likelihood. The use of predictive measures of fit offers greater protection against in-sample overfitting when uninformative priors on the model parameters are used and improves forecast performance. For the predictive likelihood we argue that the forecast weights have good large and small sample properties. This is confirmed in a simulation study and in an application to forecasts of the Swedish inflation rate, where forecast combination using the predictive likelihood outperforms standard Bayesian model averaging using the marginal likelihood.     

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.     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

Likelihood-free stochastic approximation EM for inference in complex models

A maximum likelihood methodology for the parameters of models with an intractable likelihood is introduced. We produce a likelihood-free version of the stochastic approximation expectation-maximization (SAEM) algorithm to maximize the likelihood function of model parameters. While SAEM is best suited for models having a tractable “complete likelihood” function, its application to moderately complex models is a difficult or even impossible task. We show how to construct a likelihood-free version of SAEM by using the “synthetic likelihood” paradigm. Our method is completely plug-and-play, requires almost no tuning and can be applied to both static and dynamic models.     

MODEL SELECTION FOR PENALIZED SPLINE SMOOTHING USING AKAIKE INFORMATION CRITERIA

Two different forms of Akaike's information criterion (AIC) are compared for selecting the smooth terms in penalized spline additive mixed models. The conditional AIC (cAIC) has been used traditionally as a criterion for both estimating penalty parameters and selecting covariates in smoothing, and is based on the conditional likelihood given the smooth mean and on the effective degrees of freedom for a model fit. By comparison, the marginal AIC (mAIC) is based on the marginal likelihood from the mixed‐model formulation of penalized splines which has recently become popular for estimating smoothing parameters. To the best of the authors' knowledge, the use of mAIC for selecting covariates for smoothing in additive models is new. In the competing models considered for selection, covariates may have a nonlinear effect on the response, with the possibility of group‐specific curves. Simulations are used to compare the performance of cAIC and mAIC in model selection settings that have correlated and hierarchical smooth terms. In moderately large samples, both formulations of AIC perform extremely well at detecting the function that generated the data. The mAIC does better for simple functions, whereas the cAIC is more sensitive to detecting a true model that has complex and hierarchical terms.     

Weighting Method for a Linear Mixed Model

Maximum likelihood is a widely used estimation method in statistics. This method is model dependent and as such is criticized as being non robust. In this article, we consider using weighted likelihood method to make robust inferences for linear mixed models where weights are determined at both the subject level and the observation level. This approach is appropriate for problems where maximum likelihood is the basic fitting technique, but a subset of data points is discrepant with the model. It allows us to reduce the impact of outliers without complicating the basic linear mixed model with normally distributed random effects and errors. The weighted likelihood estimators are shown to be robust and asymptotically normal. Our simulation study demonstrates that the weighted estimates are much better than the unweighted ones when a subset of data points is far away from the rest. Its application to the analysis of deglutition apnea duration in normal swallows shows that the differences between the weighted and unweighted estimates are due to large amount of outliers in the data set.     

Profile likelihood in directed graphical models from BUGS output

We present a method for using posterior samples produced by the computer program BUGS (Bayesian inference Using Gibbs Sampling) to obtain approximate profile likelihood functions of parameters or functions of parameters in directed graphical models with incomplete data. The method can also be used to approximate integrated likelihood functions. It is easily implemented and it performs a good approximation. The profile likelihood represents an aspect of the parameter uncertainty which does not depend on the specification of prior distributions, and it can be used as a worthwhile supplement to BUGS that enable us to do both Bayesian and likelihood based analyses in directed graphical models.