Empirical and Simulated Adjustments of Composite Likelihood Ratio Statistics

Composite likelihood inference has gained much popularity thanks to its computational manageability and its theoretical properties. Unfortunately, performing composite likelihood ratio tests is inconvenient because of their awkward asymptotic distribution. There are many proposals for adjusting composite likelihood ratio tests in order to recover an asymptotic chi square distribution, but they all depend on the sensitivity and variability matrices. The same is true for Wald-type and score-type counterparts. In realistic applications sensitivity and variability matrices usually need to be estimated, but there are no comparisons of the performance of composite likelihood based statistics in such an instance. A comparison of the accuracy of inference based on the statistics considering two methods typically employed for estimation of sensitivity and variability matrices, namely an empirical method that exploits independent observations, and Monte Carlo simulation, is performed. The results in two examples involving the pairwise likelihood show that a very large number of independent observations should be available in order to obtain accurate coverages using empirical estimation, while limited simulation from the full model provides accurate results regardless of the availability of independent observations.Comment: 15 page

Approximate Bayesian Computation with composite score functions

Both Approximate Bayesian Computation (ABC) and composite likelihood methods are useful for Bayesian and frequentist inference, respectively, when the likelihood function is intractable. We propose to use composite likelihood score functions as summary statistics in ABC in order to obtain accurate approximations to the posterior distribution. This is motivated by the use of the score function of the full likelihood, and extended to general unbiased estimating functions in complex models. Moreover, we show that if the composite score is suitably standardised, the resulting ABC procedure is invariant to reparameterisations and automatically adjusts the curvature of the composite likelihood, and of the corresponding posterior distribution. The method is illustrated through examples with simulated data, and an application to modelling of spatial extreme rainfall data is discussed.Comment: Statistics and Computing (final version

A note on marginal posterior simulation via higher-order tail area approximations

We explore the use of higher-order tail area approximations for Bayesian simulation. These approximations give rise to an alternative simulation scheme to MCMC for Bayesian computation of marginal posterior distributions for a scalar parameter of interest, in the presence of nuisance parameters. Its advantage over MCMC methods is that samples are drawn independently with lower computational time and the implementation requires only standard maximum likelihood routines. The method is illustrated by a genetic linkage model, a normal regression with censored data and a logistic regression model

Robust approximate Bayesian inference

We discuss an approach for deriving robust posterior distributions from $M$-estimating functions using Approximate Bayesian Computation (ABC) methods. In particular, we use $M$-estimating functions to construct suitable summary statistics in ABC algorithms. The theoretical properties of the robust posterior distributions are discussed. Special attention is given to the application of the method to linear mixed models. Simulation results and an application to a clinical study demonstrate the usefulness of the method. An R implementation is also provided in the robustBLME package.Comment: This is a revised and personal manuscript version of the article that has been accepted for publication by Journal of Statistical Planning and Inferenc

Monte Carlo modified profile likelihood in models for clustered data

The main focus of the analysts who deal with clustered data is usually not on the clustering variables, and hence the group-specific parameters are treated as nuisance. If a fixed effects formulation is preferred and the total number of clusters is large relative to the single-group sizes, classical frequentist techniques relying on the profile likelihood are often misleading. The use of alternative tools, such as modifications to the profile likelihood or integrated likelihoods, for making accurate inference on a parameter of interest can be complicated by the presence of nonstandard modelling and/or sampling assumptions. We show here how to employ Monte Carlo simulation in order to approximate the modified profile likelihood in some of these unconventional frameworks. The proposed solution is widely applicable and is shown to retain the usual properties of the modified profile likelihood. The approach is examined in two instances particularly relevant in applications, i.e. missing-data models and survival models with unspecified censoring distribution. The effectiveness of the proposed solution is validated via simulation studies and two clinical trial applications

Objective Bayesian higher-order asymptotics in models with nuisance parameters

We discuss higher-order approximations to the marginal posterior distribution for a scalar parameter of interest in the presence of nuisance parameters. These higher-order approximations are obtained using a suitable matching prior. The proposed procedure has several advantages since it does not require the elicitation on the nuisance parameter, neither numerical integration or MCMC simulation, and it enables us to perform accurate Bayesian inference even for very small sample sizes. Numerical illustrations are given for models of practical interest, such as linear non-normal models and logistic regression. We also illustrate how the proposed accurate approximation can routinely be applied in practice using results from likelihood asymptotics and the R package bundle ho

Median bias reduction in random-effects meta-analysis and meta-regression

The reduction of the mean or median bias of the maximum likelihood estimator in regular parametric models can be achieved through the additive adjustment of the score equations. In this paper, we derive the adjusted score equations for median bias reduction in random-effects meta-analysis and meta-regression models and derive efficient estimation algorithms. The median bias-reducing adjusted score functions are found to be the derivatives of a penalised likelihood. The penalised likelihood is used to form a penalised likelihood ratio statistic which has known limiting distribution and can be used for carrying out hypothesis tests or for constructing confidence intervals for either the fixed-effect parameters or the variance component. Simulation studies and real data applications are used to assess the performance of estimation and inference based on the median bias-reducing penalised likelihood and compare it to recently proposed alternatives. The results provide evidence on the effectiveness of median bias reduction in improving estimation and likelihood-based inference