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.     

A simulated annealing version of the EM algorithm for non-Gaussian deconvolution

The Expectation–Maximization (EM) algorithm is a very popular technique for maximum likelihood estimation in incomplete data models. When the expectation step cannot be performed in closed form, a stochastic approximation of EM (SAEM) can be used. Under very general conditions, the authors have shown that the attractive stationary points of the SAEM algorithm correspond to the global and local maxima of the observed likelihood. In order to avoid convergence towards a local maxima, a simulated annealing version of SAEM is proposed. An illustrative application to the convolution model for estimating the coefficients of the filter is given.     

A parameter expansion version of the SAEM algorithm

The EM algorithm and its extensions are very popular tools for maximum likelihood estimation in incomplete data setting. However, one of the limitations of these methods is their slow convergence. The PX-EM (parameter-expanded EM) algorithm was proposed by Liu, Rubin and Wu to make EM much faster. On the other hand, stochastic versions of EM are powerful alternatives of EM when the E-step is untractable in a closed form. In this paper we propose the PX-SAEM which is a parameter expansion version of the so-called SAEM (Stochastic Approximation version of EM). PX-SAEM is shown to accelerate SAEM and improve convergence toward the maximum likelihood estimate in a parametric framework. Numerical examples illustrate the behavior of PX-SAEM in linear and nonlinear mixed effects models.     

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.     

Analysis of generalized linear mixed models via a stochastic approximation algorithm with Markov chain Monte-Carlo method

In recent years much effort has been devoted to maximum likelihood estimation of generalized linear mixed models. Most of the existing methods use the EM algorithm, with various techniques in handling the intractable E-step. In this paper, a new implementation of a stochastic approximation algorithm with Markov chain Monte Carlo method is investigated. The proposed algorithm is computationally straightforward and its convergence is guaranteed. A simulation and three real data sets, including the challenging salamander data, are used to illustrate the procedure and to compare it with some existing methods. The results indicate that the proposed algorithm is an attractive alternative for problems with a large number of random effects or with high dimensional intractable integrals in the likelihood function.     

Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation

We propose a two-stage algorithm for computing maximum likelihood estimates for a class of spatial models. The algorithm combines Markov chain Monte Carlo methods such as the Metropolis–Hastings–Green algorithm and the Gibbs sampler, and stochastic approximation methods such as the off-line average and adaptive search direction. A new criterion is built into the algorithm so stopping is automatic once the desired precision has been set. Simulation studies and applications to some real data sets have been conducted with three spatial models. We compared the algorithm proposed with a direct application of the classical Robbins–Monro algorithm using Wiebe's wheat data and found that our procedure is at least 15 times faster.     

Variational Bayes for estimating the parameters of a hidden Potts model

Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets.     

An improved SAEM algorithm for maximum likelihood estimation in mixtures of non linear mixed effects models

We propose a new methodology for maximum likelihood estimation in mixtures of non linear mixed effects models (NLMEM). Such mixtures of models include mixtures of distributions, mixtures of structural models and mixtures of residual error models. Since the individual parameters inside the NLMEM are not observed, we propose to combine the EM algorithm usually used for mixtures models when the mixture structure concerns an observed variable, with the Stochastic Approximation EM (SAEM) algorithm, which is known to be suitable for maximum likelihood estimation in NLMEM and also has nice theoretical properties. The main advantage of this hybrid procedure is to avoid a simulation step of unknown group labels required by a “full” version of SAEM. The resulting MSAEM (Mixture SAEM) algorithm is now implemented in the Monolix software. Several criteria for classification of subjects and estimation of individual parameters are also proposed. Numerical experiments on simulated data show that MSAEM performs well in a general framework of mixtures of NLMEM. Indeed, MSAEM provides an estimator close to the maximum likelihood estimator in very few iterations and is robust with regard to initialization. An application to pharmacokinetic (PK) data demonstrates the potential of the method for practical applications.     

Estimation in nonlinear mixed-effects models using heavy-tailed distributions

Nonlinear mixed-effects models are very useful to analyze repeated measures data and are used in a variety of applications. Normal distributions for random effects and residual errors are usually assumed, but such assumptions make inferences vulnerable to the presence of outliers. In this work, we introduce an extension of a normal nonlinear mixed-effects model considering a subclass of elliptical contoured distributions for both random effects and residual errors. This elliptical subclass, the scale mixtures of normal (SMN) distributions, includes heavy-tailed multivariate distributions, such as Student-t, the contaminated normal and slash, among others, and represents an interesting alternative to outliers accommodation maintaining the elegance and simplicity of the maximum likelihood theory. We propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed-effects and variance components, using a stochastic approximation of the EM algorithm. We compare the performance of the normal and the SMN models with two real data sets.     

Automatic choice of driving values in Monte Carlo likelihood approximation via posterior simulations

For models with random effects or missing data, the likelihood function is sometimes intractable analytically but amenable to Monte Carlo approximation. To get a good approximation, the parameter value that drives the simulations should be sufficiently close to the maximum likelihood estimate (MLE) which unfortunately is unknown. Introducing a working prior distribution, we express the likelihood function as a posterior expectation and approximate it using posterior simulations. If the sample size is large, the sample information is likely to outweigh the prior specification and the posterior simulations will be concentrated around the MLE automatically, leading to good approximation of the likelihood near the MLE. For smaller samples, we propose to use the current posterior as the next prior distribution to make the posterior simulations closer to the MLE and hence improve the likelihood approximation. By using the technique of data duplication, we can simulate from the sharpened posterior distribution without actually updating the prior distribution. The suggested method works well in several test cases. A more complex example involving censored spatial data is 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.     

Geostatistical Modelling Using Non‐Gaussian Matérn Fields

This work provides a class of non‐Gaussian spatial Matérn fields which are useful for analysing geostatistical data. The models are constructed as solutions to stochastic partial differential equations driven by generalized hyperbolic noise and are incorporated in a standard geostatistical setting with irregularly spaced observations, measurement errors and covariates. A maximum likelihood estimation technique based on the Monte Carlo expectation‐maximization algorithm is presented, and a Monte Carlo method for spatial prediction is derived. Finally, an application to precipitation data is presented, and the performance of the non‐Gaussian models is compared with standard Gaussian and transformed Gaussian models through cross‐validation.     

Regression models for binary longitudinal responses

Some conditional models to deal with binary longitudinal responses are proposed, extending random effects models to include serial dependence of Markovian form, and hence allowing for quite general association structures between repeated observations recorded on the same individual. The presence of both these components implies a form of dependence between them, and so a complicated expression for the resulting likelihood. To handle this problem, we introduce, as a first instance, what Follmann and Wu (1995) called, in a different setting, an approximate conditional model, which represents an optimal choice for the general framework of categorical longitudinal responses. Then we define two more formally correct models for the binary case, with no assumption about the distribution of the random effect. All of the discussed models are estimated by means of an EM algorithm for nonparametric maximum likelihood. The algorithm, an adaptation of that used by Aitkin (1996) for the analysis of overdispersed generalized linear models, is initially derived as a form of Gaussian quadrature, and then extended to a completely unknown mixing distribution. A large scale simulation work is described to explore the behaviour of the proposed approaches in a number of different situations.     

Likelihood-based inference for censored linear regression models with scale mixtures of skew-normal distributions

In many studies, the data collected are subject to some upper and lower detection limits. Hence, the responses are either left or right censored. A complication arises when these continuous measures present heavy tails and asymmetrical behavior; simultaneously. For such data structures, we propose a robust-censored linear model based on the scale mixtures of skew-normal (SMSN) distributions. The SMSN is an attractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash, skew-contaminated normal and the entire family of scale mixtures of normal (SMN) distributions as special cases. We propose a fast estimation procedure to obtain the maximum likelihood (ML) estimates of the parameters, using a stochastic approximation of the EM (SAEM) algorithm. This approach allows us to estimate the parameters of interest easily and quickly, obtaining as a byproducts the standard errors, predictions of unobservable values of the response and the log-likelihood function. The proposed methods are illustrated through real data applications and several simulation studies.     

Likelihood Inference for Unions of Interacting Discs

Abstract. This is probably the first paper which discusses likelihood inference for a random set using a germ‐grain model, where the individual grains are unobservable, edge effects occur and other complications appear. We consider the case where the grains form a disc process modelled by a marked point process, where the germs are the centres and the marks are the associated radii of the discs. We propose to use a recent parametric class of interacting disc process models, where the minimal sufficient statistic depends on various geometric properties of the random set, and the density is specified with respect to a given marked Poisson model (i.e. a Boolean model). We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analysing Peter Diggle's heather data set, where we discuss the results of simulation‐based maximum likelihood inference and the effect of specifying different reference Poisson models.     

A Mixed Model Approach for Geoadditive Hazard Regression

Abstract.  Mixed model based approaches for semiparametric regression have gained much interest in recent years, both in theory and application. They provide a unified and modular framework for penalized likelihood and closely related empirical Bayes inference. In this article, we develop mixed model methodology for a broad class of Cox-type hazard regression models where the usual linear predictor is generalized to a geoadditive predictor incorporating non-parametric terms for the (log-)baseline hazard rate, time-varying coefficients and non-linear effects of continuous covariates, a spatial component, and additional cluster-specific frailties. Non-linear and time-varying effects are modelled through penalized splines, while spatial components are treated as correlated random effects following either a Markov random field or a stationary Gaussian random field prior. Generalizing existing mixed model methodology, inference is derived using penalized likelihood for regression coefficients and (approximate) marginal likelihood for smoothing parameters. In a simulation we study the performance of the proposed method, in particular comparing it with its fully Bayesian counterpart using Markov chain Monte Carlo methodology, and complement the results by some asymptotic considerations. As an application, we analyse leukaemia survival data from northwest England.     

Stochastic Differential Mixed-Effects Models

Abstract.  Stochastic differential equations have been shown useful in describing random continuous time processes. Biomedical experiments often imply repeated measurements on a series of experimental units and differences between units can be represented by incorporating random effects into the model. When both system noise and random effects are considered, stochastic differential mixed-effects models ensue. This class of models enables the simultaneous representation of randomness in the dynamics of the phenomena being considered and variability between experimental units, thus providing a powerful modelling tool with immediate applications in biomedicine and pharmacokinetic/pharmacodynamic studies. In most cases the likelihood function is not available, and thus maximum likelihood estimation of the unknown parameters is not possible. Here we propose a computationally fast approximated maximum likelihood procedure for the estimation of the non-random parameters and the random effects. The method is evaluated on simulations from some famous diffusion processes and on real data sets.     

Identifiability and Comparison of Estimation Methods on Weibull Mixture Models

In this article, the finite mixture model of Weibull distributions is studied, the identifiability of the model with m components is proven, and the parameter estimators for the case of two components resulted by several algorithms are compared. The parameter estimators are obtained with maximum likelihood performing calculations with different algorithms: expectation-maximization (EM), Fisher scoring, backfitting, optimization of k-nearest neighbor approach, and random walk algorithm using Monte Carlo simulation. The Akaike information criterion and the log-likelihood value are used to compare models. In general, the proposed random walk algorithm shows better performance in mean square error and bias. Finally, the results are applied to electronic component lifetime data.     

On a convergent stochastic estimation algorithm for frailty models

A maximum likelihood estimation procedure is presented for the frailty model. The procedure is based on a stochastic Expectation Maximization algorithm which converges quickly to the maximum likelihood estimate. The usual expectation step is replaced by a stochastic approximation of the complete log-likelihood using simulated values of unobserved frailties whereas the maximization step follows the same lines as those of the Expectation Maximization algorithm. The procedure allows to obtain at the same time estimations of the marginal likelihood and of the observed Fisher information matrix. Moreover, this stochastic Expectation Maximization algorithm requires less computation time. A wide variety of multivariate frailty models without any assumption on the covariance structure can be studied. To illustrate this procedure, a Gaussian frailty model with two frailty terms is introduced. The numerical results based on simulated data and on real bladder cancer data are more accurate than those obtained by using the Expectation Maximization Laplace algorithm and the Monte-Carlo Expectation Maximization one. Finally, since frailty models are used in many fields such as ecology, biology, economy, …, the proposed algorithm has a wide spectrum of applications.     

Likelihood Inference for Spatial Generalized Linear Mixed Models

Spatial modeling is widely used in environmental sciences, biology, and epidemiology. Generalized linear mixed models are employed to account for spatial variations of point-referenced data called spatial generalized linear mixed models (SGLMMs). Frequentist analysis of these type of data is computationally difficult. On the other hand, the advent of the Markov chain Monte Carlo algorithm has made the Bayesian analysis of SGLMM computationally convenient. Recent introduction of the method of data cloning, which leads to maximum likelihood estimate, has made frequentist analysis of mixed models also equally computationally convenient. Recently, the data cloning was employed to estimate model parameters in SGLMMs, however, the prediction of spatial random effects and kriging are also very important. In this article, we propose a frequentist approach based on data cloning to predict (and provide prediction intervals) spatial random effects and kriging. We illustrate this approach using a real dataset and also by a simulation study.