A simulation comparison on spatial Poisson mixed models

The statistical methods for analyzing spatial count data have often been based on random fields so that a latent variable can be used to specify the spatial dependence. In this article, we introduce two frequentist approaches for estimating the parameters of model-based spatial count variables. The comparison has been carried out by a simulation study. The performance is also evaluated using a real dataset and also by the simulation study. The simulation results show that the maximum likelihood estimator appears to be with the better sampling properties.     

Optimal predictive design augmentation for spatial generalised linear mixed models

A typical model for geostatistical data when the observations are counts is the spatial generalised linear mixed model. We present a criterion for optimal sampling design under this framework which aims to minimise the error in the prediction of the underlying spatial random effects. The proposed criterion is derived by performing an asymptotic expansion to the conditional prediction variance. We argue that the mean of the spatial process needs to be taken into account in the construction of the predictive design, which we demonstrate through a simulation study where we compare the proposed criterion against the widely used space-filling design. Furthermore, our results are applied to the Norway precipitation data and the rhizoctonia disease data.     

Laplace approximations for censored linear regression models

We consider approximate Bayesian inference about scalar parameters of linear regression models with possible censoring. A second-order expansion of their Laplace posterior is seen to have a simple and intuitive form for logconcave error densities with nondecreasing hazard functions. The accuracy of the approximations is assessed for normal and Gumbel errors when the number of regressors increases with sample size. Perturbations of the prior and the likelihood are seen to be easily accommodated within our framework. Links with the work of DiCiccio et al. (1990) and Viveros and Sprott (1987) extend the applicability of our results to conditional frequentist inference based on likelihood-ratio statistics.     

Approximate composite marginal likelihood inference in spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using spatial generalized linear mixed model with spatial random effects. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. There are numerical ways to maximize the likelihood function, such as Monte Carlo Expectation Maximization and Quadrature Pairwise Expectation Maximization algorithms. They can be applied but may in such cases be computationally very slow or even prohibitive. Gauss–Hermite quadrature approximation only suitable for low-dimensional latent variables and its accuracy depends on the number of quadrature points. Here, we propose a new approximate pairwise maximum likelihood method to the inference of the spatial generalized linear mixed model. This approximate method is fast and deterministic, using no sampling-based strategies. The performance of the proposed method is illustrated through two simulation examples and practical aspects are investigated through a case study on a rainfall data set.     

Robust estimation in generalized linear mixed models

Generalized linear mixed models (GLMMs) are widely used to analyse non-normal response data with extra-variation, but non-robust estimators are still routinely used. We propose robust methods for maximum quasi-likelihood and residual maximum quasi-likelihood estimation to limit the influence of outlying observations in GLMMs. The estimation procedure parallels the development of robust estimation methods in linear mixed models, but with adjustments in the dependent variable and the variance component. The methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach. When applied to a set of epileptic seizure data, the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates. Simulation shows that one of the residual maximum quasi-likelihood proposals has a smaller bias than those of the other estimation methods. We further discuss the equivalence of two GLMM formulations when the response variable follows an exponential family. Their extensions to robust GLMMs and their comparative advantages in modelling are described. Some possible modifications of the robust GLMM estimation methods are given to provide further flexibility for applying the method.     

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Estimation using penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models

We consider two estimation schemes based on penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models. The asymptotic bias in regression coefficients and variance components estimated by penalized quasilikelihood (PQL) is studied for small values of the variance components. We show the PQL estimators of both regression coefficients and variance components in Poisson mixed models have a smaller order of bias compared to those for binomial data. Unbiased estimating equations based on quasi-pseudo-likelihood are proposed and are shown to yield consistent estimators under some regularity conditions. The finite sample performance of these two methods is compared through a simulation study.     

Robust inference in generalized linear models for longitudinal data

The author develops a robust quasi‐likelihood method, which appears to be useful for down‐weighting any influential data points when estimating the model parameters. He illustrates the computational issues of the method in an example. He uses simulations to study the behaviour of the robust estimates when data are contaminated with outliers, and he compares these estimates to those obtained by the ordinary quasi‐likelihood method.     

Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

Summary.  Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models , where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.     

Maximum likelihood estimation for generalized conditionally autoregressive models of spatial data

Conditionally autoregressive (CAR) models are often used to analyze a spatial process observed over a lattice or a set of irregular regions. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. To accommodate directional and inherent anisotropy variation, a new class of spatial models is proposed that adaptively determines neighbors based on a bivariate kernel using the distances and angles between the centroid of the regions. The newly proposed model generalizes the usual CAR model in a sense of accounting for adaptively determined weights. Maximum likelihood estimators are derived and simulation studies are presented for the sampling properties of the estimates on the new model, which is compared to the CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Risk-reducing shrinkage estimation for generalized linear models

Summary.  Empirical Bayes techniques for normal theory shrinkage estimation are extended to generalized linear models in a manner retaining the original spirit of shrinkage estimation, which is to reduce risk. The investigation identifies two classes of simple, all-purpose prior distributions, which supplement such non-informative priors as Jeffreys's prior with mechanisms for risk reduction. One new class of priors is motivated as optimizers of a core component of asymptotic risk. The methodology is evaluated in a numerical exploration and application to an existing data set.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

Weighted empirical likelihood for generalized linear models with longitudinal data

In this paper, we introduce the empirical likelihood (EL) method to longitudinal studies. By considering the dependence within subjects in the auxiliary random vectors, we propose a new weighted empirical likelihood (WEL) inference for generalized linear models with longitudinal data. We show that the weighted empirical likelihood ratio always follows an asymptotically standard chi-squared distribution no matter which working weight matrix that we have chosen, but a well chosen working weight matrix can improve the efficiency of statistical inference. Simulations are conducted to demonstrate the accuracy and efficiency of our proposed WEL method, and a real data set is used to illustrate the proposed method.     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.     

Likelihood analysis for a class of beta mixed models

Beta regression is a suitable choice for modelling continuous response variables taking values on the unit interval. Data structures such as hierarchical, repeated measures and longitudinal typically induce extra variability and/or dependence and can be accounted for by the inclusion of random effects. In this sense, Statistical inference typically requires numerical methods, possibly combined with sampling algorithms. A class of Beta mixed models is adopted for the analysis of two real problems with grouped data structures. We focus on likelihood inference and describe the implemented algorithms. The first is a study on the life quality index of industry workers with data collected according to an hierarchical sampling scheme. The second is a study assessing the impact of hydroelectric power plants upon measures of water quality indexes up, downstream and at the reservoirs of the dammed rivers, with a nested and longitudinal data structure. Results from different algorithms are reported for comparison including from data-cloning, an alternative to numerical approximations which also allows assessing identifiability. Confidence intervals based on profiled likelihoods are compared with those obtained by asymptotic quadratic approximations, showing relevant differences for parameters related to the random effects. In both cases, the scientific hypothesis of interest was investigated by comparing alternative models, leading to relevant interpretations of the results within each context.     

Prediction in linear mixed models

Following estimation of effects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. The process has been well defined for linear models, but the introduction of random effects into the model means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions. This paper discusses the interpretation of predictions formed including or excluding random terms. Four datasets are used to illustrate circumstances where different prediction strategies may be appropriate: in an orthogonal design, an unbalanced nested structure, a model with cubic smoothing spline terms and for kriging after spatial analysis. The examples also show the need for different weighting schemes that recognize nesting and aliasing during prediction, and the necessity of being able to detect inestimable predictions.     

Double hierarchical generalized linear models (with discussion)

Summary.  We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h -likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities.     

Monte Carlo approximation of likelihood function in spatial GLMMs through an empirical Bayes method

In spatial generalized linear mixed models (SGLMMs), statistical inference encounters problems, since random effects in the model imply high-dimensional integrals to calculate the marginal likelihood function. In this article, we temporarily treat parameters as random variables and express the marginal likelihood function as a posterior expectation. Hence, the marginal likelihood function is approximated using the obtained samples from the posterior density of the latent variables and parameters given the data. However, in this setting, misspecification of prior distribution of correlation function parameter and problems associated with convergence of Markov chain Monte Carlo (MCMC) methods could have an unpleasant influence on the likelihood approximation. To avoid these challenges, we utilize an empirical Bayes approach to estimate prior hyperparameters. We also use a computationally efficient hybrid algorithm by combining inverse Bayes formula (IBF) and Gibbs sampler procedures. A simulation study is conducted to assess the performance of our method. Finally, we illustrate the method applying a dataset of standard penetration test of soil in an area in south of Iran.     

Estimation and inference for generalized semi-varying coefficient models

This paper is concerned with the estimation and inference in generalized semi-varying coefficient models. An orthogonal projection local quasi-likelihood estimation is investigated, which can easily be used to estimate the model parametric and nonparametric parts. Then an empirical likelihood logarithmic approach to construct the confidence regions/intervals of the nonparametric parts is developed. Under some mild conditions, the asymptotic properties of the resulting estimators are studied explicitly, respectively. Some simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by a real data set.