Using Spherical–Radial Quadrature to Fit Generalized Linear Mixed Effects Models

Although generalized linear mixed effects models have received much attention in the statistical literature, there is still no computationally efficient algorithm for computing maximum likelihood estimates for such models when there are a moderate number of random effects. Existing algorithms are either computationally intensive or they compute estimates from an approximate likelihood. Here we propose an algorithm—the spherical–radial algorithm—that is computationally efficient and computes maximum likelihood estimates. Although we concentrate on two-level, generalized linear mixed effects models, the same algorithm can be applied to many other models as well, including nonlinear mixed effects models and frailty models. The computational difficulty for estimation in these models is in integrating the joint distribution of the data and the random effects to obtain the marginal distribution of the data. Our algorithm uses a multidimensional quadrature rule developed in earlier literature to integrate the joint density. This article discusses how this rule may be combined with an optimization algorithm to efficiently compute maximum likelihood estimates. Because of stratification and other aspects of the quadrature rule, the resulting integral estimator has significantly less variance than can be obtained through simple Monte Carlo integration. Computational efficiency is achieved, in part, because relatively few evaluations of the joint density may be required in the numerical integration.     

Nonlife ratemaking and risk management with Bayesian generalized additive models for location,scale, and shape

Generalized additive models for location, scale and, shape define a flexible, semi-parametric class of regression models for analyzing insurance data in which the exponential family assumption for the response is relaxed. This approach allows the actuary to include risk factors not only in the mean but also in other key parameters governing the claiming behavior, like the degree of residual heterogeneity or the no-claim probability. In this broader setting, the Negative Binomial regression with cell-specific heterogeneity and the zero-inflated Poisson regression with cell-specific additional probability mass at zero are applied to model claim frequencies. New models for claim severities that can be applied either per claim or aggregated per year are also presented. Bayesian inference is based on efficient Markov chain Monte Carlo simulation techniques and allows for the simultaneous estimation of linear effects as well as of possible nonlinear effects, spatial variations and interactions between risk factors within the data set. To illustrate the relevance of this approach, a detailed case study is proposed based on the Belgian motor insurance portfolio studied in Denuit and Lang (2004).     

Generalized Nonparametric Mixed Effects Models

Generalized linear mixed effects models (GLMM) provide useful tools for correlated and/or over-dispersed non-Gaussian data. This article considers generalized nonparametric mixed effects models (GNMM), which relax the rigid linear assumption on the conditional predictor in a GLMM. We use smoothing splines to model fixed effects. The random effects are general and may also contain stochastic processes corresponding to smoothing splines. We show how to construct smoothing spline ANOVA (SS ANOVA) decompositions for the predictor function. Components in a SS ANOVA decomposition have nice interpretations as main effects and interactions. Experimental design considerations help determine which components are fixed or random. We estimate all parameters and spline functions using stochastic approximation with Markov chain Monte Carlo (MCMC). As iteration increases we increase the MCMC sample size and decrease the step-size of the parameter update. This approach guarantees convergence of the estimates to the expected fixed points. We evaluate our methods through a simulation study.     

Conditional Logistic Regression With Longitudinal Follow-up and Individual-Level Random Coefficients: A Stable and Efficient Two-Step Estimation Method

The analysis of data generated by animal habitat selection studies, by family studies of genetic diseases, or by longitudinal follow-up of households often involves fitting a mixed conditional logistic regression model to longitudinal data composed of clusters of matched case-control strata. The estimation of model parameters by maximum likelihood is especially difficult when the number of cases per stratum is greater than one. In this case, the denominator of each cluster contribution to the conditional likelihood involves a complex integral in high dimension, which leads to convergence problems in the numerical maximization. In this article we show how these computational complexities can be bypassed using a global two-step analysis for nonlinear mixed effects models. The first step estimates the cluster-specific parameters and can be achieved with standard statistical methods and software based on maximum likelihood for independent data. The second step uses the EM-algorithm in conjunction with conditional restricted maximum likelihood to estimate the population parameters. We use simulations to demonstrate that the method works well when the analysis is based on a large number of strata per cluster, as in many ecological studies. We apply the proposed two-step approach to evaluate habitat selection by pairs of bison roaming freely in their natural environment. This article has supplementary material online.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

A semiparametric Bayesian generalized linear mixed model for the reliability of Kevlar fibers

We analyze the reliability of NASA composite pressure vessels by using a new Bayesian semiparametric model. The data set consists of lifetimes of pressure vessels, wrapped with a Kevlar fiber, grouped by spool, subject to different stress levels; 10% of the data are right censored. The model that we consider is a regression on the log‐scale for the lifetimes, with fixed (stress) and random (spool) effects. The prior of the spool parameters is nonparametric, namely they are a sample from a normalized generalized gamma process, which encompasses the well‐known Dirichlet process. The nonparametric prior is assumed to robustify inferences to misspecification of the parametric prior. Here, this choice of likelihood and prior yields a new Bayesian model in reliability analysis. Via a Bayesian hierarchical approach, it is easy to analyze the reliability of the Kevlar fiber by predicting quantiles of the failure time when a new spool is selected at random from the population of spools. Moreover, for comparative purposes, we review the most interesting frequentist and Bayesian models analyzing this data set. Our credibility intervals of the quantiles of interest for a new random spool are narrower than those derived by previous Bayesian parametric literature, although the predictive goodness‐of‐fit performances are similar. Finally, as an original feature of our model, by means of the discreteness of the random‐effects distribution, we are able to cluster the spools into three different groups. Copyright © 2012 John Wiley & Sons, Ltd.     

An Empirical Comparison of Methods for Computing Bayes Factors in Generalized Linear Mixed Models

Generalized linear mixed models (GLMM) are used in situations where a number of characteristics (covariates) affect a nonnormal response variable and the responses are correlated due to the existence of clusters or groups. For example, the responses in biological applications may be correlated due to common genetic factors or environmental factors. The clustering or grouping is addressed by introducing cluster effects to the model; the associated parameters are often treated as random effects parameters. In many applications, the magnitude of the variance components corresponding to one or more of the sets of random effects parameters are of interest, especially the point null hypothesis that one or more of the variance components is zero. A Bayesian approach to test the hypothesis is to use Bayes factors comparing the models with and without the random effects in question—this work reviews a number of approaches for estimating the Bayes factor. We perform a comparative study of the different approaches to compute Bayes factors for GLMMs by applying them to two different datasets. The first example employs a probit regression model with a single variance component to data from a natural selection study on turtles. The second example uses a disease mapping model from epidemiology, a Poisson regression model with two variance components. Bridge sampling and a recent improvement known as warp bridge sampling, importance sampling, and Chib's marginal likelihood calculation are all found to be effective. The relative advantages of the different approaches are discussed.     

Inferences in semi-parametric dynamic mixed models for longitudinal count data

This paper considers a semi-parametric mixed model for longitudinal counts under the assumption that for conditional on a common random effect over time the repeated count responses of an individual follow a Poisson AR(1) (auto-regressive order 1) non-stationary correlation structure. A step-by-step estimation approach is developed which provides consistent estimators for the non-parametric function, regression parameters, variance of the random effects, and auto-correlation structure of the model. Proofs for the consistency properties of the estimators along with their convergence rates are derived. A simulation study is conducted to examine first the estimation effects on parameters when the non-parametric function is ignored, and then an overall estimation study is carried out in the presence of the non-parametric function by including its estimation as well.     

Two-component Poisson mixture regression modelling of count data with bivariate random effects

Two-component Poisson mixture regression is typically used to model heterogeneous count outcomes that arise from two underlying sub-populations. Furthermore, a random component can be incorporated into the linear predictor to account for the clustering data structure. However, when including random effects in both components of the mixture model, the two random effects are often assumed to be independent for simplicity. A two-component Poisson mixture regression model with bivariate random effects is proposed to deal with the correlated situation. A restricted maximum quasi-likelihood estimation procedure is provided to obtain the parameter estimates of the model. A simulation study shows both fixed effects and variance component estimates perform well under different conditions. An application to childhood gastroenteritis data demonstrates the usefulness of the proposed methodology, and suggests that neglecting the inherent correlation between random effects may lead to incorrect inferences concerning the count outcomes.     

Principal Component Analysis of Two-Dimensional Functional Data

This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation–maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online.     

Structured additive regression for overdispersed and zero‐inflated count data

In count data regression there can be several problems that prevent the use of the standard Poisson log‐linear model: overdispersion, caused by unobserved heterogeneity or correlation, excess of zeros, non‐linear effects of continuous covariates or of time scales, and spatial effects. We develop Bayesian count data models that can deal with these issues simultaneously and within a unified inferential approach. Models for overdispersed or zero‐inflated data are combined with semiparametrically structured additive predictors, resulting in a rich class of count data regression models. Inference is fully Bayesian and is carried out by computationally efficient MCMC techniques. Simulation studies investigate performance, in particular how well different model components can be identified. Applications to patent data and to data from a car insurance illustrate the potential and, to some extent, limitations of our approach. Copyright © 2006 John Wiley & Sons, Ltd.     

A Computationally Efficient Projection-Based Approach for Spatial Generalized Linear Mixed Models

Inference for spatial generalized linear mixed models (SGLMMs) for high-dimensional non-Gaussian spatial data is computationally intensive. The computational challenge is due to the high-dimensional random effects and because Markov chain Monte Carlo (MCMC) algorithms for these models tend to be slow mixing. Moreover, spatial confounding inflates the variance of fixed effect (regression coefficient) estimates. Our approach addresses both the computational and confounding issues by replacing the high-dimensional spatial random effects with a reduced-dimensional representation based on random projections. Standard MCMC algorithms mix well and the reduced-dimensional setting speeds up computations per iteration. We show, via simulated examples, that Bayesian inference for this reduced-dimensional approach works well both in terms of inference as well as prediction; our methods also compare favorably to existing “reduced-rank” approaches. We also apply our methods to two real world data examples, one on bird count data and the other classifying rock types. Supplementary material for this article is available online.     

On the Positive Definiteness of the Information Matrix Under the Binary and Poisson Mixed Models

Binary and Poisson generalized linear mixed models are used to analyse over/under-dispersed proportion and count data, respectively. As the positive definiteness of the information matrix is a required property for valid inference about the fixed regression vector and the variance components of the random effects, this paper derives the relevant necessary and sufficient conditions under both these models. It is found that the conditions for the positive definiteness are not identical for these two nonlinear mixed models and that a mere analogy with the usual linear mixed model does not dictate these conditions.     

Estimating Frailty Models via Poisson Hierarchical Generalized Linear Models

Frailty models extend proportional hazards models to multivariate survival data. Hierarchical-likelihood provides a simple unified framework for various random effect models such as hierarchical generalized linear models, frailty models, and mixed linear models with censoring. Wereview the hierarchical-likelihood estimation methods for frailty models. Hierarchical-likelihood for frailty models can be expressed as that for Poisson hierarchical generalized linear models. Frailty models can thus be fitted using Poisson hierarchical generalized linear models. Properties of the new methodology are demonstrated by simulation. The new method reduces the bias of maximum likelihood and penalized likelihood estimates.     

Hierarchical mixture models for zero-inflated correlated count data

Count data with excess zeros are often encountered in many medical, biomedical and public health applications. In this paper, an extension of zero-inflated Poisson mixed regression models is presented for dealing with multilevel data set, referred as hierarchical mixture zero-inflated Poisson mixed regression models. A stochastic EM algorithm is developed for obtaining the ML estimates of interested parameters and a model comparison is also considered for comparing models with different latent classes through BIC criterion. An application to the analysis of count data from a Shanghai Adolescence Fitness Survey and a simulation study illustrate the usefulness and effectiveness of our methodologies.     

带有不完全信息随机截尾试验下混合泊松分布参数的点估计

首先,通过添加数据,得到了带有不完全信息随机截尾试验下混合泊松分布的完全数据似然函数,然后分别利用EM算法和MCMC方法,对参数进行了估计,最后进行了随机模拟试验.结果表明参数点估计的精度比较高,     

Bayesian analysis of definitive screening designs when the response is nonnormal

Definitive screening designs (DSDs) are a class of experimental designs that allow the estimation of linear, quadratic, and interaction effects with little experimental effort if there is effect sparsity. The number of experimental runs is twice the number of factors of interest plus one. Many industrial experiments involve nonnormal responses. Generalized linear models (GLMs) are a useful alternative for analyzing these kind of data. The analysis of GLMs is based on asymptotic theory, something very debatable, for example, in the case of the DSD with only 13 experimental runs. So far, analysis of DSDs considers a normal response. In this work, we show a five‐step strategy that makes use of tools coming from the Bayesian approach to analyze this kind of experiment when the response is nonnormal. We consider the case of binomial, gamma, and Poisson responses without having to resort to asymptotic approximations. We use posterior odds that effects are active and posterior probability intervals for the effects and use them to evaluate the significance of the effects. We also combine the results of the Bayesian procedure with the lasso estimation procedure to enhance the scope of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust-efficient credibility models with heavy-tailed claims: A mixed linear models perspective

In actuarial practice, regression models serve as a popular statistical tool for analyzing insurance data and tariff ratemaking. In this paper, we consider classical credibility models that can be embedded within the framework of mixed linear models. For inference about fixed effects and variance components, likelihood-based methods such as (restricted) maximum likelihood estimators are commonly pursued. However, it is well-known that these standard and fully efficient estimators are extremely sensitive to small deviations from hypothesized normality of random components as well as to the occurrence of outliers. To obtain better estimators for premium calculation and prediction of future claims, various robust methods have been successfully adapted to credibility theory in the actuarial literature. The objective of this work is to develop robust and efficient methods for credibility when heavy-tailed claims are approximately log-location-scale distributed. To accomplish that, we first show how to express additive credibility models such as Bühlmann-Straub and Hachemeister ones as mixed linear models with symmetric or asymmetric errors. Then, we adjust adaptively truncated likelihood methods and compute highly robust credibility estimates for the ordinary but heavy-tailed claims part. Finally, we treat the identified excess claims separately and find robust-efficient credibility premiums. Practical performance of this approach is examined-via simulations-under several contaminating scenarios. A widely studied real-data set from workers’ compensation insurance is used to illustrate functional capabilities of the new robust credibility estimators.     

Some properties for the estimators in linear mixed models

Linear mixed models (LMMs) have become an important statistical method for analyzing cluster or longitudinal data. In most cases, it is assumed that the distributions of the random effects and the errors are normal. This paper removes this restrictions and replace them by the moment conditions. We show that the least square estimators of fixed effects are consistent and asymptotically normal in general LMMs. A closed-form estimator of the covariance matrix for the random effect is constructed and its consistent is shown. Based on this, the consistent estimate for the error variance is also obtained. A simulation study and a real data analysis show that the procedure is effective.     

Semi-parametric Bayesian estimation of mixed-effects models using the multivariate skew-normal distribution

In this paper, we develop a semi-parametric Bayesian estimation approach through the Dirichlet process (DP) mixture in fitting linear mixed models. The random-effects distribution is specified by introducing a multivariate skew-normal distribution as base for the Dirichlet process. The proposed approach efficiently deals with modeling issues in a wide range of non-normally distributed random effects. We adopt Gibbs sampling techniques to achieve the parameter estimates. A small simulation study is conducted to show that the proposed DP prior is better at the prediction of random effects. Two real data sets are analyzed and tested by several hypothetical models to illustrate the usefulness of the proposed approach.