Utilizing the Generalized Linear Mixed Model for Specification and Simulation of Transient Vibration Environments

Markov Chain Monte Carlo (MCMC) is widely applied to fit complex, practical models in various disciplines, such as engineering, physics, agriculture, etc. Indeed, when the target density $f$ is not available in closed form, a Markov chain is constructed with stationary density $f$ for forming Monte Carlo estimators of means with respect to $f$. Although a Harris ergodic Markov chain will converge to the target distribution as the Monte Carlo sample size tends to infinity, and the population means can be consistently estimated by the corresponding MCMC estimators, in practice, it is important to estimate the standard errors associated with these Monte Carlo estimates due to the availability of only a finite sample. Therefore, the geometric ergodicity of a Markov Chain is highly desired as it is the most standard method for guaranteeing a central limit theorem (CLT) for MCMC estimators and finding its standard errors. Among the different MCMC algorithms, Metropolis-Hastings (MH) algorithms are predominant. This thesis establishes conditions under which MH algorithms with a position-dependent proposal covariance matrix will or will not have the geometric convergence rate. Indeed, we provide these results for MH algorithms where the normal proposal density has a general mean function $c(x)$ and a covariance matrix $G(x)$ depending on the current position $x$ of the Markov chain. For modern variants of the popular Metropolis adjusted Langevin algorithms (MALA), like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for the geometric ergodicity of different variations of the Langevin algorithms, e.g., MMALA, position-dependent MALA (PMALA), and the pre-conditioned MALA (PCMALA). We also provide conditions guaranteeing geometric ergodicity of the pre-conditioned unadjusted Langevin algorithms (PCULA). Generalized linear mixed models (GLMMs), where the response variable is allowed to follow a non-Gaussian distribution, are popular statistical models, although their likelihood functions are not available in closed form. GLMMs can handle the over-dispersion often present in real-world data, and MCMC methods are generally used to fit frequentist and Bayesian GLMMs. Since the likelihood function for GLMMs is not available in closed form, Monte Carlo EM or Monte Carlo maximum likelihood methods are often used for making inferences from GLMMs. Simulation from the conditional (on the data) distribution of the random effects is, in turn, needed for implementing the Monte Carlo EM or Monte Carlo maximum likelihood methods. In this thesis, our general conditions for geometric ergodicity of variants of Langevin algorithms are also verified in the context of conditional simulation from the two most popular GLMMs, namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. These results have important practical implications as users can now confidently fit these models by attaching asymptotically valid standard errors with their Markov chain-based Monte Carlo estimates. Spatial generalized linear mixed models (SGLMMs) are GLMMs where the random effects form a spatial process and are useful for modeling spatially correlated data. We compare the performance of different variants of MALA mentioned above as well as other popular MH algorithms like Random Walk Metropolis in the context of conditional simulation from SGLMMs. Next, we consider MCMC algorithms for full Bayesian analysis of GLMMs. Blocking and reparameterization are techniques used to improve the efficiency and convergence properties of MCMC algorithms. We combine these strategies with pre-conditioning to develop efficient Langevin algorithms for Bayesian GLMMs and SGLMMs. We illustrate our algorithms using extensive real and simulated data analyses.

ObjectiveShow the benefits of using a generalized linear mixed model (GLMM) to examine long-term trends in asthma syndrome data.IntroductionOver the last decade, the application of syndromic surveillance systems has expanded beyond early event detection to include long-term disease trend monitoring. However, statistical methods employed for analyzing syndromic data tend to focus on early event detection. Generalized linear mixed models (GLMMs) may be a useful statistical framework for examining long-term disease trends because, unlike other models, GLMMs account for clustering common in syndromic data, and GLMMs can assess disease rates at multiple spatial and temporal levels (1). We show the benefits of the GLMM by using a GLMM to estimate asthma syndrome rates in New York City from 2007 to 2012, and to compare high and low asthma rates in Harlem and the Upper East Side (UES) of Manhattan.MethodsAsthma related emergency department (ED) visits, and patient age and ZIP code were obtained from data reported daily to the NYC Department of Health and Mental Hygiene. Demographic data were obtained from 2010 US Census. ZIP codes that represented high and low asthma rates in Harlem and the UES of Manhattan were chosen for closer inspection. The ratio of weekly asthma syndrome visits to total ED visits was modeled with a Poisson GLMM with week and ZIP code random intercepts (2). Age and ethnicity were adjusted for because of their association with asthma rates (3).ResultsThe GLMM showed citywide asthma rates remained stable from 2007 to 2012, but seasonal differences and significant inter-ZIP code variation were present. The Harlem ZIP code asthma rate that was estimated with the GLMM was significantly higher (5.83%, 95% CI: 3.65%, 9.49%) than the asthma rate in UES ZIP code (0.78%, 95% CI: 0.50%, 1.21%). A linear time component to the GLMM showed no appreciable change over time despite the seasonal fluctuations in asthma rate. GLMM based asthma rates are shown over time (Figure 1).ConclusionsGLMMs have several strengths as statistical frameworks for monitoring trends including:Disease rates can be estimated at multiple spatial and temporal levels,Standard error adjustment for clustering in syndromic data allows for accurate, statistical assessment of changes over time and differences between subgroups,“Strength borrowed” (4) from the aggregated data informs small subgroups and smooths trends,Integration of covariate data reduces bias in estimated rates.GLMMs have previously been suggested for early event detection with syndromic surveillance data (5), but the versatility of GLMM makes them useful for monitoring long-term disease trends as well. In comparison to GLMMs, standard errors from single level GLMs do not account for clustering and can lead to inaccurate statistical hypothesis testing. Bayesian hierarchical models (6), share many of the strengths of GLMMS, but are more complicated to fit. In the future, GLMMs could provide a framework for grouping similar ZIP codes based on their model estimates (e.g. seasonal trends and influence on overall trend), and analyzing long-term disease trends with syndromic data.

The classical model only provides a correct analysis if all the effects are fixed. For experiments that include fixed and random effects, the general linear mixed model is appropriate for handling the non-normal distributed response variables. The aim of this study is to perform the genotype selection through a generalized linear mixed model and identify the impact of treatment and the related traits on grain yield. The data were collected using a lattice square design and measured the phenotype traits of sorghum. The result of PCA was used as an input variable for the general linear mixed model. The data analysis was performed using a general linear mixed model with maximum likelihood methods to estimate the parameters of the model. The result showed that the grain yield had a gamma distribution and a treatment effect on grain yield. The first principal component was significant for grain yield. The variability of grain yield due to the random effects of replication within treatment, genotype, and the interaction of genotype by treatment were significant. The best genotypes effective for the mass production of sorghum were G137, G66 and G156 under stress conditions and G55, G41 and G78 under irrigated conditions. Overall, genotype selection using a general linear mixed model for grain yield is recommended for genotype selection of plant breeding.

Based on a hybrid road approach, a reliable process of fatigue load prediction is proposed. This approach known as Time Waveform Replication (TWR) relies on test data and multi-body model. The test data is used to back-calculate the hybrid road profile that will cause the same forces on the multi-body model. The road load is measured on the proving ground, and simulation of the multi-body model is ran by TWR to back-calculate the hybrid road profile. The simulation signals are verified by the experimental signals collected by wheel force transducers (WFT) and acceleration sensors.

In this chapter we consider dependent data but move from the linear models of Chap.8 to general regression models. As in Chap.6, we consider generalized linear models (GLMs) and, more briefly, nonlinear models. We first give an outline of this chapter. In Sect.9.2 we describe three motivating datasets to which we return throughout the chapter. The GLMs discussed in Sect.6.3 can be extended to incorporate dependences in observations on the same unit; as with the linear model, an obvious way to carry out modeling in this case is to introduce unit-specific random effects. Within a GLM a natural approach is for these random effects to be included on the linear predictor scale. The resultant conditional models are known as generalized linear mixed models (GLMMs), and these are introduced in Sect.9.3. In Sects.9.4 and9.5 we describe likelihood and conditional likelihood methods of estimation, with Sect.9.6 devoted to a Bayesian treatment. Section9.7 illustrates some of the flexibility of GLMMs by describing and applying a particular model for spatial dependence. An alternative random effects specification, based on conjugacy, is described in Sect.9.8. An important approach to the modeling and analysis of dependent data that is philosophically different from the random effects formulation is via marginal models and generalized estimating equations (GEE), and these are the subject of Sect.9.9. In Sect.9.10, a second GEE approach is described in which the estimating equations for the mean are supplemented with a second set for the variances/covariances. For GLMMs, extra care must be taken with parameter interpretation, and Sect.9.11 discusses this issue, emphasizing how interpretation differs between conditional and marginal models. In PartII of the book, which focused on independent data, Chap.7 was devoted to models for binary data. For dependent data, models binary data are less well developed, and so we do not devote a complete chapter to their description. However, Sect.9.12 introduces the modeling of dependent binary data, and, subsequently, Sects.9.13 and 9.14 describe conditional (mixed) and marginal models for binary data. Section 9.15 considers how nonlinear models, as defined in Sect.6.10, can be extended to the dependent data case. For such models, many applications concentrate on inference for units, and so the introduction of random effects is again suggested. We refer to the resultant class of models as nonlinear mixed models (NLMMs). Section9.16 considers issues related to the parameterization of the nonlinear model. Inference for nonlinear mixed models via likelihood and Bayes approaches is covered in Sects.9.17 and9.18, while GEE is briefly considered in Sect.9.19. The assessment of assumptions for general regression models is described in Sect.9.20, with concluding comments contained in Sect.9.21. Additional references appear in Sect.9.22.

BackgroundIn meta-analyses of a binary outcome, double zero events in some studies cause a critical methodology problem. The generalized linear mixed model (GLMM) has been proposed as a valid statistical tool for pooling such data. Three parameter estimation methods, including the Laplace approximation (LA), penalized quasi-likelihood (PQL) and adaptive Gauss–Hermite quadrature (AGHQ) were frequently used in the GLMM. However, the performance of GLMM via these estimation methods is unclear in meta-analysis with zero events.MethodsA simulation study was conducted to compare the performance. We fitted five random-effects GLMMs and estimated the results through the LA, PQL and AGHQ methods, respectively. Each scenario conducted 20,000 simulation iterations. The data from Cochrane Database of Systematic Reviews were collected to form the simulation settings. The estimation methods were compared in terms of the convergence rate, bias, mean square error, and coverage probability.ResultsOur results suggested that when the total events were insufficient in either of the arms, the GLMMs did not show good point estimation to pool studies of rare events. The AGHQ method did not show better properties than the LA estimation in terms of convergence rate, bias, coverage, and possibility to produce very large odds ratios. In addition, although the PQL had some advantages, it was not the preferred option due to its low convergence rate in some situations, and the suboptimal point and variance estimation compared to the LA.ConclusionThe GLMM is an alternative for meta-analysis of rare events and is especially useful in the presence of zero-events studies, while at least 10 total events in both arms is recommended when employing GLMM for meta-analysis. The penalized quasi-likelihood and adaptive Gauss–Hermite quadrature are not superior to the Laplace approximation for rare events and thus they are not recommended.

ObjectivesSeveral methods are commonly used for meta-analyses of diagnostic studies, such as the bivariate linear mixed model (LMM). It estimates the overall sensitivity, specificity, their correlation, diagnostic OR (DOR) and...

Modeling and understanding the catch rate dynamics of marine species is extremely important for fisheries management and conservation. For oceanic highly migratory species in particular, usually only fishery‐dependent data are available which have limitations in the assumption of independence and if often zero‐inflated and/or overdispersed. We tested different modeling approaches applied to the case study of blue shark in the South Atlantic, by using generalized linear models (GLMs), generalized linear mixed models (GLMMs), and generalized estimating equations (GEEs), as well as different error distributions to deal with the presence of zeros in the data. We used fractional polynomials to deal with non‐linearity in some of the explanatory variables. Operational (set level) data collected by onboard fishery observers, covering 762 longline sets (1,014,527 hooks) over a 9‐year period (2008–2016), were used. One of the most important variables affecting catch rates is leader material, with increasing catches when wire leaders are used. Spatial and seasonal variables are also important, with higher catch rates expected toward temperate southern waters and eastern longitudes, particularly between July and September. Environmental variables, especially SST, also affect catches. There were no major differences in the parameters estimated with GLMs, GLMMs, or GEEs; however, the use of GLMMs or GEEs should be more appropriate due to fishery dependence in the data. Comparing those two approaches, GLMMs seem to perform better in terms of goodness‐of‐fit and model validation.

Foot-and-mouth disease (FMD) is one of the most economically important livestock diseases worldwide. Following the clinical phase of FMD, a large proportion of ruminants remain persistently infected for extended periods. Although extinction of this carrier state occurs continuously at the animal and population levels, studies vary widely in their estimates of the duration of persistent infection. There is a need for robust statistical models to capture the dynamics of persistent infection for the sake of guiding FMD control and trade policies. The goal of the current study was to develop and assess statistical models to describe the extinction of FMD virus (FMDV) persistent infection using data from primary longitudinal studies of naturally infected cattle and Asian buffalo in Vietnam and India. Specifically, accelerated failure time (AFT) models and generalized linear mixed models (GLMM) were developed to predict the probability of persistent infection in seropositive animals and identified carriers at the individual animal level at sequential time points after outbreaks. The primary studies were analyzed by country and combined using an individual-participant data meta-analysis approach. The models estimated similar trends in the duration of persistent infection for the study/species groups included in the analyses, however the significance of the trends differed between the models. The overall probabilities of persistent infection were similar as predicted by the AFT and GLMM models: 6 months: 99% (AFT) /80% (GLMM), 12 months: 51% (AFT) /32% (GLMM), 18 months: 6% (AFT) /5% (GLMM), 24 months: 0.8% (AFT) /0.6% (GLMM). These models utilizing diverse and robust data sets predict higher probabilities of persistence than previously published, suggesting greater endurance of carriers subsequent to an outbreak. This study demonstrates the utility of statistical models to investigate the dynamics of persistent infection and the importance of large datasets, which can be achieved by combining data from several smaller studies in meta-analyses. Results of this study enhance current knowledge of the FMDV carrier state and may inform policy decisions regarding FMDV persistent infection.

It is well known that in a traditional outlier-free situation, the generalized quasi-likelihood (GQL) approach [B.C. Sutradhar, On exact quasilikelihood inference in generalized linear mixed models, Sankhya: Indian J. Statist. 66 (2004), pp. 261–289] performs very well to obtain the consistent as well as the efficient estimates for the parameters involved in the generalized linear mixed models (GLMMs). In this paper, we first examine the effect of the presence of one or more outliers on the GQL estimation for the parameters in such GLMMs, especially in two important models such as count and binary mixed models. The outliers appear to cause serious biases and hence inconsistency in the estimation. As a remedy, we then propose a robust GQL (RGQL) approach in order to obtain the consistent estimates for the parameters in the GLMMs in the presence of one or more outliers. An extensive simulation study is conducted to examine the consistency performance of the proposed RGQL approach.

Data in forestry are often spatially and (or) serially correlated. In the last two decades, mixed models have become increasingly popular for the analysis of such data because they can relax the assumption of independent observations. However, when the relationship between the response variable and the covariates is nonlinear, as is the case in generalized linear mixed models (GLMMs), population-averaged predictions cannot be obtained from the fixed effects alone. This study proposes an estimator, which is based on a five-point Gauss−Hermite quadrature, for population-averaged predictions in the context of GLMM. The estimator was tested through Monte Carlo simulation and compared with a regular generalized linear model (GLM). The estimator was also applied to a real-world case study, a harvest model. The results showed that GLM predictions were unbiased but that their confidence intervals did not achieve their nominal coverage. On the other hand, the proposed estimator yielded unbiased predictions with reliable confidence intervals. The predictions based on the fixed effects of a GLMM exhibited the largest biases. If statistical inferences are needed, the proposed estimator should be used. It is easily implemented as long as the random effect specification does not contain multiple random effects for the same hierarchical level.

MotivationThe generalized linear mixed model (GLMM) is an extension of the generalized linear model (GLM) in which the linear predictor takes random effects into account. Given its power of precisely modeling the mixed effects from multiple sources of random variations, the method has been widely used in biomedical computation, for instance in the genome-wide association studies (GWASs) that aim to detect genetic variance significantly associated with phenotypes such as human diseases. Collaborative GWAS on large cohorts of patients across multiple institutions is often impeded by the privacy concerns of sharing personal genomic and other health data. To address such concerns, we present in this paper a privacy-preserving Expectation–Maximization (EM) algorithm to build GLMM collaboratively when input data are distributed to multiple participating parties and cannot be transferred to a central server. We assume that the data are horizontally partitioned among participating parties: i.e. each party holds a subset of records (including observational values of fixed effect variables and their corresponding outcome), and for all records, the outcome is regulated by the same set of known fixed effects and random effects.ResultsOur collaborative EM algorithm is mathematically equivalent to the original EM algorithm commonly used in GLMM construction. The algorithm also runs efficiently when tested on simulated and real human genomic data, and thus can be practically used for privacy-preserving GLMM construction. We implemented the algorithm for collaborative GLMM (cGLMM) construction in R. The data communication was implemented using the rsocket package.Availability and implementationThe software is released in open source at https://github.com/huthvincent/cGLMM.Supplementary informationSupplementary data are available at Bioinformatics online.

Linear Mixed models (LMMs) that incorporate genetic and spatial covariance structures have been used for many years to estimate genetic parameters and to predict breeding values in animal and plant breeding. Although the theoretical aspects for extending LMM to generalised linear mixed models (GLMMs) have been around for some time, suitable software has been developed only within the last decade or so. The GLIMMIX procedure in SAS® is becoming popular for fitting GLMMs in various disciplines. Applications of GLMMs to genetic analysis have been limited, probably because of the complexity of the models used. This is particularly so for Proc GLIMMIX because, unlike ASReml software, it is not specifically tailored for analysis of breeding data and some pre-procedure coding is necessary. Binary data that fits the GLMM framework is commonly encountered in breeding experiments, such as when evaluating individuals for resistance by observing the presence or absence of disease. Bacterial canker (Psa) caused by Pseudomonas syringae pv. actinidiae is a serious disease of kiwifruit in New Zealand and other kiwifruit-producing countries. Data from a progeny test trial was available to identify parents with high breeding values for resistance. We successfully applied the GLIMMIX procedure for this purpose. Heritability for resistance was moderate, and we identified two parents and their family as having high potential for Psa resistance breeding. There are several potential pitfalls when using GLMMs with binary data and these are briefly discussed.

Many outcome variables in oral research are characterized by positive values and heavy skewness in the right tail. Examples are provided by many distributions of dental variables such as DMF (decayed, missing, filled teeth) scores, oral health impact profile score, gingival index scores, and microbiologic counts. Moreover, heterogeneity in data arises when more than one tooth is studied for each patient, due to the clusterization.Over the past decade, linear mixed models (LMEs) have become a common statistical tool to account for within-subject correlation in data with repeated measures. When a normal error is reasonably assumed, estimates of LMEs are supported by many statistical packages. Such is not the case for skewed data, where generalized linear mixed models (GLMMs) are required. However, the current software available supports only special cases of GLMMs or relies on crude Laplace-type approximation of integrals. In this study, a Bayesian approach is taken to estimate GLMMs for clustered skewed dental data. A Gamma GLMM and a log-normal model are employed to allow for heterogeneity across clusters, deriving from the patient-operator-tooth susceptibility typical of this clinical context. A comparison to the frequentist framework is also provided. In our case, Gamma GLMM fits data better than the log-normal distribution, while providing more precise estimates compared with the likelihood approach. A key advantage of the Bayesian framework is its ability to readily provide a flexible approach for implementation while simultaneously providing a formal procedure for solving inference problems.

Root-cause identification for quality-related problems is a key issue in quality and productivity improvement for a manufacturing process. Unfortunately, root-cause identification is also a very challenging engineering problem, particularly for a multistage manufacturing process. In this paper, root-cause identification is formulated as a problem of estimation and hypothesis testing of a general linear mixed model. First, a linear mixed fault-quality model is built to describe the cause-effect relationship between the process faults and product quality. Then, the estimation algorithms developed for a general linear mixed model are adapted to estimate the process mean and variance. Finally, a hypothesis testing method is developed to determine if process faults exist in terms of statistical significance. A detailed experimental study illustrated the effectiveness of the proposed methodology.<br><br>Note to Practitioners-Economic globalization brings intense competition among manufacturing enterprises. The key to succeed in this competitive climate is to rapidly respond to fast-changing market demands with high-quality and competitively priced products. To achieve this, we need to quickly identify root causes of quality-related problems in a complicated manufacturing system. However, the current widely adopted quality-control techniques focus more on monitoring than on root-cause identification. These techniques can efficiently detect the changes in the process but the root cause identification is often left to the plant engineers or operators. In this paper, a systematic estimation and testing method is proposed to identify the variational root causes in multistage manufacturing processes. First, a linear model is built based on the design information to describe the cause-effect relationship between the process faults and product quality. Then, an algorithm is developed to estimate the mean and variance of the process faults from the quality measurements of products. Finally, a statistical testing method is developed to determine if process faults (i.e. root causes) exist in terms of statistical significance. A detailed experimental study illustrates the effectiveness of this method. The method presented in this paper is a new quality-control technique and can be used for quality improvement of multistage manufacturing processes.