General Linear Mixed Model for Analysing Longitudinal Data in Developmental Research

The dissertation proposes contributions to longitudinal data analysis and moderated regression analysis and is structured into three parts. The first part develops a decorated parallel coordinate plot for longitudinal categorical data, featuring a jitter mechanism revealing the diversity of observed longitudinal patterns, allowing the tracking of each individual pattern and filter options for highlighting typical patterns. The second and the third parts develop semi-parametric methods for moderated regression that combine linear regression models with tree-based algorithms. Specifically, the method of the second part allows fitting tree-structured varying coefficients in multivariate generalized linear mixed models for longitudinal data, and the method of the third part implements coefficient-wise tree-structured varying coefficients in generalized linear models. For general use, the methods are implemented in the freely available packages TraMineR and vcrpart for the statistical software environment R.

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.

BackgroundCOVID-19 has resulted in substantial global upheaval. Resilience is important in protecting wellbeing, however few studies have investigated changes in resilience over time, and associations between resilience with depression, anxiety, stress, and physical activity during the COVID-19 pandemic.MethodsOnline surveys were conducted to collect both longitudinal and cross-sectional data at three time points during 2020. Australian adults aged 18 years and over were invited to complete the online surveys. Measures include the six-item Brief Resilience Scale, the 21-item Depression, Anxiety and Stress Scale, and the Active Australia Survey which have eight items identifying the duration and frequency of walking, and moderate and vigorous physical activities (MVPA), over the past 7 days. General linear mixed models and general linear models were used in the analysis.ResultsIn the longitudinal sample, adjusted differences (aDif) in resilience scores did not significantly change over time (time 2 vs. time 1 [aDif = − 0.02, 95% CI = − 0.08, 0.03], and time 3 vs. time 1 [aDif = < 0.01, 95% CI = − 0.07, 0.06]). On average, those engaging in at least 150 min of MVPA per week (aDif = 0.10, 95% CI = 0.04, 0.16), and having depression (aDif = 0.40, 95% CI = 0.33), anxiety (aDif = 0.34, 95% CI = 0.26, 0.41), and stress scores (aDif = 0.30, 95% CI = 0.23, 0.37) within the normal range had significantly higher resilience scores. The association between resilience and physical activity was independent of depression, anxiety, and stress levels. All results were similar for the cross-sectional sample.ConclusionsResilience scores did not change significantly during the COVID-19 pandemic. However, there were significant associations between resilience with physical activity and psychological distress. This research helps inform future interventions to enhance or nurture resilience, particularly targeted at people identified as at risk of psychological distress.

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.

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.

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.

This research data file contains data from a study conducted autumn term 2018 on vegetarian, meat, fish and vegan sales, total sales; summaries of individual meal selections, and model estimates from a college cafeterias at the University of Cambridge. The aim of the study was to investigate the impact of a small change in price on vegetarian sales, meat sales and overall sales. The vegetarian option price decreased by 20p from £2.05 to £1.85, and meat option price increased by 20p from £2.52 to £2.72. The study comprised of 106 mealtimes with 13,840 meal selections. We summarized the sales transaction data into 1) aggregate data, summarizing the total vegetarian, meat, vegan and fish meal sales at each lunch and dinner and 2) individual-level data on whether each diner at a meal selected a vegetarian, meat, fish or vegan meal. Purchases made with university cards enabled anonymized individual diner-level purchases to be tracked; this is useful in evaluating how diners with different pre-study levels of purchasing vegetarian&vegan meals responded to the change in price. We used the total number of meals sold at a mealtime to analyse total sales. We estimated the effect of the price change on vegetarian, meat and total meal sales, adjusting for other predetermined variables, including day of the week, ambient temperature, and time since the start of the study. We used linear models (LMs) and binomial generalized linear models (GLMs) for aggregate data. Binomial generalized linear mixed models (GLMMs) were used for the individual-level data, with individual diner fitted as a random effect, which allows each diner to have a different likelihood of selecting a vegetarian meal. A 95% confidence level was used to calculate confidence intervals (CIs). Models were evaluated using model diagnostics.

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.

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.

The impact of vitamin D on fall incidence remains controversial. We studied the effect of 5 years of vitamin D3 supplementation on the risk of falls in a double-blind, placebo-controlled randomized trial with generally healthy, community-dwelling men and women in Finland. The study included 2495 participants, men aged ≥ 60 and women aged ≥ 65, who were randomized into three arms: 1600 IU/day or 3200 IU/day of vitamin D3 or placebo. A random subgroup of 551 participants underwent more detailed examinations. Falls and fall-related injuries were collected with questionnaires at months 0, 12, 24, 36, and 60. General linear mixed models and generalized linear models were used for analyses. Over the 5-year follow-up, a similar fall risk of 55% and fall-injury risk of 11% were observed in the placebo, 1600 IU/day, and 3200 IU/day arms, with the mean number of falls and fall-injuries per person-year of 1.26 (95% CI 1.14-1.38) and 0.07 (95% CI 0.06-0.08), respectively. Age, sex, or BMI did not modify the results. In the random subgroup, the mean baseline serum 25(OH)D concentration was 75 nmol/L (SD 18). After 12 months, the concentrations were 73, 100, and 120 nmol/L in the placebo, 1600 IU/day, and 3200 IU/day arms, respectively. Five-year vitamin D3 supplementation of 1600 IU/day or 3200 IU/day did not affect the overall risk of falls or fall injuries among generally healthy, largely vitamin D sufficient men and women. The findings do not support the use of high vitamin D doses for fall prevention in such populations. ClinicalTrials.gov: NCT01463813, https://clinicaltrials.gov/ct2/show/NCT01463813.

Visitors are a prominent feature of the zoo environment and lives of zoo animals. The COVID-19 pandemic led to repeated and extended closure periods for zoos worldwide. This unique period in zoological history enabled the opportunity to investigate the consistency of behavioural responses of zoo animals to closures and subsequent reopenings. Bennett’s wallabies (Notamacropus rufogriseus), meerkats (Suricata suricatta), macaws (red and green: Ara chloropterus; blue and yellow: Ara ararauna; military: Ara militaris) and rabbits (Oryctolagus cuniculus domesticus) held at four zoological collections in the United Kingdom were studied during COVID-19 closures and subsequent reopening periods. Facilities were closed for three time periods during 2020 and 2021: March–June/July 2020; November–December 2020; January–April/May 2021. Behavioural data were captured during closures (maximum n = 3) and reopening periods (maximum n = 3) during five-min scans using instantaneous scan sampling with a one-minute inter-scan interval. General linear models (GLMs) and general linear mixed models (GLMMs) were used to investigate the relationship between observed behaviours and open/closed periods. Changes were observed in behaviour between open and closure periods in all species, and in some instances changes were also observed over time, with animals responding differently to different closure and reopening periods. However, no overt positive or negative impacts of the closures or reopening periods were identified for these species. The study species may have different relationships with zoo visitors, but no clear differences were seen across the species studied. The unique opportunity to study animals over a long period of time during repeated closure periods enabled a greater understanding of the impact of zoo visitors on animals. As with other work in this sphere, these data support the adaptability of zoo animals to zoo visitors. This work contributes to the growing field of research undertaken during the COVID-19 periods and enhances our understanding of the impact that these zoological closures had on a wider body of species in a number of facilities.

The catch per unit of effort (CPUE) data of cockfish, Callorhinchus callorynchus, during 1986-2011 was evaluated for the bottom trawl fishery of the San Matías gulf (Patagonia, Argentina). The objective of this work was to detect what are the factors related to fishery dynamic that affect catch rate of cockfish and to assess standardized CPUE by General linear models (GLMs) and General linear mixed models (GLMMs) as a relative abundance index. The annual trend of the catch rate indicated an increase during the evaluated period. The nominal CPUE and the indices standardized by the Delta-GLM and Delta-GLMM showed the same annual trend, with increases of 57%, 61.1% and 60.7%, respectively. The LogNormal models have the best-fit model and explained 23.5% of the total variability. The factors year, month, depth and hake CPUE explained the highest variability. The analysis of the models indicated that the catch rate of cockfish is subject to the lack of an homogeneous distribution of the fishing effort during different years, months and vessels. This was related to the fleet dynamic searching the targets species. The increasing trend of CPUE and landings would be indicating an increase in effort directionality to the cockfish in the last decade.

Chapter 5 - Fitting Models Using the Bayesian Modeling Software BUGS and JAGS

Longitudinal data occur frequently in practice such as medical studies and life sciences. Generalized linear mixed models (GLMMs) are commonly used to analyze such data. It is typically assumed that the random effects covariance matrix is constant among subjects in these models. In many situations, however, the correlation structure may differ among subjects and ignoring this heterogeneity can lead to biases in model parameters estimate. Recently, Lee et al developed a heterogeneous random effects covariance matrix for GLMMs for error-free covariates. Covariates measured with error also happen frequently in the longitudinal data set-up (eg, blood pressure and cholesterol level). Ignoring this issue in the data may produce bias in model parameters estimate and lead to wrong conclusions. In this paper, we propose an approach to properly model the random effects covariance matrix based on covariates in the class of GLMMs, where we also have covariates measured with error. The resulting parameters from the decomposition of random effects covariance matrix have a sensible interpretation and can be easily modeled without the concern of positive definiteness of the resulting estimator. The performance of the proposed approach is evaluated through simulation studies, which show that the proposed method performs very well in terms of bias, mean squared error, and coverage rate. An application of the proposed method is also provided using a longitudinal data from Manitoba follow-up study.

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.