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We propose a multivariate regression model to handle multiple continuous bounded outcomes. We adopted the maximum likelihood approach for parameter estimation and inference. The model is specified by the product of univariate probability distributions and the correlation between the response variables is obtained through the correlation matrix of the random intercepts. For modeling continuous bounded variables on the interval [Formula: see text] we considered the beta and unit gamma distributions. The main advantage of the proposed model is that we can easily combine different marginal distributions for the response variable vector. The computational implementation is performed using Template Model Builder, which combines the Laplace approximation with automatic differentiation. Therefore, the proposed approach allows us to estimate the model parameters quickly and efficiently. We conducted a simulation study to evaluate the computational implementation and the properties of the maximum likelihood estimators under different scenarios. Moreover, we investigate the impact of distribution misspecification in the proposed model. Our model was motivated by a data set with multiple continuous bounded outcomes, which refer to the body fat percentage measured at five regions of the body. Simulation studies and data analysis showed that the proposed model provides a general and rich framework to deal with multiple continuous bounded outcomes.

Small area estimation (SAE) methods can provide information that conventional direct survey estimation methods cannot. The use of small area estimates based on linear and generalized linear mixed models is still very limited, possibly because of the perceived complexity of estimating the root mean square errors (RMSEs) of the estimates. This paper outlines a study used to determine the conditions under which the estimated RMSEs, produced as part of statistical output (‘plug-in’ estimates of RMSEs) could be considered appropriate for a practical application of SAE methods where one of the main requirements was to use SAS software. We first show that the estimated RMSEs created using an EBLUP model in SAS and those obtained using a parametric bootstrap are similar to the published estimated RMSEs for the corn data in the seminal paper by Battese, Harter and Fuller. We then compare plug-in estimates of RMSEs from SAS procedures used to create EBLUP and EBP estimators against estimates of RMSEs obtained from a parametric bootstrap. For this comparison we created estimates of current smoking in males for 153 local government areas (LGAs) using data from the NSW Population Health Survey in Australia. Demographic variables from the survey data were included as covariates, with LGA-level population proportions, obtained mainly from the Australian Census used for prediction. For the EBLUP, the estimated plug-in estimates of RMSEs can be used, provided the sample size for the small area is more than seven. For the EBP, the plug-in estimates of RMSEs are suitable for all in-sample areas, out-of-sample areas need to use estimated RMSEs that use the parametric bootstrap.

In this paper, we concentrate on the generalized maximum entropy (GME) estimators. The aim is to improve the problem of multicollinearity in the linear mixed models (LMMs). Then the asymptotic prop...

The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.

This study sought to examine differences in measures of intense periods of physical performance during competitive match-play, between one-match (1 match weeks) and two-match microcycles (2 match week). 1, 3 and 5 min "peak" and mean averages for total distance (TD), high-speed running (HSR) and metabolic power (MP) were analysed for each 15 min period of match-play. Linear mixed models were employed to examine the differences in dependent variables for each method of measurement between the 1 and 2 game microcycles. No differences were reported for "peak" values for all epoch lengths, however, results revealed significantly reduced "average" values, during periods of fixture congestion, for 1, 3 and 5 min epochs for average TD, and 3 min epochs for average HSR towards the end of the match (75-90 min split). There was, however, a trend for the opposite response to occur in the 60-75-min period. The current data suggests that players potentially display altered pacing strategies during periods of fixture congestion, with these observed responses being dependent on sampling method and epoch length.

Fay-Herriot model assumes that the random effects between regions (areas) are independent of each other. This allows the regions to be mutually independent so that the estimators obtained are unbiased estimators. In cases where the regions are not mutually independent, it can develop a model in which the assumptions are violated (not fulfilled) or allow the regions to be dependent. The development of this model is known as small area estimation (SAE) with spatial effects. In small area estimation with spatial effects, one of the important parameters is the spatial autocorrelation parameter. In various small area estimation with spatial effects, it is still very rare to generate estimators of the spatial autocorrelation parameter. Most of the parameter values used are known, that is by trying to enter several spatial autocorrelation parameter values to show that the addition of regional aspects can increase the accuracy of the small area estimation. In addition, they have also tried a restricted maximum likelihood approach in estimating the spatial autocorrelation and component variance but they still assume that the sampling variance is known (assigned). Therefore, this research proposes a concentrated log-likelihood function by means of numerical procedure to find an optimum estimate value for spatial autocorrelation coefficient where both a sampling variance and a component variance are unknown. Parameters estimators obtained in the models, fixed and random effects parameters, are proved to be consistent.

Any average pattern observed at the population level (cross-sectional analysis) may confound two different types of processes: some processes that occur among individuals and others that occur within individuals. Separating within- from among-individual processes is critical for our understanding of ecological and evolutionary dynamics. The within-individual centring method allows distinguishing within- from among-individual processes and this method has been widely used in ecology to investigate both linear and quadratic patterns. Here we show that two alternative equations could be used for the investigation of quadratic within-individual patterns. We explain the different assumptions and constraints of both equations. Reviewing the literature, we found that mainly one of these two equations has been used in studies investigating quadratic patterns. Yet this equation might not be the most appropriate in all circumstances leading to bias and imprecision. We show that these two alternative equations make different assumptions about the shape of the within-individual pattern. One equation assumes that the within-individual effect is related to an absolute process whereas the other assumes the effect arises from an individual relative process. The choice of using one equation instead of the other should depend upon the biological process investigated. Using simulations, we showed that a mismatch between the assumptions made by the equation used to analyse the data and the biological process investigated might led to flawed inference affecting output of model selection and accuracy of estimates. We stress that the equation used should be chosen carefully. We provide step by step guidelines for choosing an equation when studying quadratic pattern with the within-individual centring approach. We encourage the use of the within-individual centring method, promoting its relevant application for nonlinear relationships.

In the extant literature, hierarchical models typically assume a flexible distribution for the random-effects. The random-effects approach has been used in the inferential procedure of the generalized linear mixed models . In this paper, we propose a random intercept gamma mixed model to fit correlated asymmetric positive continuous outcomes. The generalized log-gamma (GLG) distribution is assumed as an alternative to the normality assumption for the random intercept. Numerical results demonstrate the impact on the maximum likelihood (ML) estimator when the random-effect distribution is misspecified. The extended inverted Dirichlet (EID) distribution is derived from the random intercept gamma-GLG model that leads to the EID regression model by supposing a particular parameter setting of the hierarchical model. Monte Carlo simulation studies are performed to evaluate the asymptotic behavior of the ML estimators from the proposed models. Analysis of diagnostic methods based on quantile residual and COVARATIO statistic are used to assess departures from the EID regression model and identify atypical subjects. Two applications with real data are presented to illustrate the proposed methodology.

This paper considers D-optimal designs for linear mixed models involving random effects with unknown distributions. From Bayesian point of view, the Dirichlet process as a prior distribution on the...

The research conducted on overtaking maneuver for evaluating drivers’ safety showed adverse effects of urgency on driving performance and decision making. Therefore, a driving simulator study was designed to examine driving performance of the drivers and its implication on overtaking and crash probabilities under increasing time pressure conditions. Eighty-eight participants data were analyzed in the current study. Three different time pressure conditions: No Time Pressure (NTP), Low Time Pressure (LTP), and High Time Pressure (HTP) were considered for analyzing driving performance of the drivers while executing overtaking maneuvers. The driving performance was assessed using minimum time-to-line crossing and coefficient of variation in speed to dissect the safety margin adopted by the drivers while overtaking the lead vehicle. Further, minimum time-to-line crossing and coefficient of variation in speed were considered as explanatory variables to investigate their influence on overtaking and crash probabilities. Parametric survival analysis and Generalized Linear Mixed Models (GLMM) were used to assess the driving performance, overtaking and crash probabilities. The parametric survival analysis showed that minimum time-to-line crossing reduced by 36.7% and 63.8% in LTP and HTP driving conditions, respectively. The GLMM results revealed that coefficient of variation in speed increased by 3.437% in HTP (no significant effect in LTP) as compared to NTP driving conditions. Further, the GLMM results showed that overtaking and crash probabilities decreased with increment in minimum time-to-line crossing and coefficient of variation in speed values. Additionally, it was observed that male drivers took risky decisions than female drivers. Nevertheless, the comparative analysis revealed that male drivers were less prone to crashes than female drivers. Overall, it can be inferred that the drivers take risky decisions with increment in time pressure to complete the driving task, even at the expense of their own safety which exposed them to high likelihood of crashes.

Clustered count data, common in health-related research, are routinely analyzed using generalized linear mixed models. There are two well-known challenges in small-sample inference in mixed modelin...

In cluster-randomized trials, sometimes the effect of the intervention being studied differs between clusters, commonly referred to as treatment effect heterogeneity. In the analysis of stepped wedge and cluster-randomized crossover trials, it is possible to include terms in outcome regression models to allow for such treatment effect heterogeneity yet this is not frequently considered. Outside of some simulation studies of specific cases where the outcome is binary, the impact of failing to include terms for treatment effect heterogeneity on the variance of the treatment effect estimator is unknown. We analytically examine the impact of failing to include terms for treatment effect heterogeneity on the variance of the treatment effect estimator, when outcomes are continuous. Using analysis of variance and feasible generalized least squares we provide expressions for this variance. For both the cluster-randomized crossover design and the stepped wedge design, our analytic derivations indicate that failing to include treatment effect heterogeneity results in the estimates for variance of the treatment effect that are too small, leading to inflation of type I error rates. We therefore recommend assessing the sensitivity of sample size calculations and conclusions drawn from the analysis of cluster randomized trials to the inclusion of treatment effect heterogeneity.

This paper is concerned with comparison problems of predictors between a linear mixed model (LMM) that includes both fixed and random effects and its transformed model under general assumptions. Ou...

It is quite appealing to extend existing theories in classical linear models to correlated responses where linear mixed-effects models are utilized and the dependency in the data is modeled by random effects. In the mixed modeling framework, missing values occur naturally due to dropouts or non-responses, which is frequently encountered when dealing with real data. Motivated by such problems, we aim to investigate the estimation and model selection performance in linear mixed models when missing data are present. Inspired by the property of the indicator function for missingness and its relation to missing rates, we propose an approach that records missingness in an indicator-based matrix and derive the likelihood-based estimators for all parameters involved in the linear mixed-effects models. Based on the proposed method for estimation, we explore the relationship between estimation and selection behavior over missing rates. Simulations and a real data application are conducted for illustrating the effectiveness of the proposed method in selecting the most appropriate model and in estimating parameters.

Hyungmi An,1 Dongseong Shin2,3 1Institute of Convergence Medicine, Ewha Womans University Mokdong Hospital, Seoul, Korea; 2Department of Pharmacology, Gachon University College of Medicine, Incheon, Korea; 3Clinical Trials Center, Gachon University Gil Medical Center, Incheon, KoreaCorrespondence: Dongseong ShinDepartment of Pharmacology, Gachon University College of Medicine, 21, Namdong-daero 774 Beon-gil, Namdong-gu, Incheon, 21565, KoreaTel +82-32-460-9459Fax +82-32-460-9443Email dsshin@gilhospital.comBackground and Objective: Bioequivalence tests are fundamental step in assessing the equivalence in bioavailability between a test and reference product. In practice, two separate linear mixed models (LMMs) with random subject effects, which have an area under the concentration-time curve (AUC) and the peak concentration (Cmax) as the responses, have become the gold standard for evaluating bioequivalence. Recently, Lee et al developed a multivariate hierarchical generalized linear model (HGLM) for several responses that modeled correlations among multivariate responses via correlated random effects. The objective of this study was to apply this multivariate analysis to the bioequivalence test in practice and to compare the performance of multivariate HGLM and separate LMMs.Methods: Three pharmacokinetic datasets, fixed-dose combination (naproxen and esomeprazole), tramadol and fimasartan data were analyzed. We compared the 90% confidence interval (CI) for the geometric mean ratio (GMR) of a test product to a reference product using the multivariate HGLM and two conventional separate LMMs.Results: We found that the 90% CIs for the GMRs of both AUC and Cmax from the multivariate HGLM were narrower than those from the separate LMMs: (0.843, 1.152) vs (0.825, 1.177) for Cmax of esomeprazole in fixed-dose combination data; (0.805, 0.931) vs (0.797, 0.941) for Cmax in tramadol data; (0.801, 1.501) vs (0.762, 1.578) for Cmax and (1.163, 1.332) vs (1.009, 1.341) for AUC in fimasartan data, consistent with the random subject effects from two separate LMMs being highly correlated in the three datasets (correlation coefficient r = 0.883; r = 0.966; r = 0.832).Conclusion: This multivariate HGLM had good performance in the bioequivalence test with multiple endpoints. This method would provide a more reasonable option to reduce the 90% CI by adding correlation parameters and thus an advantage especially in evaluating the bioequivalence of highly variable drugs with broad 90% CIs.Keywords: bioequivalence test, correlated multiple responses, multivariate random effects model, H-likelihood, multivariate HGLM

In this article, we propose the stochastic restricted Liu predictors by augmenting the stochastic restrictions to the linear mixed models. The Liu biasing parameter is selected via generalized cross validation (GCV) criterion. Comparisons between the stochastic restricted Liu estimators and several other estimators, namely the BLUE, the mixed and Liu estimators are made through the mean square error matrix criterion. Finally, a numerical example and a simulation study are done to show the performance of the estimators.

The hierarchical Bayes predictor of small area proportions (HBP) under an area level version of generalized linear mixed model with logit link function is widely used in small area estimation for b...

Spatial confounding, that is, collinearity between fixed effects and random effects in a spatial generalized linear mixed model, can adversely affect estimates of the fixed effects. Restricted spat...

We explore the possibility of modeling clustered count data using the Poisson Inverse Gaussian distribution. We develop a regression model, which relates the number of mastitis cases in a sample of dairy farms in Ontario, Canada, to various farm level covariates, to illustrate the methodology. Residual plots are constructed to explore the quality of the fit. We compare the results with a negative binomial regression model using maximum likelihood estimation, and to the generalized linear mixed regression model fitted in SAS.

Normality (symmetric) of the random effects and the within-subject errors is a routine assumptions for the linear mixed model, but it may be unrealistic, obscuring important features of among- and within-subjects variation. We relax this assumption by considering that the random effects and model errors follow a skew-normal distributions, which includes normality as a special case and provides flexibility in capturing a broad range of non-normal behavior. The marginal distribution for the observed quantity is derived which is expressed in closed form, so inference may be carried out using existing statistical software and standard optimization techniques. We also implement an EM type algorithm which seem to provide some advantages over a direct maximization of the likelihood. Results of simulation studies and applications to real data sets are reported.

Receiver operating characteristic (ROC) curve is an effective and widely used method for evaluating the discriminating power of a diagnostic test or statistical model. As a useful statistical method, a wealth of literature about its theories and computation methods has been established. The research on ROC curves, however, has focused mainly on cross-sectional design. Very little research on estimating ROC curves and their summary statistics, especially significance testing, has been conducted for repeated measures design. Due to the complexity of estimating the standard error of a ROC curve, there is no currently established statistical method for testing the significance of ROC curves under a repeated measures design. In this paper, we estimate the area of a ROC curve under a repeated measures design through generalized linear mixed model (GLMM) using the predicted probability of a disease or positivity of a condition and propose a bootstrap method to estimate the standard error of the area under a ROC curve for such designs. Statistical significance testing of the area under a ROC curve is then conducted using the bootstrapped standard error. The validity of bootstrap approach and the statistical testing of the area under the ROC curve was validated through simulation analyses. A special statistical software written in SAS/IML/MACRO v8 was also created for implementing the bootstrapping algorithm, conducting the calculations and statistical testing.

Longitudinal studies represent one of the principal research strategies employed in medical and social research. These studies are the most appropriate for studying individual change over time. The prematurely withdrawal of some subjects from the study (dropout) is termed nonrandom when the probability of missingness depends on the missing value. Nonran- dom dropout is common phenomenon associated with longitudinal data and it complicates statistical inference. Linear mixed effects model is used to fit longitudinal data in the presence of nonrandom dropout. The stochastic EM algorithm is developed to obtain the model parameter estimates. Also, parameter estimates of the dropout model have been obtained. Standard errors of estimates have been calculated using the developed Monte Carlo method. All these methods are applied to two data sets.

In this paper, we propose a nonparametric approach using the Dirichlet processes (DP) as a class of prior distributions for the distribution G of the random effects in the hierarchical generalized linear mixed model (GLMM). The support of the prior distribution (and the posterior distribution) is large, allowing for a wide range of shapes for G. This provides great flexibility in estimating G and therefore produces a more flexible estimator than does the parametric analysis. We present some computation strategies for posterior computations involved in DP modeling. The proposed method is illustrated with real examples as well as simulations.

Generalized Linear Mixed Model (GLMM) is a framework that has a response variable, fixed effects, and random effects. The response variable comes from an exponential family, whereas random effects have a normal distribution. Estimating parameters can be calculated using the maximum likelihood method using the Laplace approach or the Gauss-Hermite Quadrature (GHQ) approach. The purpose of this study was to identify factors that trigger student's interest to continue studying at Universitas Syiah Kuala (USK) using both techniques. The GLMM is suitable for the data because the variable response has a Bernoulli distribution, and the random effects are assumed to be having a normal distribution. Also, the model helps identify the relationship between the dependent variable and the predictors. This study utilizes data from six high schools in Banda Aceh city drawn using a two-stage sampling technique. Stage 1, we randomly chose six out of sixteen public senior high schools in Banda Aceh. Stage 2, we selected students from each school from four different major classes. The GLMM model includes one binary response variable, five numerical fixed-effects, and two random effects. The response variable is the interest of high school students to continue study at USK (yes or no). The five fixed effects in the model including scores of collaboration (C), Action (A), Emotion (E), Purposes (P), and Hope (H). Finally, the random effects are schools (S) and majors (M). In this study, both Laplace and GHQ techniques produce identical results. The predictors that can explain student interest are A, E, and H. These predictors have a positive effect. The random effects of schools and majors are not significantly different from zero. The model with three significant predictors is better than the complete predictor model.

In this paper, we introduce a new ridge-type estimator/predictor called the weighted mixed ridge estimator/predictor (WMRE/P) by unifying the sample and prior information in a linear mixed model wi...

This paper considers the optimal design problem for predicting a linear combination of fixed and random effects when the variance components in the linear mixed model are known or unknown. New desi...

In this work it is proposed a modeling for operating room times based on a Bayesian Hierarchical structure. Specifically, it is employed a Bayesian generalized linear mixed model with an additional hierarchical level on the random effects. This configuration allows the estimation of operating room times (ORT) with few or no historical observations, without requiring a prior surgeon's estimate. In addition to the widely used lognormal distribution, it is also studied the gamma distribution to model the operating room times. For the scale parameters related to the random effects (surgeon and surgical group), which are important quantities in this type of modeling, different kinds of prior distributions such as Half-Cauchy, Sbeta2, and uniform are studied. A Bayesian version of the classical ANOVA is implemented to identify relevant predictors for the operating room times. We find that lognormal models outperform the gamma models in estimating upper prediction bounds (UB). Especially, the best ORT predictions for cases with few or no historical data (i.e., between 0 and 3 historical cases) are obtained with the [Formula: see text], SBeta2 model. With a deviation of less than 1% with respect to the nominal coverage of the upper bound predictions UB80% and UB90% and an average absolute percentage error of 38.5% in the point estimate.

Rainfall is one of the climatic elements in the tropics which is very influential in agriculture, especially in determining the growing season. Thus, proper rainfall modeling is needed to help determine the best time to start cultivating the soil. Rainfall modeling can be done using the Statistical Downscaling (SDS) method. SDS is a statistical model in the field of climatology to analyze the relationship between large-scale and small-scale climate data. This study uses response variables as a small-scale climate data in the form of rainfall and explanatory variables as a large-scale climate data of the General Circulation Model (GCM) output in the form of precipitation. However, the application of SDS modeling is known to cause several problems, including correlated and not stationary response variables, multi-dimensional explanatory variables, multicollinearity, and spatial correlation between grids. Modeling with some of these problems will cause violations of the assumptions of independence and multicollinearity. This research aims to model the rainfall in Indramayu Regency, West Java Province using a combined regression model between the Generalized linear mixed model (GLMM) and Least Absolute Selection and Shrinkage Operator ( LASSO) regulation (L1 ). GLMM was used to deal with the problem of independence and Lasso Regulation (L1) was used to deal with multicollinearity problems or the number of explanatory variables that is greater than the response variable. Several models were formed to find the best model for modeling rainfall. This research used the GLMM-Lasso model with Normal spread compared to the GLMM model with Gamma response (Gamma-GLMM). The results showed that the RMSE and R-square GLMM-Lasso models were smaller than the Gamma-GLMM models. Thus, it can be concluded that GLMM-Lasso model can be used to model statistical downscaling and solve the previously mentioned constraints. Received February 10, 2021 Revised March 29, 2021 Accepted March 29, 2021

Mixed‐effects models for repeated measures (MMRMs) with an ‘unstructured’ (UN) covariance structure are frequently used in primary analyses for group comparisons of incomplete continuous longitudinal data from drug development trials. However, MMRM‐UN analysis could lead to convergence problems in numerical optimisation, especially in trials with a small sample size or high dropout rate. Although the so‐called sandwich covariance estimator is robust against the misspecification of the covariance structure, its performance deteriorates for small sample sizes. We review eight modified covariance estimators adjusted for small‐sample bias and compare their performances in the framework of MMRM analysis through simulations. In terms of the type 1 error rate and coverage probability of confidence intervals for group comparisons, Mancl and DeRouen's covariance estimator (MD) shows the best performance among the modified covariance estimators, followed by Fay and Graubard's estimator. The performance of MD is nearly equivalent to that of the Kenward–Roger method with a UN structure. The Kenward–Roger method with first‐order autoregressive structure results in substantial inflation of the type 1 error rate in the scenario where the variance of measurements increases across visits. In summary, we recommend the use of MD in MMRM analysis if the convergence problem involving a UN structure occurs in small clinical trials.

Existing studies on spatial linear mixed models typically assume a normal distribution for the random error components. However, such an assumption may not be appropriate in many applications. This...

When designing repeated measures studies, both the amount and the pattern of missing outcome data can affect power. The chance that an observation is missing may vary across measurements, and missingness may be correlated across measurements. For example, in a physiotherapy study of patients with Parkinson’s disease, increasing intermittent dropout over time yielded missing measurements of physical function. In this example, we assume data are missing completely at random, since the chance that a data point was missing appears to be unrelated to either outcomes or covariates. For data missing completely at random, we propose noncentral F power approximations for the Wald test for balanced linear mixed models with Gaussian responses. The power approximations are based on moments of missing data summary statistics. The moments were derived assuming a conditional linear missingness process. The approach provides approximate power for both complete-case analyses, which include independent sampling units where all measurements are present, and observed-case analyses, which include all independent sampling units with at least one measurement. Monte Carlo simulations demonstrate the accuracy of the method in small samples. We illustrate the utility of the method by computing power for proposed replications of the Parkinson’s study.

Mixed models are a useful way of analysing longitudinal data. Random effects terms allow modelling of patient specific deviations from the overall trend over time. Correlation between repeated measurements are captured by specifying a joint distribution for all random effects in a model. Typically, this joint distribution is assumed to be a multivariate normal distribution. For Gaussian outcomes misspecification of the random effects distribution usually has little impact. However, when the outcome is discrete (e.g. counts or binary outcomes) generalised linear mixed models (GLMMs) are used to analyse longitudinal trends. Opinion is divided about how robust GLMMs are to misspecification of the random effects. Previous work explored the impact of random effects misspecification on the bias of model parameters in single outcome GLMMs. Accepting that these model parameters may be biased, we investigate whether this affects our ability to classify patients into clinical groups using a longitudinal discriminant analysis. We also consider multiple outcomes, which can significantly increase the dimensions of the random effects distribution when modelled simultaneously. We show that when there is severe departure from normality, more flexible mixture distributions can give better classification accuracy. However, in many cases, wrongly assuming a single multivariate normal distribution has little impact on classification accuracy.

We group approaches to modeling correlated binary data according to data recorded cross-sectionally as opposed to data recorded longitudinally; according to models that are population-averaged as opposed to subject-specific; and according to data with time-dependent covariates as opposed to time-independent covariates. Standard logistic regression models are appropriate for cross-sectional data. However, for longitudinal data, methods such as generalized estimating equations (GEE) and generalized method of moments (GMM) are commonly used to fit population-averaged models, while random-effects models such as generalized linear mixed models (GLMM) are used to fit subject-specific models. Some of these methods account for time-dependence in covariates while others do not. This paper addressed these approaches with an illustration using a Medicare dataset as it relates to rehospitalization. In particular, we compared results from standard logistic models, GEE models, GMM models, and random-effects models by analyzing a binary outcome for four successive hospitalizations. We found that these procedures address differently the correlation among responses and the feedback from response to covariate. We found marginal GMM logistic regression models to be more appropriate when covariates are classified as time-dependent in comparison to GEE models. We also found conditional random-intercept models with time-dependent covariates decomposed into components to be more appropriate when time-dependent covariates are present in comparison to ordinary random-effects models. We used the SAS procedures GLIMMIX, NLMIXED, IML, GENMOD, and LOGISTIC to analyze the illustrative dataset, as well as unique programs written using the R language.

Finite mixture models are a widely known method for modelling data that arise from a heterogeneous population. Within the family of mixtures of regression models, mixtures of linear mixed models have also been applied in different areas since, besides taking into consideration the heterogeneity in the population, they also allow to take into account the correlation between observations from the same individual. One of the main issues in mixture models concerns the estimation of the parameters. Maximum likelihood estimation is one of the most used methods in the estimation of the parameters for mixture models. However, the maximization of the log-likelihood function in mixture models is complex, producing in many cases infinite solutions whereby the maximum likelihood estimator may not exist, at least globally. For this reason, it is common to resort to iterative methods, in particular to the Expectation-Maximization (EM) algorithm. However, the slow convergence and the selection of initial values are two of biggest issues of the EM algorithm, the reason why some modified versions of this algorithm have been developed over the years. In this article we compare the performance of the EM, Classification EM (CEM) and Stochastic EM (SEM) algorithms in the estimation of the parameters for mixtures of linear mixed models. In order to evaluate their performance, we carry out a simulation study and a real data application. The results show that the CEM algorithm is the least computationally demanding algorithm, although the three algorithms provide similar maximum likelihood estimates for the parameters., The research of L. Novais was financed by FCT - Fundacao para a Ciencia e a Tecnologia, through the PhD scholarship with reference SFRH/BD/139121/2018.

Rathouz and Gao [2] and Luo and Tsai [3] proposed valuable extensions to the generalized linear model for modeling a nonlinear monotonic relationship between the mean response and a set of covariates. In their extensions for discrete data the baseline response distribution is unspecified and is estimated from the data. We propose to extend this model for the analysis of longitudinal data by incorporating random effects into the linear predictor, and using maximum likelihood for estimation and inference. Motivated in particular by longitudinal studies of clinical scale outcomes, we developed an estimation procedure for a finite-support response using a generalized expectation-maximization algorithm where Gauss-Hermite quadrature is employed to approximate the integrals in the E step of the algorithm. Upon convergence, the observed information matrix is estimated through second-order numerical differentiation of the log-likelihood function. Asymptotic properties of the maximum likelihood estimates follow under the usual regularity conditions. Simulation studies are conducted to assess its finite-sample properties and compare the proposed model to the generalized linear mixed model. The proposed method is illustrated in an analysis of data from a longitudinal study of Huntington's disease.

The main purpose of this paper is to investigate D-optimal population designs in multi-response linear mixed models for longitudinal data. Observations of each response variable within subjects are...

We propose a latent linear mixed model to analyze multivariate longitudinal data of multiple ordinal variables, which are manifestations of fewer continuous latent variables. We focus on the latent level where the effects of observed covariates on the latent variables are of interest. We incorporate serial correlation into the variance component rather than assuming independent residuals. We show that misleading inference may be drawn when misspecifying the variance component. Furthermore, we provide a graphical tool depicting latent empirical semi-variograms to detect serial correlation for latent stationary linear mixed models. We apply our proposed model to examine the treatment effect on patients having the amyotrophic lateral sclerosis disease. The result shows that the treatment can slow down progression of latent cervical and lumbar functions.

Multi-regional clinical trials have a hierarchical data structure because several regions form a patient population and individual patients are nested within their own regions. Data are obtained fr...

The connection between regularization and min–max robustification in the presence of unobservable covariate measurement errors in linear mixed models is addressed. We prove that regularized model parameter estimation is equivalent to robust loss minimization under a min–max approach. On the example of the LASSO, Ridge regression, and the Elastic Net, we derive uncertainty sets that characterize the feasible noise that can be added to a given estimation problem. These sets allow us to determine measurement error bounds without distribution assumptions. A conservative Jackknife estimator of the mean squared error in this setting is proposed. We further derive conditions under which min-max robust estimation of model parameters is consistent. The theoretical findings are supported by a Monte Carlo simulation study under multiple measurement error scenarios.

In economic studies researchers are often interested in the estimation of the distribution function or certain functions of the distribution function such as quantiles. This work focuses on the estimation quantiles as inverses of the estimates of the distribution function in the presence of auxiliary information that is correlated with the study variable. In the paper a plug-in estimator of the distribution function is proposed which is used to obtain quantiles in the population and in the small areas. Performance of the proposed method is compared with other estimators of the distribution function and quantiles using the simulation study. The obtained results show that the proposed method usually has smaller relative biases and relative RMSE comparing to other methods of obtaining quantiles based on inverting the distribution function.

We study robust designs for generalized linear mixed models (GLMMs) with protections against possible departures from underlying model assumptions. Among various types of model departures, an imprecision in the assumed linear predictor or the link function has a great impact on predicting the conditional mean response function in a GLMM. We develop methods for constructing adaptive sequential designs when the fitted mean response or the link function is possibly of an incorrect parametric form. We adopt the maximum likelihood method for estimating the parameters in GLMMs and investigate both I-optimal and D-optimal design criteria for the construction of robust sequential designs. To study the empirical properties of these sequential designs, we ran a series of simulations using both logistic and Poisson mixed models. As indicated in the simulation results, the I-optimal design generally outperforms the D-optimal design for all scenarios considered. Both designs are more efficient than the conventionally used uniform design and the classical D-optimal design obtained under the assumption that the fitted models are correctly specified. The proposed designs are also illustrated in an example using actual data from a dose–response experiment.

Background: Accurate and reliable prediction of changes in the severity of cerebellar ataxia (CA) will be necessary for trials of disease-modifying therapies. Cerebellar dysarthria (CD) is a common feature of CA. This study demonstrated that objective acoustic measures were more sensitive than perceptive analysis through the Scale for the Assessment and Rating of Ataxia (SARA) in assessing the progression of CD, within a time window of two years (mean). Method: Thirty-seven people with CA were tested at baseline (time point 1, TP1) and two years later (time point 2, TP2). A machine-learning framework with a robust three-step feature selection criterion and a Bayesian data-driven clustering technique based on the multivariate mixture extension of the generalized linear mixed model (GLMM) was used. The outcomes included two (time and cepstral-based) objective speech parameters recorded at TP1 and TP2. Subject testing involved dynamic prediction and was conducted using samples from the posterior distributions of parameter estimates and random effects. This study further employed the penalized expected deviance (PED) criterion for model comparison and the selection of the number of groups in the clustering procedure. Results: First, the selected objective speech metrics in the individual patients showed a significant worsening of the speech impairment (p $PED_{Fea1+Fea2}=5175$ ; number of groups = 2) performed more reliably than the univariate models ( $PED_{Fea1}=4225$ , $PED_{Fea2}=3850$ ; number of groups = 2) in discriminating the CA patients. Fourth, the individual-level predictions of the change in profiles of the objective measures over time were performed for the testing data. Conclusion: Such a framework using objective speech metrics indeed holds promise to predict the rate of clinical progression of CD in individuals with CA.

Today, Linear Mixed Models (LMMs) are fitted, mostly, by assuming that random effects and errors have Gaussian distributions, therefore using Maximum Likelihood (ML) or REML estimation. However, for many data sets, that double assumption is unlikely to hold, particularly for the random effects, a crucial component in which assessment of magnitude is key in such modeling. Alternative fitting methods not relying on that assumption (as ANOVA ones and Rao’s MINQUE) apply, quite often, only to the very constrained class of variance components models. In this paper, a new computationally feasible estimation methodology is designed, first for the widely used class of 2-level (or longitudinal) LMMs with only assumption (beyond the usual basic ones) that residual errors are uncorrelated and homoscedastic, with no distributional assumption imposed on the random effects. A major asset of this new approach is that it yields nonnegative variance estimates and covariance matrices estimates which are symmetric and, at least, positive semi-definite. Furthermore, it is shown that when the LMM is, indeed, Gaussian, this new methodology differs from ML just through a slight variation in the denominator of the residual variance estimate. The new methodology actually generalizes to LMMs a well known nonparametric fitting procedure for standard Linear Models. Finally, the methodology is also extended to ANOVA LMMs, generalizing an old method by Henderson for ML estimation in such models under normality.

In this paper, we introduce stochastic-restricted Liu predictors which will be defined by combining in a special way the two approaches followed in obtaining the mixed predictors and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictor to the Liu and mixed predictors are done in the sense of mean square error matrix criterion. Finally, numerical examples and a simulation study are done to illustrate the findings. In numerical examples, we took some arbitrary observations from the data as the prior information since we did not have historical data or additional information about the data sets. The results show that this case does the new estimator gain efficiency over the constituent estimators and provide accurate estimation and prediction of the data.