Robust estimation in mixed linear models with non-monotone missingness

We introduce a model to account for abrupt changes among repeated measures with non-monotone missingness. Development of likelihood inferences for such models is hard because it involves intractable integration to obtain the marginal likelihood. We use hierarchical likelihood to overcome such difficulty. Abrupt changes among repeated measures can be well described by introducing random effects in the dispersion. A simulation study shows that the resulting estimator is efficient, robust against misspecification of fatness of tails. For illustration we use a schizophrenic behaviour data presented by Rubin and Wu.     

Strategies for analyzing multilevel cluster-randomized studies with binary outcomes collected at varying intervals of time

Frequently, studies are conducted in a real clinic setting. When the outcome of interest is collected longitudinally over a specified period of time, this design can lead to unequally spaced intervals and varying numbers of assessments. In our study, these features were embedded in a randomized, factorial design in which interventions to improve blood pressure control were delivered to both patients and providers. We examine the effect of the intervention and compare methods of estimation of both fixed effects and variance components in the multilevel generalized linear mixed model. Methods of comparison include penalized quasi-likelihood (PQL), adaptive quadrature, and Bayesian Monte Carlo methods. We also investigate the implications of reducing the data and analysis to baseline and final measurements. In the full analysis, the PQL fixed-effects estimates were closest to zero and confidence intervals were generally narrower than those of the other methods. The adaptive quadrature and Bayesian fixed-effects estimates were similar, but the Bayesian credible intervals were consistently wider. Variance component estimation was markedly different across methods, particularly for the patient-level random effects. In the baseline and final measurement analysis, we found that estimates and corresponding confidence intervals for the adaptive quadrature and Bayesian methods were very similar. However, the time effect was diminished and other factors also failed to reach statistical significance, most likely due to decreased power. When analyzing data from this type of design, we recommend using either adaptive quadrature or Bayesian methods to fit a multilevel generalized linear mixed model including all available measurements. Published in 2008 by John Wiley & Sons, Ltd.     

Assessing correlation of clustered mixed outcomes from a multivariate generalized linear mixed model

The classic concordance correlation coefficient measures the agreement between two variables. In recent studies, concordance correlation coefficients have been generalized to deal with responses from a distribution from the exponential family using the univariate generalized linear mixed model. Multivariate data arise when responses on the same unit are measured repeatedly by several methods. The relationship among these responses is often of interest. In clustered mixed data, the correlation could be present between repeated measurements either within the same observer or between different methods on the same subjects. Indices for measuring such association are needed. This study proposes a series of indices, namely, intra‐correlation, inter‐correlation, and total correlation coefficients to measure the correlation under various circumstances in a multivariate generalized linear model, especially for joint modeling of clustered count and continuous outcomes. The proposed indices are natural extensions of the concordance correlation coefficient. We demonstrate the methodology with simulation studies. A case example of osteoarthritis study is provided to illustrate the use of these proposed indices. Copyright © 2014 John Wiley & Sons, Ltd.     

Multicomponent variance estimation for binary traits in family-based studies

In biometrical genetic analyses of binary traits, the use of family data overcomes some limitations of twin studies, particularly in terms of sample size and types of genetic or environmental factors that can be estimated. However, because of computational problems, recent methods in the application of generalized linear mixed models for family data structure have limited the ability to handle large data sets with general covariates. In this paper, we investigate the use of the hierarchical likelihood approach to the analysis of binary traits from family data. In a simulation study, the method is shown to be highly accurate for the estimation of both the variance components and fixed regression parameters, even for small family sizes. For illustration, we analyze a real data set of familial aggregation of preeclampsia, a pregnancy-induced hypertension. When possible, the analysis is compared with the exact maximum likelihood approach.     

Model comparison of generalized linear mixed models

Generalized linear mixed models (GLMMs) have been widely appreciated in biological and medical research. Maximum likelihood estimation has received a great deal of attention. Comparatively, not much has been done on model comparison or hypotheses testing. In this article, we propose a path sampling procedure to compute the observed-data log-likelihood function, so that the Bayesian information criterion (BIC) can be applied to model comparison or hypothesis testing. Advantages of the proposed path sampling procedure are discussed. Two medical data sets are analysed for providing illustrative examples of the proposed methodology.     

Adjusting for confounding by neighborhood using generalized linear mixed models and complex survey data

When investigating health disparities, it can be of interest to explore whether adjustment for socioeconomic factors at the neighborhood level can account for, or even reverse, an unadjusted difference. Recently, we proposed new methods to adjust the effect of an individual‐level covariate for confounding by unmeasured neighborhood‐level covariates using complex survey data and a generalization of conditional likelihood methods. Generalized linear mixed models (GLMMs) are a popular alternative to conditional likelihood methods in many circumstances. Therefore, in the present article, we propose and investigate a new adaptation of GLMMs for complex survey data that achieves the same goal of adjusting for confounding by unmeasured neighborhood‐level covariates. With the new GLMM approach, one must correctly model the expectation of the unmeasured neighborhood‐level effect as a function of the individual‐level covariates. We demonstrate using simulations that even if that model is correct, census data on the individual‐level covariates are sometimes required for consistent estimation of the effect of the individual‐level covariate. We apply the new methods to investigate disparities in recency of dental cleaning, treated as an ordinal outcome, using data from the 2008 Florida Behavioral Risk Factor Surveillance System (BRFSS) survey. We operationalize neighborhood as zip code and merge the BRFSS data with census data on ZIP Code Tabulated Areas to incorporate census data on the individual‐level covariates. We compare the new results to our previous analysis, which used conditional likelihood methods. We find that the results are qualitatively similar. Copyright © 2012 John Wiley & Sons, Ltd.     

Penalized likelihood estimation for semiparametric mixed models,with application to alcohol treatment research

In this article, we implement a practical computational method for various semiparametric mixed effects models, estimating nonlinear functions by penalized splines. We approximate the integration of the penalized likelihood with respect to random effects with the use of adaptive Gaussian quadrature, which we can conveniently implement in SAS procedure NLMIXED. We carry out the selection of smoothing parameters through approximated generalized cross‐validation scores. Our method has two advantages: (1) the estimation is more accurate than the current available quasi‐likelihood method for sparse data, for example, binary data; and (2) it can be used in fitting more sophisticated models. We show the performance of our approach in simulation studies with longitudinal outcomes from three settings: binary, normal data after Box–Cox transformation, and count data with log‐Gamma random effects. We also develop an estimation method for a longitudinal two‐part nonparametric random effects model and apply it to analyze repeated measures of semicontinuous daily drinking records in a randomized controlled trial of topiramate. Copyright © 2012 John Wiley & Sons, Ltd.     

Longitudinal and repeated cross-sectional cluster-randomization designs using mixed effects regression for binary outcomes: bias and coverage of frequentist and Bayesian methods

As medical applications for cluster randomization designs become more common, investigators look for guidance on optimal methods for estimating the effect of group-based interventions over time. This study examines two distinct cluster randomization designs: (1) the repeated cross-sectional design in which centres are followed over time but patients change, and (2) the longitudinal design in which individual patients are followed over time within treatment clusters. Simulations of each study design stipulated a multiplicative treatment effect (on the log odds scale), between 5 and 15 clusters in each of two treatment arms, and followed over two time periods. Estimation options included linear mixed effects models using restricted maximum likelihood (REML), generalized estimating equations (GEE), mixed effects logistic regression using both penalized quasi likelihood (PQL) and numerical integration, and Bayesian Monte Carlo analysis. For the repeated cross-sectional designs, most methods performed well in terms of bias and coverage when clusters were numerous (30) and variability across clusters of baseline risk and treatment effect was modest. With few clusters (two groups of five) and higher variability, only the Bayesian methods maintained coverage. In the longitudinal designs, the common methods of REML, GEE, or PQL performed poorly when compared to numerical integration, while Bayesian methods demonstrated less bias and better coverage for estimates of both log odds ratios and risk differences. The performance of common statistical tools for the analysis of cluster randomization designs depends heavily on the precise design, the number of clusters, and the variability of baseline outcomes and treatment effects across centres.     

Estimation of genetic and environmental factors for binary traits using family data

While the family-based analysis of genetic and environmental contributions to continuous or Gaussian traits is now straightforward using the linear mixed models approach, the corresponding analysis of complex binary traits is still rather limited. In the latter we usually rely on twin studies or pairs of relatives, but these studies often have limited sample size or have difficulties in dealing with the dependence between the pairs. Direct analysis of extended family data can potentially overcome these limitations. In this paper, we will describe various genetic models that can be analysed using an extended family structure. We use the generalized linear mixed model to deal with the family structure and likelihood-based methodology for parameter inference. The method is completely general, accommodating arbitrary family structures and incomplete data. We illustrate the methodology in great detail using the Swedish birth registry data on pre-eclampsia, a hypertensive condition induced by pregnancy. The statistical challenges include the specification of sensible models that contain a relatively large number of variance components compared to standard mixed models. In our illustration the models will account for maternal or foetal genetic effects, environmental effects, or a combination of these and we show how these effects can be readily estimated using family data.     

Analysis of ulcer data using hierarchical generalized linear models

In multi-centre clinical trials, heterogeneities in individual hospital treatment effects can be modelled as random effects. Estimates of the individual hospital treatment effects and estimate of the mean treatment effect, allowing for the presence of overall hospital differences, are required, together with some measure of their uncertainty. Systematic inferences from the hierarchical-likelihood are now possible, using hierarchical generalized linear models. We show how to construct profile likelihoods for the treatment effects of individual hospitals.     

Correcting for nonrandom ascertainment in generalized linear mixed models (GLMMs), fitted using Gibbs sampling

Gibbs sampling-based generalized linear mixed models (GLMMs) provide a convenient and flexible way to extend variance components models for multivariate normally distributed continuous traits to other classes of phenotype. This includes binary traits and right-censored failure times such as age-at-onset data. The approach has applications in many areas of genetic epidemiology. However, the required GLMMs are sensitive to nonrandom ascertainment. In the absence of an appropriate correction for ascertainment, they can exhibit marked positive bias in the estimated grand mean and serious shrinkage in the estimated magnitude of variance components. To compound practical difficulties, it is currently difficult to implement a conventional adjustment for ascertainment because of the need to undertake repeated integration across the distribution of random effects. This is prohibitively slow when it must be repeated at every iteration of the Markov chain Monte Carlo (MCMC) procedure. This paper motivates a correction for ascertainment that is based on sampling random effects rather than integrating across them and can therefore be implemented in a general-purpose Gibbs sampling environment such as WinBUGS. The approach has the characteristic that it returns ascertainment-adjusted parameter estimates that pertain to the true distribution of determinants in the ascertained sample rather than in the general population. The implications of this characteristic are investigated and discussed. This paper extends the utility of Gibbs sampling-based GLMMs to a variety of settings in which family data are ascertained nonrandomly.     

Generalized linear mixed model for binary outcomes when covariates are subject to measurement errors and detection limits

Longitudinal measurement of biomarkers is important in determining risk factors for binary endpoints such as infection or disease. However, biomarkers are subject to measurement error, and some are also subject to left‐censoring due to a lower limit of detection. Statistical methods to address these issues are few. We herein propose a generalized linear mixed model and estimate the model parameters using the Monte Carlo Newton‐Raphson (MCNR) method. Inferences regarding the parameters are made by applying Louis's method and the delta method. Simulation studies were conducted to compare the proposed MCNR method with existing methods including the maximum likelihood (ML) method and the ad hoc approach of replacing the left‐censored values with half of the detection limit (HDL). The results showed that the performance of the MCNR method is superior to ML and HDL with respect to the empirical standard error, as well as the coverage probability for the 95% confidence interval. The HDL method uses an incorrect imputation method, and the computation is constrained by the number of quadrature points; while the ML method also suffers from the constrain for the number of quadrature points, the MCNR method does not have this limitation and approximates the likelihood function better than the other methods. The improvement of the MCNR method is further illustrated with real‐world data from a longitudinal study of local cervicovaginal HIV viral load and its effects on oncogenic HPV detection in HIV‐positive women.     

Bayesian inference for generalized linear mixed models with predictors subject to detection limits: an approach that leverages information from auxiliary variables

This paper is motivated from a retrospective study of the impact of vitamin D deficiency on the clinical outcomes for critically ill patients in multi‐center critical care units. The primary predictors of interest, vitamin D2 and D3 levels, are censored at a known detection limit. Within the context of generalized linear mixed models, we investigate statistical methods to handle multiple censored predictors in the presence of auxiliary variables. A Bayesian joint modeling approach is proposed to fit the complex heterogeneous multi‐center data, in which the data information is fully used to estimate parameters of interest. Efficient Monte Carlo Markov chain algorithms are specifically developed depending on the nature of the response. Simulation studies demonstrate the outperformance of the proposed Bayesian approach over other existing methods. An application to the data set from the vitamin D deficiency study is presented. Possible extensions of the method regarding the absence of auxiliary variables, semiparametric models, as well as the type of censoring are also discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

HGLM versus conditional estimators for the analysis of clustered binary data

Clustered binary data arise frequently in medical research such as cross-over clinical trials and twin studies. For the analysis of such data either a random-effects model or a conditional likelihood approach can be used. In this paper, we compare numerically the random-effects model estimator and the conditional likelihood estimator and discuss their relative merits for the analysis of binary data.     

Measuring the individual benefit of a medical or behavioral treatment using generalized linear mixed‐effects models

We propose statistical definitions of the individual benefit of a medical or behavioral treatment and of the severity of a chronic illness. These definitions are used to develop a graphical method that can be used by statisticians and clinicians in the data analysis of clinical trials from the perspective of personalized medicine. The method focuses on assessing and comparing individual effects of treatments rather than average effects and can be used with continuous and discrete responses, including dichotomous and count responses. The method is based on new developments in generalized linear mixed‐effects models, which are introduced in this article. To illustrate, analyses of data from the Sequenced Treatment Alternatives to Relieve Depression clinical trial of sequences of treatments for depression and data from a clinical trial of respiratory treatments are presented. The estimation of individual benefits is also explained. Copyright © 2016 John Wiley & Sons, Ltd.     

Extensions to Bayesian generalized linear mixed effects models for household tuberculosis transmission

Household contact studies, a mainstay of tuberculosis transmission research, often assume that tuberculosis‐infected household contacts of an index case were infected within the household. However, strain genotyping has provided evidence against this assumption. Understanding the household versus community infection dynamic is essential for designing interventions. The misattribution of infection sources can also bias household transmission predictor estimates. We present a household‐community transmission model that estimates the probability of community infection, that is, the probability that a household contact of an index case was actually infected from a source outside the home and simultaneously estimates transmission predictors. We show through simulation that our method accurately predicts the probability of community infection in several scenarios and that not accounting for community‐acquired infection in household contact studies can bias risk factor estimates. Applying the model to data from Vitória, Brazil, produced household risk factor estimates similar to two other standard methods for age and sex. However, our model gave different estimates for sleeping proximity to index case and disease severity score. These results show that estimating both the probability of community infection and household transmission predictors is feasible and that standard tuberculosis transmission models likely underestimate the risk for two important transmission predictors. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimation of covariate effects in generalized linear mixed models with a misspecified distribution of random intercepts and slopes

Generalized linear mixed models with random intercepts and slopes provide useful analyses of clustered and longitudinal data and typically require the specification of the distribution of the random effects. Previous work for models with only random intercepts has shown that misspecifying the shape of this distribution may bias estimates of the intercept, but typically leads to little bias in estimates of covariate effects. Very few papers have examined the effects of misspecifying the joint distribution of random intercepts and slopes. However, simulation results in a recent paper suggest that misspecifying the shape of the random slope distribution can yield severely biased estimates of all model parameters. Using analytic results, simulation studies and fits to example data, this paper examines the bias in parameter estimates due to misspecification of the shape of the joint distribution of random intercepts and slopes. Consistent with results for models with only random intercepts, and contrary to the claims of severe bias in a recent paper, we show that misspecification of the joint distribution typically yields little bias in estimates of covariate effects and is restricted to covariates associated with the misspecified random effects distributions. We also show that misspecification of the distribution of random effects has little effect on confidence interval performance. Coverage rates based on the model‐based standard errors from fitted likelihoods were generally quite close to nominal. Copyright © 2012 John Wiley & Sons, Ltd.     

Constrained S-estimators for linear mixed effects models with covariance components

Linear mixed effects (LME) models are increasingly used for analyses of biological and biomedical data. When the multivariate normal assumption is not adequate for an LME model, then a robust estimation approach is preferable to the maximum likelihood one. M-estimators were considered before for robust estimation of the LME models, and recently a constrained S-estimator was proposed. This S-estimator cannot be applied directly to LME models with correlated error terms and vector random effects with correlated dimensions. Therefore, a modification is proposed, which extends application of the constrained S-estimator to the LME models for multivariate responses with correlated dimensions and to longitudinal data. Also, a new computational algorithm is developed for computing constrained S-estimators. Performance of the S-estimators based on the original Tukey's biweight and translated biweight is evaluated in a small simulation study with repeated multivariate responses with correlated dimensions. The proposed methodology is applied to jointly analyze repeated measures on three cholesterol components, HDL, LDL, and triglycerides.     

Conditional Monte Carlo randomization tests for regression models

We discuss the computation of randomization tests for clinical trials of two treatments when the primary outcome is based on a regression model. We begin by revisiting the seminal paper of Gail, Tan, and Piantadosi (1988), and then describe a method based on Monte Carlo generation of randomization sequences. The tests based on this Monte Carlo procedure are design based, in that they incorporate the particular randomization procedure used. We discuss permuted block designs, complete randomization, and biased coin designs. We also use a new technique by Plamadeala and Rosenberger (2012) for simple computation of conditional randomization tests. Like Gail, Tan, and Piantadosi, we focus on residuals from generalized linear models and martingale residuals from survival models. Such techniques do not apply to longitudinal data analysis, and we introduce a method for computation of randomization tests based on the predicted rate of change from a generalized linear mixed model when outcomes are longitudinal. We show, by simulation, that these randomization tests preserve the size and power well under model misspecification. Copyright © 2014 John Wiley & Sons, Ltd.     

有序多分类重复测量资料的广义线性混合效应模型分析

目的 探讨广义线性混合效应模型在有序多分类重复测量资料分析中的应用及SAS9.1的GLIMMIX和NLMIXED过程实现.方法 为了评价某新药治疗糖尿病神经病变的临床疗效,采用以安慰剂为对照的随机双盲临床试验.在各个随访时间记录各受试者的神经病变主觉症状总分,并根据减分率评定疗效.建立广义线性混合效应模型,并分别用线性化法和数值法积分近似法进行参数估计,利用SAS中的GLIMMIX和NLMIXED过程得以实现.结果 2种参数估计方法 结果 很接近.疗效的组间差别有统计学意义（P〈0.000 1）,试验组疗效优于安慰剂组;各个疗程间的疗效差别有统计学意义（P〈0.000 1）,且疗程越大疗效越好; 治疗前神经病变主觉症状总分对疗效有影响（P=0.061 3,接近显著性水平）,其值越高,越容易治愈,提示病情严重的患者相比病情轻微的患者治愈效果更好.另外用数值法积分近似法还给出了随机截距和随机斜率的统计显著性检验.结论 采用广义线性混合效应模型对有序多分类重复测量临床资料进行统计分析,可以更客观的进行药物疗效评价.