Random effects are often used in generalized linear models to explain the serial dependence for longitudinal categorical data. Marginalized random effects models (MREMs) for the analysis of longitudinal binary data have been proposed to permit likelihood-based estimation of marginal regression parameters. In this paper, we introduce an extension of the MREM to accommodate longitudinal ordinal data. Maximum marginal likelihood estimation is implemented utilizing quasi-Newton algorithms with Monte Carlo integration of the random effects. Our approach is applied to analyze the quality of life data from a recent colorectal cancer clinical trial. Dropout occurs at a high rate and is often due to tumor progression or death. To deal with progression/death, we use a mixture model for the joint distribution of longitudinal measures and progression/death times and principal stratification to draw causal inferences about survivors.
marginalized likelihood-based models; ordinal data models; dropout
Longitudinal studies with binary repeated measures are widespread in biomedical research. Marginal regression approaches for balanced binary data are well developed, while for binary process data, where measurement times are irregular and may differ by individuals, likelihood-based methods for marginal regression analysis are less well developed. In this article, we develop a Bayesian regression model for analyzing longitudinal binary process data, with emphasis on dealing with missingness. We focus on the settings where data are missing at random, which require a correctly specified joint distribution for the repeated measures in order to draw valid likelihood-based inference about the marginal mean. To provide maximum flexibility, the proposed model specifies both the marginal mean and serial dependence structures using nonparametric smooth functions. Serial dependence is allowed to depend on the time lag between adjacent outcomes as well as other relevant covariates. Inference is fully Bayesian. Using simulations, we show that adequate modeling of the serial dependence structure is necessary for valid inference of the marginal mean when the binary process data are missing at random. Longitudinal viral load data from the HIV Epidemiology Research Study (HERS) are analyzed for illustration.
Repeated measures; Marginal model; Nonparametric regression; Penalized splines; HIV/AIDS; Antiviral treatment
Existing joint models for longitudinal and survival data are not applicable for longitudinal ordinal outcomes with possible non-ignorable missing values caused by multiple reasons. We propose a joint model for longitudinal ordinal measurements and competing risks failure time data, in which a partial proportional odds model for the longitudinal ordinal outcome is linked to the event times by latent random variables. At the survival endpoint, our model adopts the competing risks framework to model multiple failure types at the same time. The partial proportional odds model, as an extension of the popular proportional odds model for ordinal outcomes, is more flexible and at the same time provides a tool to test the proportional odds assumption. We use a likelihood approach and derive an EM algorithm to obtain the maximum likelihood estimates of the parameters. We further show that all the parameters at the survival endpoint are identifiable from the data. Our joint model enables one to make inference for both the longitudinal ordinal outcome and the failure times simultaneously. In addition, the inference at the longitudinal endpoint is adjusted for possible non-ignorable missing data caused by the failure times. We apply the method to the NINDS rt-PA stroke trial. Our study considers the modified Rankin Scale only. Other ordinal outcomes in the trial, such as the Barthel and Glasgow scales can be treated in the same way.
In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel (Prentice et al., 1978, Biometrics 34, 541-554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.
Cause-specific hazard; Competing risks; EM algorithm; Joint modeling; Longitudinal data; Mixed effects model
In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women.
Bayesian model averaging; Incomplete data; Latent variable; Marginal model; Random effects
Longitudinal ordinal data are common in many scientific studies, including those of multiple sclerosis (MS), and are frequently modeled using Markov dependency. Several authors have proposed random-effects Markov models to account for heterogeneity in the population. In this paper, we go one step further and study prediction based on random-effects Markov models. In particular, we show how to calculate the probabilities of future events and confidence intervals for those probabilities, given observed data on the ordinal outcome and a set of covariates, and how to update them over time. We discuss the usefulness of depicting these probabilities for visualization and interpretation of model results and illustrate our method using data from a phase III clinical trial that evaluated the utility of interferon beta-1a (trademark Avonex) to MS patients of type relapsing–remitting.
Markov model; Ordinal response; Prediction; Transition model
Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outcomes, where observations are observed clustered within subjects. Random effects are included in the model to account for the correlation of the clustered observations. Typically, the error variance and the variance of the random effects are considered to be homogeneous. These variance terms characterize the within-subjects (i.e., error variance) and between-subjects (i.e., random-effects variance) variation in the data. In this article, we describe how covariates can influence these variances, and also extend the standard logistic mixed model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their responses. Additionally, we allow the random effects to be correlated. We illustrate application of these models for ordinal data using Ecological Momentary Assessment (EMA) data, or intensive longitudinal data, from an adolescent smoking study. These mixed-effects ordinal location scale models have useful applications in mental health research where outcomes are often ordinal and there is interest in subject heterogeneity, both between- and within-subjects.
Complex variation; Mood variation; Heterogeneity; Variance modeling
Within the pattern-mixture modeling framework for informative dropout, conditional linear models (CLMs) are a useful approach to deal with dropout that can occur at any point in continuous time (not just at observation times). However, in contrast with selection models, inferences about marginal covariate effects in CLMs are not readily available if nonidentity links are used in the mean structures. In this article, we propose a CLM for long series of longitudinal binary data with marginal covariate effects directly specified. The association between the binary responses and the dropout time is taken into account by modeling the conditional mean of the binary response as well as the dependence between the binary responses given the dropout time. Specifically, parameters in both the conditional mean and dependence models are assumed to be linear or quadratic functions of the dropout time; and the continuous dropout time distribution is left completely unspecified. Inference is fully Bayesian. We illustrate the proposed model using data from a longitudinal study of depression in HIV-infected women, where the strategy of sensitivity analysis based on the extrapolation method is also demonstrated.
Bayesian analysis; HIV/AIDS; Marginal model; Missing data; Sensitivity analysis
Two models for the analysis of longitudinal binary data are discussed: the marginal model and the random intercepts model. In contrast to the linear mixed model (LMM), the two models for binary data are not subsumed under a single hierarchical model. The marginal model provides group-level information whereas the random intercepts model provides individual-level information including information about heterogeneity of growth. It is shown how a type of numerical averaging can be used with the random intercepts model to obtain group-level information, thus approximating individual and marginal aspects of the LMM. The types of inferences associated with each model are illustrated with longitudinal criminal offending data based on N = 506 males followed over a 22-year period. Violent offending indexed by official records and self-report were analyzed, with the marginal model estimated using generalized estimating equations and the random intercepts model estimated using maximum likelihood. The results show that the numerical averaging based on the random intercepts can produce prediction curves almost identical to those obtained directly from the marginal model parameter estimates. The results provide a basis for contrasting the models and the estimation procedures and key features are discussed to aid in selecting a method for empirical analysis.
In this review, we explore recent developments in the area of linear and nonlinear generalized mixed-effects regression models and various alternatives, including generalized estimating equations for analysis of longitudinal data. Methods are described for continuous and normally distributed as well as categorical (binary, ordinal, nominal) and count (Poisson) variables. Extensions of the model to three and four levels of clustering, multivariate outcomes, and incorporation of design weights are also described. Linear and nonlinear models are illustrated using an example involving a study of the relationship between mood and smoking.
mixed-effects models; logistic regression; Poisson regression; marginal maximum likelihood; generalized estimating equations; multilevel models
Most epidemiological studies of major depression report period prevalence estimates. These are of limited utility in characterizing the longitudinal epidemiology of this condition. Markov models provide a methodological framework for increasing the utility of epidemiological data. Markov models relating incidence and recovery to major depression prevalence have been described in a series of prior papers. In this paper, the models are extended to describe the longitudinal course of the disorder.
Data from three national surveys conducted by the Canadian national statistical agency (Statistics Canada) were used in this analysis. These data were integrated using a Markov model. Incidence, recurrence and recovery were represented as weekly transition probabilities. Model parameters were calibrated to the survey estimates.
The population was divided into three categories: low, moderate and high recurrence groups. The size of each category was approximated using lifetime data from a study using the WHO Mental Health Composite International Diagnostic Interview (WMH-CIDI). Consistent with previous work, transition probabilities reflecting recovery were high in the initial weeks of the episodes, and declined by a fixed proportion with each passing week.
Markov models provide a framework for integrating psychiatric epidemiological data. Previous studies have illustrated the utility of Markov models for decomposing prevalence into its various determinants: incidence, recovery and mortality. This study extends the Markov approach by distinguishing several recurrence categories.
Depressive Disorder; Epidemiologic Methods; Markov Chain
Previous research has compared methods of estimation for multilevel models fit to binary data but there are reasons to believe that the results will not always generalize to the ordinal case. This paper thus evaluates (a) whether and when fitting multilevel linear models to ordinal outcome data is justified and (b) which estimator to employ when instead fitting multilevel cumulative logit models to ordinal data, Maximum Likelihood (ML) or Penalized Quasi-Likelihood (PQL). ML and PQL are compared across variations in sample size, magnitude of variance components, number of outcome categories, and distribution shape. Fitting a multilevel linear model to ordinal outcomes is shown to be inferior in virtually all circumstances. PQL performance improves markedly with the number of ordinal categories, regardless of distribution shape. In contrast to binary data, PQL often performs as well as ML when used with ordinal data. Further, the performance of PQL is typically superior to ML when the data includes a small to moderate number of clusters (i.e., ≤ 50 clusters).
Multilevel Models; Random Effects; Ordinal; Categorical; Cumulative Logit Model; Proportional Odds Model
This article addresses modeling and inference for ordinal outcomes nested within categorical responses. We propose a mixture of normal distributions for latent variables associated with the ordinal data. This mixture model allows us to fix without loss of generality the cutpoint parameters that link the latent variable with the observed ordinal outcome. Moreover, the mixture model is shown to be more flexible in estimating cell probabilities when compared to the traditional Bayesian ordinal probit regression model with random cutpoint parameters. We extend our model to take into account possible dependence among the outcomes in different categories. We apply the model to a randomized phase III study to compare treatments on the basis of toxicities recorded by type of toxicity and grade within type. The data include the different (categorical) toxicity types exhibited in each patient. Each type of toxicity has an (ordinal) grade associated to it. The dependence among the different types of toxicity exhibited by the same patient is modeled by introducing patient-specific random effects.
Adverse events; Clinical trial; Gibbs sampling; Latent variable; Ordinal data nested within categories; Probit model
Analysis of longitudinal ordered categorical efficacy or safety data in clinical trials using mixed models is increasingly performed. However, algorithms available for maximum likelihood estimation using an approximation of the likelihood integral, including LAPLACE approach, may give rise to biased parameter estimates. The SAEM algorithm is an efficient and powerful tool in the analysis of continuous/count mixed models. The aim of this study was to implement and investigate the performance of the SAEM algorithm for longitudinal categorical data. The SAEM algorithm is extended for parameter estimation in ordered categorical mixed models together with an estimation of the Fisher information matrix and the likelihood. We used Monte Carlo simulations using previously published scenarios evaluated with NONMEM. Accuracy and precision in parameter estimation and standard error estimates were assessed in terms of relative bias and root mean square error. This algorithm was illustrated on the simultaneous analysis of pharmacokinetic and discretized efficacy data obtained after a single dose of warfarin in healthy volunteers. The new SAEM algorithm is implemented in MONOLIX 3.1 for discrete mixed models. The analyses show that for parameter estimation, the relative bias is low for both fixed effects and variance components in all models studied. Estimated and empirical standard errors are similar. The warfarin example illustrates how simple and rapid it is to analyze simultaneously continuous and discrete data with MONOLIX 3.1. The SAEM algorithm is extended for analysis of longitudinal categorical data. It provides accurate estimates parameters and standard errors. The estimation is fast and stable.
categorical data; mixed models; MONOLIX; proportional odds model; SAEM
Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis (2003) proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper, we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall’s τ. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.
Correlated binary data; multivariate normal distribution; probability integral transformation
The generalized linear mixed-effects model (GLMM) is a popular paradigm to extend models for cross-sectional data to a longitudinal setting. When applied to modeling binary responses, different software packages and even different procedures within a package may give quite different results. In this report, we describe the statistical approaches that underlie these different procedures and discuss their strengths and weaknesses when applied to fit correlated binary responses. We then illustrate these considerations by applying these procedures implemented in some popular software packages to simulated and real study data. Our simulation results indicate a lack of reliability for most of the procedures considered, which carries significant implications for applying such popular software packages in practice.
Integral Approximation; Linearization; GLIMMIX; lme4; NLMIXED; R; SAS; ZELIG
We present a Bayesian variable selection method for the setting in which the number of independent variables or predictors in a particular dataset is much larger than the available sample size. While most existing methods allow some degree of correlations among predictors but do not consider these correlations for variable selection, our method accounts for correlations among the predictors in variable selection. Our correlation-based stochastic search (CBS) method, the hybrid-CBS algorithm, extends a popular search algorithm for high-dimensional data, the stochastic search variable selection (SSVS) method. Similar to SSVS, we search the space of all possible models using variable addition, deletion or swap moves. However, our moves through the model space are designed to accommodate correlations among the variables. We describe our approach for continuous, binary, ordinal, and count outcome data. The impact of choices of prior distributions and hyper-parameters is assessed in simulation studies. We also examined performance of variable selection and prediction as the correlation structure of the predictors varies. We found that the hybrid-CBS resulted in lower prediction errors and better identified the true outcome associated predictors than SSVS when predictors were moderately to highly correlated. We illustrate the method on data from a proteomic profiling study of melanoma, a skin cancer.
Correlated predictors; correlation-based search; proteomic data
Latinos in the United States have been identified as a high-risk group for depression, anxiety, and substance abuse. HIV/AIDS has disproportionately impacted Latinos. Review findings suggest that HIV-risk behaviors among persons with severe mental illness (SMI) are influenced by a multitude of factors including psychiatric illness, cognitive-behavioral factors, substance use, childhood abuse, and social relationships.
To examine the impact of psychiatric and social correlates of HIV sexual risk behavior in Puerto Rican women with SMI.
Data collected longitudinally (from 2002 to 2005) in semi-structured interviews and from non-continuous participant observation was analyzed using a cross-sectional design. Bivariate associations between predictor variables and sexual risk behaviors were examined using binary and ordinal logistic regression. Linear regression was used to examine the association between significant predictor variables and the total number of risk behaviors the women engaged in during the 6 months prior to baseline.
Just over one-third (35.9%) of the study population (N = 53) was diagnosed with bipolar disorder and GAF scores ranged from 30 to 80 with a median score of 60. Participants ranged in age from 18 to 50 years (M = 32.6 ± 8.7), three-fourths reported a history of either sexual or physical abuse or of both in childhood, and one-fourth had abused substances in their lifetimes. Bivariate analyses indicated that psychiatric and social factors were differentially associated with sexual risk behaviors. Multivariate linear regression models showed that suffering from increased severity of psychiatric symptoms and factors and living below the poverty line are predictive of engagement in a greater number of HIV sexual risk behaviors.
Puerto Rican women with SMI are at high risk for HIV infection and are in need of targeted sexual risk reduction interventions that simultaneously address substance abuse prevention and treatment, childhood abuse, and the indirect effects associated with SMI such as living in poverty. Mental health programs should address risk behavior among adults with SMI in the context of specific symptomatology and comorbidities.
HIV sexual risk behavior; Severe mental illness; Puerto Rican; Psychiatric history profile
Positive and negative affect data are often collected over time in psychiatric care settings, yet no generally accepted means are available to relate these data to useful diagnoses or treatments. Latent class analysis attempts data reduction by classifying subjects into one of K unobserved classes based on observed data. Latent class models have recently been extended to accommodate longitudinally observed data. We extend these approaches in a Bayesian framework to accommodate trajectories of both continuous and discrete data. We consider whether latent class models might be used to distinguish patients on the basis of trajectories of observed affect scores, reported events, and presence or absence of clinical depression.
Cardiovascular disease; Depression; DIC; General growth mixture modeling; Gibbs sampling; Label switching; Model choice
We developed a generalized linear model of QTL mapping for discrete traits in line crossing experiments. Parameter estimation was achieved using two different algorithms, a mixture model-based EM (expectation–maximization) algorithm and a GEE (generalized estimating equation) algorithm under a heterogeneous residual variance model. The methods were developed using ordinal data, binary data, binomial data and Poisson data as examples. Applications of the methods to simulated as well as real data are presented. The two different algorithms were compared in the data analyses. In most situations, the two algorithms were indistinguishable, but when large QTL are located in large marker intervals, the mixture model-based EM algorithm can fail to converge to the correct solutions. Both algorithms were coded in C++ and interfaced with SAS as a user-defined SAS procedure called PROC QTL.
We propose Bayesian parametric and semiparametric partially linear regression methods to analyze the outcome-dependent follow-up data when the random time of a follow-up measurement of an individual depends on the history of both observed longitudinal outcomes and previous measurement times. We begin with the investigation of the simplifying assumptions of Lipsitz, Fitzmaurice, Ibrahim, Gelber, and Lipshultz, and present a new model for analyzing such data by allowing subject-specific correlations for the longitudinal response and by introducing a subject-specific latent variable to accommodate the association between the longitudinal measurements and the follow-up times. An extensive simulation study shows that our Bayesian partially linear regression method facilitates accurate estimation of the true regression line and the regression parameters. We illustrate our new methodology using data from a longitudinal observational study.
Bayesian cubic smoothing spline; Latent variable; Partially linear model
Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow-up. Our model consists of a linear mixed effects sub-model for the longitudinal outcome and a proportional cause-specific hazards frailty sub-model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub-model, we adopt a t-distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study.
Cause-specific hazard; EM algorithm; Joint modeling; Longitudinal data; Non-ignorable missing data; Robust inference
Two methods for point and interval estimation of relative risk for log-linear exposure-response relations in meta-analyses of published ordinal categorical exposure-response data have been proposed. The authors compared the results of a meta-analysis of published data using each of the 2 methods with the results that would be obtained if the primary data were available and investigated the circumstances under which the approximations required for valid use of each meta-analytic method break down. They then extended the methods to handle nonlinear exposure-response relations. In the present article, methods are illustrated using studies of the relation between alcohol consumption and colorectal and lung cancer risks from the ongoing Pooling Project of Prospective Studies of Diet and Cancer. In these examples, the differences between the results of a meta-analysis of summarized published data and the pooled analysis of the individual original data were small. However, incorrectly assuming no correlation between relative risk estimates for exposure categories from the same study gave biased confidence intervals for the trend and biased P values for the tests for nonlinearity and between-study heterogeneity when there was strong confounding by other model covariates. The authors illustrate the use of 2 publicly available user-friendly programs (Stata and SAS) to implement meta-analysis for dose-response data.
cohort studies; data interpretation, statistical; dose-response relationship, drug; linear models; meta-analysis; meta-analysis as topic
In longitudinal clinical trials, when outcome variables at later time points are only defined for patients who survive to those times, the evaluation of the causal effect of treatment is complicated. In this paper, we describe an approach that can be used to obtain the causal effect of three treatment arms with ordinal outcomes in the presence of death using a principal stratification approach. We introduce a set of flexible assumptions to identify the causal effect and implement a sensitivity analysis for non-identifiable assumptions which we parameterize parsimoniously. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial.
Principal stratification; QOL; Ordinal data; Sensitivity analysis
Joint models for the association of a longitudinal binary and a longitudinal continuous process are proposed for situations in which their association is of direct interest. The models are parameterized such that the dependence between the two processes is characterized by unconstrained regression coefficients. Bayesian variable selection techniques are used to parsimoniously model these coefficients. A Markov chain Monte Carlo (MCMC) sampling algorithm is developed for sampling from the posterior distribution, using data augmentation steps to handle missing data. Several technical issues are addressed to implement the MCMC algorithm efficiently. The models are motivated by, and are used for, the analysis of a smoking cessation clinical trial in which an important question of interest was the effect of the (exercise) treatment on the relationship between smoking cessation and weight gain.
Calibrated posterior predictive p-value; Data augmentation; Dependence; Joint models; Markov chain Monte Carlo; Parameter expansion; Stochastic search variable selection