A latent autoregressive model for longitudinal binary data subject to informative missingness

Longitudinal clinical trials often collect long sequences of binary data. Our application is a recent clinical trial in opiate addicts that examined the effect of a new treatment on repeated binary urine tests to assess opiate use over an extended follow-up. The dataset had two sources of missingness: dropout and intermittent missing observations. The primary endpoint of the study was comparing the marginal probability of a positive urine test over follow-up across treatment arms. We present a latent autoregressive model for longitudinal binary data subject to informative missingness. In this model, a Gaussian autoregressive process is shared between the binary response and missing-data processes, thereby inducing informative missingness. Our approach extends the work of others who have developed models that link the various processes through a shared random effect but do not allow for autocorrelation. We discuss parameter estimation using Monte Carlo EM and demonstrate through simulations that incorporating within-subject autocorrelation through a latent autoregressive process can be very important when longitudinal binary data is subject to informative missingness. We illustrate our new methodology using the opiate clinical trial data.     

Marginal analysis of incomplete longitudinal binary data: a cautionary note on LOCF imputation

In recent years there has been considerable research devoted to the development of methods for the analysis of incomplete data in longitudinal studies. Despite these advances, the methods used in practice have changed relatively little, particularly in the reporting of pharmaceutical trials. In this setting, perhaps the most widely adopted strategy for dealing with incomplete longitudinal data is imputation by the "last observation carried forward" (LOCF) approach, in which values for missing responses are imputed using observations from the most recently completed assessment. We examine the asymptotic and empirical bias, the empirical type I error rate, and the empirical coverage probability associated with estimators and tests of treatment effect based on the LOCF imputation strategy. We consider a setting involving longitudinal binary data with longitudinal analyses based on generalized estimating equations, and an analysis based simply on the response at the end of the scheduled follow-up. We find that for both of these approaches, imputation by LOCF can lead to substantial biases in estimators of treatment effects, the type I error rates of associated tests can be greatly inflated, and the coverage probability can be far from the nominal level. Alternative analyses based on all available data lead to estimators with comparatively small bias, and inverse probability weighted analyses yield consistent estimators subject to correct specification of the missing data process. We illustrate the differences between various methods of dealing with drop-outs using data from a study of smoking behavior.     

Latent pattern mixture models for informative intermittent missing data in longitudinal studies

A frequently encountered problem in longitudinal studies is data that are missing due to missed visits or dropouts. In the statistical literature, interest has primarily focused on monotone missing data (dropout) with much less work on intermittent missing data in which a subject may return after one or more missed visits. Intermittent missing data have broader applicability that can include the frequent situation in which subjects do not have common sets of visit times or they visit at nonprescheduled times. In this article, we propose a latent pattern mixture model (LPMM), where the mixture patterns are formed from latent classes that link the longitudinal response and the missingness process. This allows us to handle arbitrary patterns of missing data embodied by subjects' visit process, and avoids the need to specify the mixture patterns a priori. One assumption of our model is that the missingness process is assumed to be conditionally independent of the longitudinal outcomes given the latent classes. We propose a noniterative approach to assess this key assumption. The LPMM is illustrated with a data set from a health service research study in which homeless people with mental illness were randomized to three different service packages and measures of homelessness were recorded at multiple time points. Our model suggests the presence of four latent classes linking subject visit patterns to homeless outcomes.     

A statistical model for under- or overdispersed clustered and longitudinal count data

We propose a likelihood-based model for correlated count data that display under- or overdispersion within units (e.g. subjects). The model is capable of handling correlation due to clustering and/or serial correlation, in the presence of unbalanced, missing or unequally spaced data. A family of distributions based on birth-event processes is used to model within-subject underdispersion. A computational approach is given to overcome a parameterization difficulty with this family, and this allows use of common Markov Chain Monte Carlo software (e.g. WinBUGS) for estimation. Application of the model to daily counts of asthma inhaler use by children shows substantial within-subject underdispersion, between-subject heterogeneity and correlation due to both clustering of measurements within subjects and serial correlation of longitudinal measurements. The model provides a major improvement over Poisson longitudinal models, and diagnostics show that the model fits well.     

Marginalized transition shared random effects models for longitudinal binary data with nonignorable dropout

In longitudinal studies investigators frequently have to assess and address potential biases introduced by missing data. New methods are proposed for modeling longitudinal categorical data with nonignorable dropout using marginalized transition models and shared random effects models. Random effects are introduced for both serial dependence of outcomes and nonignorable missingness. Fisher‐scoring and Quasi–Newton algorithms are developed for parameter estimation. Methods are illustrated with a real dataset.     

A mixture model for longitudinal data with application to assessment of noncompliance

In clinical trials of a self-administered drug, repeated measures of a laboratory marker, which is affected by study medication and collected in all treatment arms, can provide valuable information on population and individual summaries of compliance. In this paper, we introduce a general finite mixture of nonlinear hierarchical models that allows estimates of component membership probabilities and random effect distributions for longitudinal data arising from multiple subpopulations, such as from noncomplying and complying subgroups in clinical trials. We outline a sampling strategy for fitting these models, which consists of a sequence of Gibbs, Metropolis-Hastings, and reversible jump steps, where the latter is required for switching between component models of different dimensions. Our model is applied to identify noncomplying subjects in the placebo arm of a clinical trial assessing the effectiveness of zidovudine (AZT) in the treatment of patients with HIV, where noncompliance was defined as initiation of AZT during the trial without the investigators' knowledge. We fit a hierarchical nonlinear change-point model for increases in the marker MCV (mean corpuscular volume of erythrocytes) for subjects who noncomply and a constant mean random effects model for those who comply. As part of our fully Bayesian analysis, we assess the sensitivity of conclusions to prior and modeling assumptions and demonstrate how external information and covariates can be incorporated to distinguish subgroups.     

Joint modeling of progression of HIV resistance mutations measured with uncertainty and failure time data

Development of HIV resistance mutations is a major cause for failure of antiretroviral treatment. This article proposes a method for jointly modeling the processes of viral genetic changes and treatment failure. Because the viral genome is measured with uncertainty, a hidden Markov model is used to fit the viral genetic process. The uncertain viral genotype is included as a time-dependent covariate in a Cox model for failure time, and an expectation-maximization algorithm is used to estimate the model parameters. This model allows simultaneous evaluation of the sequencing uncertainty and the effect of resistance mutation on the risk of virological and immunological failures. Various model checking tests are provided to assess the appropriateness of the model. Simulation studies are performed to investigate the finite-sample properties of the proposed methods, which are then applied to data collected in three phase II clinical trials testing antiretroviral treatments containing the drug efavirenz.     

An assessment of non-randomized medical treatment of long-term schizophrenia relapse using bivariate binary-response transition models

The analyses of observational longitudinal studies involving concurrent changes in treatment and medical conditions present difficulties because of the multitude of directions of potential relationships: past medication influences current symptoms; past symptoms influence current medication; and current medication is associated with current symptoms. In the context of a long-term study of non-randomized pharmacological treatment of schizophrenic relapse, we present an analysis of bivariate discrete-time transitional data with binary responses in an attempt to understand the transitional and concurrent relationships between schizophrenia relapse and medication use. A naive analysis does not show any association between previous medication and current relapse. However, we provide evidence suggesting that current treatment may impact current relapse for those who have previously taken medication, but not for those who haven't taken medication in the past. When univariate models are specified to assess these associations, the bivariate nature of the problem requires a choice of which response, relapse or medication, should be the dependent variable. In this case, the choice of relapse or medication as a dependent variable does matter. Hence, our results derive from models where both relapse and medication are treated as dependent variables. Specifically, we specify a bivariate log odds ratio for current relapse and current medication use and a separate univariate logit component for each of these outcomes. Each of these components contains transitional associations with previous relapse and medication. Such models represent extensions of univariate transitional association models (e.g. Diggle et al. (1994)) and correspond to bivariate transitional models (e.g. Zeger and Liang (1991)). We incorporate changes in transitional associations into the full-data parametric model for final inference, and investigate if these temporal changes are due to learning effects or the impact of drop-out. We also perform residual analyses and sensitivity analyses in the context of missing data patterns.     

Bayesian latent multi‐state modeling for nonequidistant longitudinal electronic health records

Large amounts of longitudinal health records are now available for dynamic monitoring of the underlying processes governing the observations. However, the health status progression across time is not typically observed directly: records are observed only when a subject interacts with the system, yielding irregular and often sparse observations. This suggests that the observed trajectories should be modeled via a latent continuous‐time process potentially as a function of time‐varying covariates. We develop a continuous‐time hidden Markov model to analyze longitudinal data accounting for irregular visits and different types of observations. By employing a specific missing data likelihood formulation, we can construct an efficient computational algorithm. We focus on Bayesian inference for the model: this is facilitated by an expectation‐maximization algorithm and Markov chain Monte Carlo methods. Simulation studies demonstrate that these approaches can be implemented efficiently for large data sets in a fully Bayesian setting. We apply this model to a real cohort where patients suffer from chronic obstructive pulmonary disease with the outcome being the number of drugs taken, using health care utilization indicators and patient characteristics as covariates.     

Spatial multistate transitional models for longitudinal event data

Summary .   Follow-up medical studies often collect longitudinal data on patients. Multistate transitional models are useful for analysis in such studies where at any point in time, individuals may be said to occupy one of a discrete set of states and interest centers on the transition process between states. For example, states may refer to the number of recurrences of an event, or the stage of a disease. We develop a hierarchical modeling framework for the analysis of such longitudinal data when the processes corresponding to different subjects may be correlated spatially over a region. Continuous-time Markov chains incorporating spatially correlated random effects are introduced. Here, joint modeling of both spatial dependence as well as dependence between different transition rates is required and a multivariate spatial approach is employed. A proportional intensities frailty model is developed where baseline intensity functions are modeled using parametric Weibull forms, piecewise-exponential formulations, and flexible representations based on cubic B-splines. The methodology is developed within the context of a study examining invasive cardiac procedures in Quebec. We consider patients admitted for acute coronary syndrome throughout the 139 local health units of the province and examine readmission and mortality rates over a 4-year period.     

A mixed-effects regression model for longitudinal multivariate ordinal data

A mixed-effects item response theory model that allows for three-level multivariate ordinal outcomes and accommodates multiple random subject effects is proposed for analysis of multivariate ordinal outcomes in longitudinal studies. This model allows for the estimation of different item factor loadings (item discrimination parameters) for the multiple outcomes. The covariates in the model do not have to follow the proportional odds assumption and can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is proposed utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher scoring solution, which provides standard errors for all model parameters, is used. An analysis of a longitudinal substance use data set, where four items of substance use behavior (cigarette use, alcohol use, marijuana use, and getting drunk or high) are repeatedly measured over time, is used to illustrate application of the proposed model.     

A bivariate pseudolikelihood for incomplete longitudinal binary data with nonignorable nonmonotone missingness

For analyzing longitudinal binary data with nonignorable and nonmonotone missing responses, a full likelihood method is complicated algebraically, and often requires intensive computation, especially when there are many follow-up times. As an alternative, a pseudolikelihood approach has been proposed in the literature under minimal parametric assumptions. This formulation only requires specification of the marginal distributions of the responses and missing data mechanism, and uses an independence working assumption. However, this estimator can be inefficient for estimating both time-varying and time-stationary effects under moderate to strong within-subject associations among repeated responses. In this article, we propose an alternative estimator, based on a bivariate pseudolikelihood, and demonstrate in simulations that the proposed method can be much more efficient than the previous pseudolikelihood obtained under the assumption of independence. We illustrate the method using longitudinal data on CD4 counts from two clinical trials of HIV-infected patients.     

Joint model selection of marginal mean regression and correlation structure for longitudinal data with missing outcome and covariates

This work develops a joint model selection criterion for simultaneously selecting the marginal mean regression and the correlation/covariance structure in longitudinal data analysis where both the outcome and the covariate variables may be subject to general intermittent patterns of missingness under the missing at random mechanism. The new proposal, termed “joint longitudinal information criterion” (JLIC), is based on the expected quadratic error for assessing model adequacy, and the second‐order weighted generalized estimating equation (WGEE) estimation for mean and covariance models. Simulation results reveal that JLIC outperforms existing methods performing model selection for the mean regression and the correlation structure in a two stage and hence separate manner. We apply the proposal to a longitudinal study to identify factors associated with life satisfaction in the elderly of Taiwan.     

Analyzing complex capture-recapture data in the presence of individual and temporal covariates and model uncertainty

SUMMARY: We consider the issue of analyzing complex ecological data in the presence of covariate information and model uncertainty. Several issues can arise when analyzing such data, not least the need to take into account where there are missing covariate values. This is most acutely observed in the presence of time-varying covariates. We consider mark-recapture-recovery data, where the corresponding recapture probabilities are less than unity, so that individuals are not always observed at each capture event. This often leads to a large amount of missing time-varying individual covariate information, because the covariate cannot usually be recorded if an individual is not observed. In addition, we address the problem of model selection over these covariates with missing data. We consider a Bayesian approach, where we are able to deal with large amounts of missing data, by essentially treating the missing values as auxiliary variables. This approach also allows a quantitative comparison of different models via posterior model probabilities, obtained via the reversible jump Markov chain Monte Carlo algorithm. To demonstrate this approach we analyze data relating to Soay sheep, which pose several statistical challenges in fully describing the intricacies of the system.     

Modeling longitudinal biomarker data from multiple assays that have different known detection limits

Summary .   Assays to measure biomarkers are commonly subject to large amounts of measurement error and known detection limits. Studies with longitudinal biomarker measurements may use multiple assays in assessing outcome. I propose an approach for jointly modeling repeated measures of multiple assays when these assays are subject to measurement error and known lower detection limits. A commonly used approach is to perform an initial assay with a larger lower detection limit on all repeated samples, followed by only performing a second more expensive assay with a lower minimum level of detection when the initial assay value is below its lower limit of detection. I show how simply replacing the initial assay measurement with the second assay measurement may be a biased approach and investigate the performance of the proposed joint model in this situation. Additionally, I compare the performance of the joint model with an approach that only uses the initial assay measurements in analysis. Further, I consider alternative designs to only performing the second assay when the initial assay measurement is below its lower detection limit. Specifically, I show that one only needs to perform the second assay on a fraction of assays that are above the lower detection limit on the first assay to substantially increase the efficiency. Further, I show the efficiency advantages of performing the second assay at random without regard to the initial assay measurement over a design in which the second assay is only performed when the initial assay is below its lower limit of detection. The methodology is illustrated with a recent study examining the use of a vaccine in treating macaques with simian immunodeficiency virus.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

Continuous time Markov models for binary longitudinal data

Longitudinal data usually consist of a number of short time series. A group of subjects or groups of subjects are followed over time and observations are often taken at unequally spaced time points, and may be at different times for different subjects. When the errors and random effects are Gaussian, the likelihood of these unbalanced linear mixed models can be directly calculated, and nonlinear optimization used to obtain maximum likelihood estimates of the fixed regression coefficients and parameters in the variance components. For binary longitudinal data, a two state, non-homogeneous continuous time Markov process approach is used to model serial correlation within subjects. Formulating the model as a continuous time Markov process allows the observations to be equally or unequally spaced. Fixed and time varying covariates can be included in the model, and the continuous time model allows the estimation of the odds ratio for an exposure variable based on the steady state distribution. Exact likelihoods can be calculated. The initial probability distribution on the first observation on each subject is estimated using logistic regression that can involve covariates, and this estimation is embedded in the overall estimation. These models are applied to an intervention study designed to reduce children's sun exposure.     

A generalized transition model for grouped longitudinal categorical data

Transition models are an important framework that can be used to model longitudinal categorical data. They are particularly useful when the primary interest is in prediction. The available methods for this class of models are suitable for the cases in which responses are recorded individually over time. However, in many areas, it is common for categorical data to be recorded as groups, that is, different categories with a number of individuals in each. As motivation we consider a study in insect movement and another in pig behaviou. The first study was developed to understand the movement patterns of female adults of Diaphorina citri, a pest of citrus plantations. The second study investigated how hogs behaved under the influence of environmental enrichment. In both studies, the number of individuals in different response categories was observed over time. We propose a new framework for considering the time dependence in the linear predictor of a generalized logit transition model using a quantitative response, corresponding to the number of individuals in each category. We use maximum likelihood estimation and present the results of the fitted models under stationarity and non-stationarity assumptions, and use recently proposed tests to assess non-stationarity. We evaluated the performance of the proposed model using simulation studies under different scenarios, and concluded that our modeling framework represents a flexible alternative to analyze grouped longitudinal categorical data.     

Model Selection for Generalized Estimating Equations Accommodating Dropout Missingness

Summary The generalized estimating equation (GEE) has been a popular tool for marginal regression analysis with longitudinal data, and its extension, the weighted GEE approach, can further accommodate data that are missing at random (MAR). Model selection methodologies for GEE, however, have not been systematically developed to allow for missing data. We propose the missing longitudinal information criterion (MLIC) for selection of the mean model, and the MLIC for correlation (MLICC) for selection of the correlation structure in GEE when the outcome data are subject to dropout/monotone missingness and are MAR. Our simulation results reveal that the MLIC and MLICC are effective for variable selection in the mean model and selecting the correlation structure, respectively. We also demonstrate the remarkable drawbacks of naively treating incomplete data as if they were complete and applying the existing GEE model selection method. The utility of proposed method is further illustrated by two real applications involving missing longitudinal outcome data.     

Meta-analysis of studies with missing data

Summary .  Consider a meta-analysis of studies with varying proportions of patient-level missing data, and assume that each primary study has made certain missing data adjustments so that the reported estimates of treatment effect size and variance are valid. These estimates of treatment effects can be combined across studies by standard meta-analytic methods, employing a random-effects model to account for heterogeneity across studies. However, we note that a meta-analysis based on the standard random-effects model will lead to biased estimates when the attrition rates of primary studies depend on the size of the underlying study-level treatment effect. Perhaps ignorable within each study, these types of missing data are in fact not ignorable in a meta-analysis. We propose three methods to correct the bias resulting from such missing data in a meta-analysis: reweighting the DerSimonian–Laird estimate by the completion rate; incorporating the completion rate into a Bayesian random-effects model; and inference based on a Bayesian shared-parameter model that includes the completion rate. We illustrate these methods through a meta-analysis of 16 published randomized trials that examined combined pharmacotherapy and psychological treatment for depression.