Marginalized transition models for longitudinal binary data with ignorable and non-ignorable drop-out

A Bayesian transition model for missing longitudinal binary outcomes and an application to a smoking cessation study

Smoking cessation intervention studies often produce data on smoking status at discrete follow-up assessments, often with missing data in different amounts at each assessment. Smoking status in these studies is a dynamic process with individuals transitioning from smoking to abstinent, as well as abstinent to smoking, at different times during the intervention. Directly assessing transitions provides an opportunity to answer important questions like 'Does the proposed intervention help smokers remain abstinent or quit smoking more effectively than other interventions?' In this article, we model changes in smoking status and examine how interventions and other covariates affect the transitions. We propose a Bayesian approach for fitting the transition model to the observed data and impute missing outcomes based on a logistic model, which accounts for both missing at random (MAR) and missing not at random (MNAR) mechanisms. The proposed Bayesian approach treats missing data as additional unknown quantities and samples them from their posterior distributions. The performance of the proposed method is investigated through simulation studies and illustrated by data from a randomized controlled trial of smoking cessation interventions. Finally, posterior predictive checking and log pseudo marginal likelihood (LPML) are used to assess model assumptions and perform model comparisons, respectively.

A Latent Transition Analysis Model for Latent-State-Dependent Nonignorable Missingness

Psychologists often use latent transition analysis (LTA) to investigate state-to-state change in discrete latent constructs involving delinquent or risky behaviors. In this setting, latent-state-dependent nonignorable missingness is a potential concern. For some longitudinal models (e.g., growth models), a large literature has addressed extensions to accommodate nonignorable missingness. In contrast, little research has addressed how to extend the LTA to accommodate nonignorable missingness. Here we present a shared parameter LTA that can reduce bias due to latent-state-dependent nonignorable missingness: a parallel-process missing-not-at-random (MNAR-PP) LTA. The MNAR-PP LTA allows outcome process parameters to be interpreted as in the conventional LTA, which facilitates sensitivity analyses assessing changes in estimates between LTA and MNAR-PP LTA. In a sensitivity analysis for our empirical example, previous and current membership in high-delinquency states predicted adolescents' membership in missingness states that had high nonresponse probabilities for some or all items. A conventional LTA overestimated the proportion of adolescents ending up in a low-delinquency state, compared to an MNAR-PP LTA.

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 Bayesian Shrinkage Model for Incomplete Longitudinal Binary Data with Application to the Breast Cancer Prevention Trial

We consider inference in randomized longitudinal studies with missing data that is generated by skipped clinic visits and loss to follow-up. In this setting, it is well known that full data estimands are not identified unless unverified assumptions are imposed. We assume a non-future dependence model for the drop-out mechanism and partial ignorability for the intermittent missingness. We posit an exponential tilt model that links non-identifiable distributions and distributions identified under partial ignorability. This exponential tilt model is indexed by non-identified parameters, which are assumed to have an informative prior distribution, elicited from subject-matter experts. Under this model, full data estimands are shown to be expressed as functionals of the distribution of the observed data. To avoid the curse of dimensionality, we model the distribution of the observed data using a Bayesian shrinkage model. In a simulation study, we compare our approach to a fully parametric and a fully saturated model for the distribution of the observed data. Our methodology is motivated by, and applied to, data from the Breast Cancer Prevention Trial.

A Semiparametric Marginalized Model for Longitudinal Data with Informative Dropout

We propose a marginalized joint-modeling approach for marginal inference on the association between longitudinal responses and covariates when longitudinal measurements are subject to informative dropouts. The proposed model is motivated by the idea of linking longitudinal responses and dropout times by latent variables while focusing on marginal inferences. We develop a simple inference procedure based on a series of estimating equations, and the resulting estimators are consistent and asymptotically normal with a sandwich-type covariance matrix ready to be estimated by the usual plug-in rule. The performance of our approach is evaluated through simulations and illustrated with a renal disease data application.

A marginalized conditional linear model for longitudinal binary data when informative dropout occurs in continuous time

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.

Estimation of average treatment effect with incompletely observed longitudinal data: application to a smoking cessation study

We study the problem of estimation and inference on the average treatment effect in a smoking cessation trial where an outcome and some auxiliary information were measured longitudinally, and both were subject to missing values. Dynamic generalized linear mixed effects models linking the outcome, the auxiliary information, and the covariates are proposed. The maximum likelihood approach is applied to the estimation and inference on the model parameters. The average treatment effect is estimated by the G-computation approach, and the sensitivity of the treatment effect estimate to the nonignorable missing data mechanisms is investigated through the local sensitivity analysis approach. The proposed approach can handle missing data that form arbitrary missing patterns over time. We applied the proposed method to the analysis of the smoking cessation trial.

Longitudinal Data with Follow-up Truncated by Death: Match the Analysis Method to Research Aims

Diverse analysis approaches have been proposed to distinguish data missing due to death from nonresponse, and to summarize trajectories of longitudinal data truncated by death. We demonstrate how these analysis approaches arise from factorizations of the distribution of longitudinal data and survival information. Models are illustrated using cognitive functioning data for older adults. For unconditional models, deaths do not occur, deaths are independent of the longitudinal response, or the unconditional longitudinal response is averaged over the survival distribution. Unconditional models, such as random effects models fit to unbalanced data, may implicitly impute data beyond the time of death. Fully conditional models stratify the longitudinal response trajectory by time of death. Fully conditional models are effective for describing individual trajectories, in terms of either aging (age, or years from baseline) or dying (years from death). Causal models (principal stratification) as currently applied are fully conditional models, since group differences at one timepoint are described for a cohort that will survive past a later timepoint. Partly conditional models summarize the longitudinal response in the dynamic cohort of survivors. Partly conditional models are serial cross-sectional snapshots of the response, reflecting the average response in survivors at a given timepoint rather than individual trajectories. Joint models of survival and longitudinal response describe the evolving health status of the entire cohort. Researchers using longitudinal data should consider which method of accommodating deaths is consistent with research aims, and use analysis methods accordingly.

Bayesian semiparametric regression for longitudinal binary processes with missing data

Longitudinal studies with binary repeated measures are widespread in biomedical research. Marginal regression approaches for balanced binary data are well developed, whereas 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 (MAR), 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 MAR. Longitudinal viral load data from the HIV Epidemiology Research Study are analyzed for illustration.

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.

Accounting for missing data in end-of-life research

End-of-life studies are likely to have missing data because sicker persons are less likely to provide information and because measurements cannot be made after death. Ignoring missing data may result in data that are too favorable, because the sickest persons are effectively dropped from the analysis. In a comparison of two groups, the group with the most deaths and missing data will tend to have the most favorable data, which is not desirable. Results based on only the available data may not be generalizable to the original study population. If most of the missing data are absent because of death, methods that account for the deaths may remove much of the bias. Imputation methods can then be used for the data that are missing for other reasons. An example is presented from a randomized trial involving frail veterans. In that dataset, only two thirds of the subjects had complete data, but 60% of the "missing" data were missing because of death. The available data alone suggested that health improved significantly over time. However, after accounting for the deaths, there was a significant decline in health over time, as had been expected. Imputation of the remaining missing data did not change the results very much. With and without the imputed data, there was never a significant difference between the treatment and control groups, but in two nonrandomized comparisons the method of handling the missing data made a substantive difference. These sensitivity analyses suggest that the main results were not sensitive to the death and missing data, but that some secondary analyses were sensitive to these problems. Similar approaches should be considered in other end-of-life studies.