Mixtures of varying coefficient models for longitudinal data with discrete or continuous nonignorable dropout

The analysis of longitudinal repeated measures data is frequently complicated by missing data due to informative dropout. We describe a mixture model for joint distribution for longitudinal repeated measures, where the dropout distribution may be continuous and the dependence between response and dropout is semiparametric. Specifically, we assume that responses follow a varying coefficient random effects model conditional on dropout time, where the regression coefficients depend on dropout time through unspecified nonparametric functions that are estimated using step functions when dropout time is discrete (e.g., for panel data) and using smoothing splines when dropout time is continuous. Inference under the proposed semiparametric model is hence more robust than the parametric conditional linear model. The unconditional distribution of the repeated measures is a mixture over the dropout distribution. We show that estimation in the semiparametric varying coefficient mixture model can proceed by fitting a parametric mixed effects model and can be carried out on standard software platforms such as SAS. The model is used to analyze data from a recent AIDS clinical trial and its performance is evaluated using simulations.     

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.     

Missing covariates in longitudinal data with informative dropouts: bias analysis and inference

We consider estimation in generalized linear mixed models (GLMM) for longitudinal data with informative dropouts. At the time a unit drops out, time-varying covariates are often unobserved in addition to the missing outcome. However, existing informative dropout models typically require covariates to be completely observed. This assumption is not realistic in the presence of time-varying covariates. In this article, we first study the asymptotic bias that would result from applying existing methods, where missing time-varying covariates are handled using naive approaches, which include: (1) using only baseline values; (2) carrying forward the last observation; and (3) assuming the missing data are ignorable. Our asymptotic bias analysis shows that these naive approaches yield inconsistent estimators of model parameters. We next propose a selection/transition model that allows covariates to be missing in addition to the outcome variable at the time of dropout. The EM algorithm is used for inference in the proposed model. Data from a longitudinal study of human immunodeficiency virus (HIV)-infected women are used to illustrate the methodology.     

Bayesian informative dropout model for longitudinal binary data with random effects using conditional and joint modeling approaches

Dropouts are common in longitudinal study. If the dropout probability depends on the missing observations at or after dropout, this type of dropout is called informative (or nonignorable) dropout (ID). Failure to accommodate such dropout mechanism into the model will bias the parameter estimates. We propose a conditional autoregressive model for longitudinal binary data with an ID model such that the probabilities of positive outcomes as well as the drop‐out indicator in each occasion are logit linear in some covariates and outcomes. This model adopting a marginal model for outcomes and a conditional model for dropouts is called a selection model. To allow for the heterogeneity and clustering effects, the outcome model is extended to incorporate mixture and random effects. Lastly, the model is further extended to a novel model that models the outcome and dropout jointly such that their dependency is formulated through an odds ratio function. Parameters are estimated by a Bayesian approach implemented using the user‐friendly Bayesian software WinBUGS. A methadone clinic dataset is analyzed to illustrate the proposed models. Result shows that the treatment time effect is still significant but weaker after allowing for an ID process in the data. Finally the effect of drop‐out on parameter estimates is evaluated through simulation studies.     

Sensitivity analysis for nonrandom dropout: a local influence approach

Diggle and Kenward (1994, Applied Statistics 43, 49-93) proposed a selection model for continuous longitudinal data subject to nonrandom dropout. It has provoked a large debate about the role for such models. The original enthusiasm was followed by skepticism about the strong but untestable assumptions on which this type of model invariably rests. Since then, the view has emerged that these models should ideally be made part of a sensitivity analysis. This paper presents a formal and flexible approach to such a sensitivity assessment based on local influence (Cook, 1986, Journal of the Royal Statistical Society, Series B 48, 133-169). The influence of perturbing a missing-at-random dropout model in the direction of nonrandom dropout is explored. The method is applied to data from a randomized experiment on the inhibition of testosterone production in rats.     

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.     

On Estimating the Relationship between Longitudinal Measurements and Time‐to‐Event Data Using a Simple Two‐Stage Procedure

Summary Ye, Lin, and Taylor (2008, Biometrics 64 , 1238–1246) proposed a joint model for longitudinal measurements and time‐to‐event data in which the longitudinal measurements are modeled with a semiparametric mixed model to allow for the complex patterns in longitudinal biomarker data. They proposed a two‐stage regression calibration approach that is simpler to implement than a joint modeling approach. In the first stage of their approach, the mixed model is fit without regard to the time‐to‐event data. In the second stage, the posterior expectation of an individual's random effects from the mixed‐model are included as covariates in a Cox model. Although Ye et al. (2008) acknowledged that their regression calibration approach may cause a bias due to the problem of informative dropout and measurement error, they argued that the bias is small relative to alternative methods. In this article, we show that this bias may be substantial. We show how to alleviate much of this bias with an alternative regression calibration approach that can be applied for both discrete and continuous time‐to‐event data. Through simulations, the proposed approach is shown to have substantially less bias than the regression calibration approach proposed by Ye et al. (2008) . In agreement with the methodology proposed by Ye et al. (2008) , an advantage of our proposed approach over joint modeling is that it can be implemented with standard statistical software and does not require complex estimation techniques.     

A Nonparametric Test for Random Dropouts

The problem of dropout is a common one in longitudinal studies. One usually assumes for the analysis that dropout is at random. There are some tests to investigate this assumption. But these tests depend on normally distributed data or lack power, cf. Listing and Schlittgen (1998). We here propose an overall test which combines several Wilcoxon rank sum tests. The alternative hypothesis states that there is a tendency for larger (smaller) values of the target variable the last time the probands show up. The test is applicable with many ties also. It proves to perform well, compared to the test developed for normally distributed data, as well as to a test for completely missing at random which is proposed by Little (1988). An application to real data is given too.     

An exploration of fixed and random effects selection for longitudinal binary outcomes in the presence of nonignorable dropout

We explore a Bayesian approach to selection of variables that represent fixed and random effects in modeling of longitudinal binary outcomes with missing data caused by dropouts. We show via analytic results for a simple example that nonignorable missing data lead to biased parameter estimates. This bias results in selection of wrong effects asymptotically, which we can confirm via simulations for more complex settings. By jointly modeling the longitudinal binary data with the dropout process that possibly leads to nonignorable missing data, we are able to correct the bias in estimation and selection. Mixture priors with a point mass at zero are used to facilitate variable selection. We illustrate the proposed approach using a clinical trial for acute ischemic stroke.     

Modeling longitudinal data with nonignorable dropouts using a latent dropout class model

In longitudinal studies with dropout, pattern-mixture models form an attractive modeling framework to account for nonignorable missing data. However, pattern-mixture models assume that the components of the mixture distribution are entirely determined by the dropout times. That is, two subjects with the same dropout time have the same distribution for their response with probability one. As that is unlikely to be the case, this assumption made lead to classification error. In addition, if there are certain dropout patterns with very few subjects, which often occurs when the number of observation times is relatively large, pattern-specific parameters may be weakly identified or require identifying restrictions. We propose an alternative approach, which is a latent-class model. The dropout time is assumed to be related to the unobserved (latent) class membership, where the number of classes is less than the number of observed patterns; a regression model for the response is specified conditional on the latent variable. This is a type of shared-parameter model, where the shared "parameter" is discrete. Parameter estimates are obtained using the method of maximum likelihood. Averaging the estimates of the conditional parameters over the distribution of the latent variable yields estimates of the marginal regression parameters. The methodology is illustrated using longitudinal data on depression from a study of HIV in women.     

Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths

For observational longitudinal studies of geriatric populations, outcomes such as disability or cognitive functioning are often censored by death. Statistical analysis of such data may explicitly condition on either vital status or survival time when summarizing the longitudinal response. For example a pattern-mixture model characterizes the mean response at time t conditional on death at time S = s (for s > t), and thus uses future status as a predictor for the time t response. As an alternative, we define regression conditioning on being alive as a regression model that conditions on survival status, rather than a specific survival time. Such models may be referred to as partly conditional since the mean at time t is specified conditional on being alive (S > t), rather than using finer stratification (S = s for s > t). We show that naive use of standard likelihood-based longitudinal methods and generalized estimating equations with non-independence weights may lead to biased estimation of the partly conditional mean model. We develop a taxonomy for accommodation of both dropout and death, and describe estimation for binary longitudinal data that applies selection weights to estimating equations with independence working correlation. Simulation studies and an analysis of monthly disability status illustrate potential bias in regression methods that do not explicitly condition on survival.     

The Milk Protein Trial: Influence Analysis of the Dropout Process

Diggle and Kenward (1994) proposed a selection model for continuous longitudinal data subject to possible non‐random dropout. Their method in general, and the milk protein example in particular, has provoked a large debate about the role of such models. The original enthusiasm was followed by skepticism about the strong but untestable assumption upon which this type of models invariably rests. Concern was raised about the very nature of incompleteness which is arguably more due to design reasons (the experiment was stopped due to insufficient feed supply), than to genuine dropout. In the meantime, the view has emerged that these models should ideally be made part of a sensitivity analysis. This paper presents a formal and flexible approach to such a sensitivity assessment, based on both global influence (Chatterjee and Hadi , 1988) as well as local influence (Cook , 1986). It will be argued that local influence is more apt to zoom in on a particular source of influence, such as the assumed non‐response mechanism. The method is applied to a set of data on milk protein contents in dairy cattle. The same data were used in the original paper by Diggle and Kenward (1994), who concluded that the dropout process was non‐random.     

Pattern‐Mixture Zero‐Inflated Mixed Models for Longitudinal Unbalanced Count Data with Excessive Zeros

Analysis of longitudinal data with excessive zeros has gained increasing attention in recent years; however, current approaches to the analysis of longitudinal data with excessive zeros have primarily focused on balanced data. Dropouts are common in longitudinal studies; therefore, the analysis of the resulting unbalanced data is complicated by the missing mechanism. Our study is motivated by the analysis of longitudinal skin cancer count data presented by Greenberg, Baron, Stukel, Stevens, Mandel, Spencer, Elias, Lowe, Nierenberg, Bayrd, Vance, Freeman, Clendenning, Kwan, and the Skin Cancer Prevention Study Group[New England Journal of Medicine 323 , 789–795]. The data consist of a large number of zero responses (83% of the observations) as well as a substantial amount of dropout (about 52% of the observations). To account for both excessive zeros and dropout patterns, we propose a pattern‐mixture zero‐inflated model with compound Poisson random effects for the unbalanced longitudinal skin cancer data. We also incorporate an autoregressive of order 1 correlation structure in the model to capture longitudinal correlation of the count responses. A quasi‐likelihood approach has been developed in the estimation of our model. We illustrated the method with analysis of the longitudinal skin cancer data.     

Model selection in the weighted generalized estimating equations for longitudinal data with dropout

We propose criteria for variable selection in the mean model and for the selection of a working correlation structure in longitudinal data with dropout missingness using weighted generalized estimating equations. The proposed criteria are based on a weighted quasi‐likelihood function and a penalty term. Our simulation results show that the proposed criteria frequently select the correct model in candidate mean models. The proposed criteria also have good performance in selecting the working correlation structure for binary and normal outcomes. We illustrate our approaches using two empirical examples. In the first example, we use data from a randomized double‐blind study to test the cancer‐preventing effects of beta carotene. In the second example, we use longitudinal CD4 count data from a randomized double‐blind study.     

Modeling tumor growth with random onset

The longitudinal assessment of tumor volume is commonly used as an endpoint in small animal studies in cancer research. Groups of genetically identical mice are injected with mutant cells from clones developed with different mutations. The interest is on comparing tumor onset (i.e., the time of tumor detection) and tumor growth after onset, between mutation groups. This article proposes a class of linear and nonlinear growth models for jointly modeling tumor onset and growth in this situation. Our approach allows for interval-censored time of onset and missing-at-random dropout due to early sacrifice, which are common situations in animal research. We show that our approach has good small-sample properties for testing and is robust to some key unverifiable modeling assumptions. We illustrate this methodology with an application examining the effect of different mutations on tumorigenesis.     

Bayesian variable selection and estimation in semiparametric joint models of multivariate longitudinal and survival data

This paper presents a novel semiparametric joint model for multivariate longitudinal and survival data (SJMLS) by relaxing the normality assumption of the longitudinal outcomes, leaving the baseline hazard functions unspecified and allowing the history of the longitudinal response having an effect on the risk of dropout. Using Bayesian penalized splines to approximate the unspecified baseline hazard function and combining the Gibbs sampler and the Metropolis–Hastings algorithm, we propose a Bayesian Lasso (BLasso) method to simultaneously estimate unknown parameters and select important covariates in SJMLS. Simulation studies are conducted to investigate the finite sample performance of the proposed techniques. An example from the International Breast Cancer Study Group (IBCSG) is used to illustrate the proposed methodologies.     

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.     

Bayesian semiparametric zero-inflated Poisson model for longitudinal count data

This paper presents new methods, using a Bayesian approach, for analyzing longitudinal count data with excess zeros and nonlinear effects of continuously valued covariates. In longitudinal count data there are many problems that can make the use of a zero-inflated Poisson (ZIP) model ineffective. These problems are unobserved heterogeneity and nonlinear effects of continuously valued covariates. Our proposed semiparametric model can simultaneously handle these problems in a unified framework. The framework accounts for heterogeneity by incorporating random effects and has two components. The parametric component of the model which deals with the linear effects of time invariant covariates and the non-parametric component which gives an arbitrary smooth function to model the effect of time or time-varying covariates on the logarithm of mean count. The proposed methods are illustrated by analyzing longitudinal count data on the assessment of an efficacy of pesticides in controlling the reproduction of whitefly.     

Bayesian Quantile Regression for Longitudinal Studies with Nonignorable Missing Data

Summary .  We study quantile regression (QR) for longitudinal measurements with nonignorable intermittent missing data and dropout. Compared to conventional mean regression, quantile regression can characterize the entire conditional distribution of the outcome variable, and is more robust to outliers and misspecification of the error distribution. We account for the within-subject correlation by introducing a   ℓ₂   penalty in the usual QR check function to shrink the subject-specific intercepts and slopes toward the common population values. The informative missing data are assumed to be related to the longitudinal outcome process through the shared latent random effects. We assess the performance of the proposed method using simulation studies, and illustrate it with data from a pediatric AIDS clinical trial.     

Mixed-effect hybrid models for longitudinal data with nonignorable dropout

Summary .  Selection models and pattern-mixture models are often used to deal with nonignorable dropout in longitudinal studies. These two classes of models are based on different factorizations of the joint distribution of the outcome process and the dropout process. We consider a new class of models, called mixed-effect hybrid models (MEHMs), where the joint distribution of the outcome process and dropout process is factorized into the marginal distribution of random effects, the dropout process conditional on random effects, and the outcome process conditional on dropout patterns and random effects. MEHMs combine features of selection models and pattern-mixture models: they directly model the missingness process as in selection models, and enjoy the computational simplicity of pattern-mixture models. The MEHM provides a generalization of shared-parameter models (SPMs) by relaxing the conditional independence assumption between the measurement process and the dropout process given random effects. Because SPMs are nested within MEHMs, likelihood ratio tests can be constructed to evaluate the conditional independence assumption of SPMs. We use data from a pediatric AIDS clinical trial to illustrate the models.