Informative Dropout Research Articles

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

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.

Joint modeling of multivariate longitudinal data and the dropout process in a competing risk setting: application to ICU data.

BackgroundJoint modeling of longitudinal and survival data has been increasingly considered in clinical trials, notably in cancer and AIDS. In critically ill patients admitted to an intensive care unit (ICU), such models also appear to be of interest in the investigation of the effect of treatment on severity scores due to the likely association between the longitudinal score and the dropout process, either caused by deaths or live discharges from the ICU. However, in this competing risk setting, only cause-specific hazard sub-models for the multiple failure types data have been used.MethodsWe propose a joint model that consists of a linear mixed effects submodel for the longitudinal outcome, and a proportional subdistribution hazards submodel for the competing risks survival data, linked together by latent random effects. We use Markov chain Monte Carlo technique of Gibbs sampling to estimate the joint posterior distribution of the unknown parameters of the model. The proposed method is studied and compared to joint model with cause-specific hazards submodel in simulations and applied to a data set that consisted of repeated measurements of severity score and time of discharge and death for 1,401 ICU patients.ResultsTime by treatment interaction was observed on the evolution of the mean SOFA score when ignoring potentially informative dropouts due to ICU deaths and live discharges from the ICU. In contrast, this was no longer significant when modeling the cause-specific hazards of informative dropouts. Such a time by treatment interaction persisted together with an evidence of treatment effect on the hazard of death when modeling dropout processes through the use of the Fine and Gray model for sub-distribution hazards.ConclusionsIn the joint modeling of competing risks with longitudinal response, differences in the handling of competing risk outcomes appear to translate into the estimated difference in treatment effect on the longitudinal outcome. Such a modeling strategy should be carefully defined prior to analysis.

Analyzing Longitudinal Data in the Presence of Informative Dropout: The Jmre1 Command

Many studies in various research areas have designs that involve repeated measurements over time of a continuous variable across a group of subjects. A frequent and serious problem in such studies is the occurrence of missing data. In many cases, missing data are caused by an event that leads to a premature termination of the series of repeated measurements on some subjects. When the probability of the occurrence of this event is related to the subject-specific underlying trend of the variable of interest, this missingness process is called informative censoring or informative dropout. Standard likelihood-based methods (for example, linear mixed models) fail to give consistent estimates. In such cases, one needs to apply methods that simultaneously model the observed data and the missingness process. In this article, we review a method proposed by Touloumi et al. (1999, Statistics in Medicine 18: 1215–1233) to adjust for informative dropout in longitudinal data analysis. We also present the jmre1 command, which can be used to fit the proposed model. The estimation method combines the restricted iterative generalized least-squares method with a nested expectation-maximization algorithm. The method is implemented mainly using Stata's matrix programming language, Mata. Our example is derived from the epidemiology of the HIV infection.

HIV Viral Dynamic Models With Censoring and Informative Dropouts

HIV viral dynamic models have received much interest in the literature in recent years. These models are useful for modeling the viral load trajectories during an anti-HIV treatment and for evaluating the efficacy of the treatment. In AIDS studies, patients may drop out of the study early due possibly to drug side-effects, and viral load measurements often have a lower limit of detection. Statistical analyses are therefore complicated by the censoring and dropouts in the data. We propose a joint likelihood method which addresses censoring and dropouts in a mixed effects model simultaneously. A real AIDS dataset is analyzed, and a simulation is conducted to evaluate the proposed method.

A joint model for incomplete data in crossover trials

Crossover designs are popular in early phases of clinical trials and in bioavailability and bioequivalence studies. Assessment of carryover effects, in addition to the treatment effects, is a critical issue in crossover trails. The observed data from a crossover trial can be incomplete because of potential dropouts. A joint model for analyzing incomplete data from crossover trials is proposed in this article; the model includes a measurement model and an outcome dependent informative model for the dropout process. The informative-dropout model is compared with the ignorable-dropout model as specific cases of the latter are nested subcases of the proposed joint model. Markov chain sampling methods are used for Bayesian analysis of this model. The joint model is used to analyze depression score data from a clinical trial in women with late luteal phase dysphoric disorder. Interestingly, carryover effect is found to have a strong effect in the informative dropout model, but it is less significant when dropout is considered ignorable.

The analysis of longitudinal quality of life measures with informative drop-out: a pattern mixture approach

PurposeThe analysis of longitudinal health-related quality of life measures (HRQOL) can be seriously hampered due to informative drop-out. Random effects models assume Missing At Random and do not take into account informative drop-out. We therefore aim to correct the bias due to informative drop-out.MethodsAnalyses of data from a trial comparing standard-dose and high-dose chemotherapy for patients with breast cancer with respect to long-term impact on HRQOL will serve as illustration. The subscale Physical Function (PF) of the SF36 will be used. A pattern mixture approach is proposed to account for informative drop-out. Patterns are defined based on events related to HRQOL, such as death and relapse. The results of this pattern mixture approach are compared to the results of the commonly used random effects model.ResultsThe findings of the pattern mixture approach are well interpretable, and different courses over time in different patterns are distinguished. In terms of estimated differences between standard dose and high dose, the results of both approaches are slightly different, but have no consequences for the clinical evaluation of both doses.ConclusionUnder the assumption that drop-out is at random within the patterns, the pattern mixture approach adjusts the estimates to a certain degree. This approach accounts in a relatively simple way for informative drop-out.

Mixtures of semiparametric varying coefficient models for longitudinal data with nonignorable dropout

Informative dropout often arise in longitudinal data. In this paper we propose a mixture model in which the responses follow a semiparametric varying coefficient random effects model and some of the regression coefficients depend on the dropout time in a non-parametric way. The local linear version of the profile-kernel method is used to estimate the parameters of the model. The proposed estimators are shown to be consistent and asymptotically normal, and the finite performance of the estimators is evaluated by numerical simulation.

A Review on Joint Models in Biometrical Research

Abstract In biometrical research joint modeling of longitudinal measures and event time data has become a very popular tool to account for informative dropouts, to incorporate longitudinal trends and to link disease processes that occur simultaneously. This article reviews the work in that area of recent fruitful research by classifying approaches on joint models in three categories: Approaches with focus on serial trends, approaches with focus on event time data and approaches with equal focus on both outcomes. Typically longitudinal measures and event time data are modeled jointly by introducing shared random effects or by considering conditional distributions together with marginal distributions. We present the approaches in a uniform nomenclature, comment on sub-models applied to longitudinal measures and event time data outcomes individually and exemplify applications in biometrical research. The increasing variety of joint model approaches are a promising framework to shed light on biometrical quest...

A Bias Correction in Testing Treatment Efficacy Under Informative Dropout in Clinical Trials

In clinical trials of drug development, patients are often followed for a certain period of time, and the outcome variables are measured at scheduled time intervals. The main interest of the trial is the treatment efficacy at a prespecified time point, which is often the last visit. In such trials, patient dropout is often the major source for missing data. With possible informative patient dropout, the missing information often causes biases in the inference of treatment efficacy. In this article, for a time-saturated treatment effect model and an informative dropout scheme that depends on the unobserved outcomes only through the random coefficients, we propose a grouping method to correct the biases in the estimation of treatment effect. The asymptotic variance estimator is also obtained for statistical inference. In a simulation study, we compare the new method with the traditional methods of the observed case (OC) analysis, the last observation carried forward (LOCF) analysis, and the mixed model repeated measurement (MMRM) approach, and find it improves the current methods and gives more stable results in the treatment efficacy inferences.

Varying-coefficient models for longitudinal processes with continuous-time informative dropout

Dropout is a common occurrence in longitudinal studies. Building upon the pattern-mixture modeling approach within the Bayesian paradigm, we propose a general framework of varying-coefficient models for longitudinal data with informative dropout, where measurement times can be irregular and dropout can occur at any point in continuous time (not just at observation times) together with administrative censoring. Specifically, we assume that the longitudinal outcome process depends on the dropout process through its model parameters. The unconditional distribution of the repeated measures is a mixture over the dropout (administrative censoring) time distribution, and the continuous dropout time distribution with administrative censoring is left completely unspecified. We use Markov chain Monte Carlo to sample from the posterior distribution of the repeated measures given the dropout (administrative censoring) times; Bayesian bootstrapping on the observed dropout (administrative censoring) times is carried out to obtain marginal covariate effects. We illustrate the proposed framework using data from a longitudinal study of depression in HIV-infected women; the strategy for sensitivity analysis on unverifiable assumption is also demonstrated.

Impact of daclizumab, low-dose cyclosporine, mycophenolate mofetil and steroids on renal function after kidney transplantation

Early and long-term use of cyclosporine A (CsA) leads to increased risks of renal toxicity. We hypothesized that administration of daclizumab in combination with mycophenolate mofetil (MMF) allows a relevant reduction in the dose of CsA. We carried out a 3-year, prospective, randomized, controlled clinical multi-centre trial in 156 patients. The patients were randomized to standard treatment (CsA, MMF, steroids) or to high-dose daclizumab (first dose: 2 mg/kg), in combination with low-dose CsA, MMF and steroids. We maintained the mean CsA levels of daclizumab patients at 57% of standard patients (132 versus 216 ng/ml) on Day 7 post-transplant, and 84% by 6 months. Primary outcome, creatinine clearance (with imputation of informative dropouts) at 12 months, was significantly better in daclizumab-treated (34 +/- 17) than standard patients (29 +/- 17; P = 0.028, two sided). Only 5 cases of BPAR were recorded in the daclizumab compared to 22 in the standard group (P = 0.0016). Daclizumab patients had 91% event-free survival after 1 year compared to 66% in standard patients (P = 0.00017). We demonstrate here that high-dose daclizumab in combination with lower CsA levels in adult renal transplant recipients is as or more effective than standard regimen (CsA, MMF, steroids) and may result in better outcomes at 12 months post-transplant with no increase in adverse reactions.

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 Two-Latent-Class Model for Smoking Cessation Data with Informative Dropouts

Non ignorable missing data is a common problem in longitudinal studies. Latent class models are attractive for simplifying the modeling of missing data when the data are subject to either a monotone or intermittent missing data pattern. In our study, we propose a new two-latent-class model for categorical data with informative dropouts, dividing the observed data into two latent classes; one class in which the outcomes are deterministic and a second one in which the outcomes can be modeled using logistic regression. In the model, the latent classes connect the longitudinal responses and the missingness process under the assumption of conditional independence. Parameters are estimated by the method of maximum likelihood estimation based on the above assumptions and the tetrachoric correlation between responses within the same subject. We compare the proposed method with the shared parameter model and the weighted GEE model using the areas under the ROC curves in the simulations and the application to the smoking cessation data set. The simulation results indicate that the proposed two-latent-class model performs well under different missing procedures. The application results show that our proposed method is better than the shared parameter model and the weighted GEE model.

Global Sensitivity Analysis for Randomized Trials with Informative Dropout: A Semiparametric Perspective

Global Sensitivity Analysis for Randomized Trials with Informative Dropout: A Semiparametric Perspective

Assessing Sexual Attitudes and Behaviors of Young Women: A Joint Model with Nonlinear Time Effects, Time Varying Covariates, and Dropouts

Understanding human sexual behaviors is essential for the effective prevention of sexually transmitted infections (STI). Analysis of longitudinally measured sexual behavioral data, however, is often complicated by zero-inflation of event counts, nonlinear time trend, time-varying covariates, and informative dropouts. Ignoring these complicating factors could undermine the validity of the study findings. In this article, we put forth a unified joint modeling structure that accommodates these features of the data. Specifically, we propose a pair of simultaneous models for the zero-inflated event counts: Each of these models contains an auto-regressive structure for the accommodation of the effect of recent event history, and a nonparametric component for the modeling of nonlinear time effect. Informative dropout and time varying covariates are modeled explicitly in the process. Model fitting and parameter estimation are carried out in a Bayesian paradigm by the use of a Markov chain Monte Carlo (MCMC) method. Analytical results showed that adolescent sexual behaviors tended to evolve nonlinearly over time, and they were strongly influenced by the day-to-day variations in mood and sexual interests. These findings suggest that adolescent sex is, to a large extent, driven by intrinsic factors rather than being compelled by circumstances, thus highlighting the need of education on self-protective measures against infection risks.

Modelling placebo response in depression trials using a longitudinal model with informative dropout

Dropouts are common events in longitudinal studies in depression. Ignoring missing information may lead to biased and inconsistent assessment of study results. A non-linear model was recently developed to describe the time-course of HAMD-17 clinical score in the placebo arms of antidepressant clinical trials. In this paper we complemented this model by introducing an informative dropout component to jointly estimate HAMD-17 time-course and dropout mechanism. The aims of this work were to: (a) characterise typical placebo response in depression trials in presence of dropouts, (b) explore which dropout mechanism better describe the time-varying probability of a subject to dropout from the trial, and (c) define a framework for the development of clinical trial simulation in depression. A meta-analytic approach was used on placebo data collected in 6 clinical trials including 695 subjects suffering from Major Depressive Disorders. Alternative hypotheses for “missingness” were evaluated using different hazard models. The “Missing Not At Random” performed statistically ( p < 0.01) better than “Missing At Random”, that in turn performed better ( p < 0.01) than “Missing Completely At Random” model. This finding provided new insights on the validity of the analyses currently used in many longitudinal clinical trials to support the registration of a new medicinal product.

Testing for the presence of multiple sources of informative dropout in longitudinal data

Longitudinal studies tracking the rate of change are subject to patient dropout. This dropout process might not only be informative but also heterogeneous in the sense that different causes might contribute to multiple patterns of informative dropout. We propose a random-effects approach to test for homogeneity of informative dropout that accommodates the realistic situation where reasons for dropout are not fully understood, or perhaps are even entirely unknown. The proposed score test is robust in that it does not depend on the underlying distribution of the informative dropout random effects. The test allows for an additional level of clustering among participating subjects, as might be found in a family study, provided the informative dropout random effects have a known correlation structure.

Bias in the last observation carried forward method under informative dropout

Subject dropout is an inevitable problem in longitudinal studies. It makes the analysis challenging when the main interest is the change in outcome from baseline to endpoint of study. The last observation carried forward (LOCF) method is a very common approach for handling this problem. It assumes that the last measured outcome is frozen in time after the point of dropout, an unrealistic assumption given any time trends. Though existence and direction of the bias can sometimes be anticipated, the more important statistical question involves the actual magnitude of the bias and this requires computation. This paper provides explicit expressions for the exact bias in the LOCF estimates of mean change and its variance when the longitudinal data follow a linear mixed-effects model with linear time trajectories. General dropout patterns are considered that may depend on treatment group, subject-specific trajectories and follow different time to dropout distributions. In our case studies, the magnitude of bias for mean change estimators linearly increases as time to dropout decreases. The bias depends heavily on the dropout interval. The variance term is always underestimated.

Pattern Mixture Models and Latent Class Models for the Analysis of Multivariate Longitudinal Data with Informative Dropouts

Missing data and especially dropouts frequently arise in longitudinal data. Maximum likelihood estimates are consistent when data are missing at random (MAR) but, as this assumption is not checkable, pattern mixture models (PMM) have been developed to deal with informative dropout. More recently, latent class models (LCM) have been proposed as a way to relax PMM assumptions. The aim of this paper is to compare PMM and LCM in order to tackle informative dropout in a longitudinal study of cognitive ageing measured by several psychometric tests. Using a multivariate longitudinal model with a latent process, a sensitivity analysis was performed to compare estimates under the MAR assumption, from a PMM and from two LCM. In the PMM, dropout patterns are included as covariates in the multivariate longitudinal model. In the simple LCM, they are predictors of the class membership probabilities while, in the joint LCM, the dropout time is jointly modeled using a proportional hazard model depending on latent classes. We show that parameter interpretation is different in the two kinds of models and thus can lead to different estimated values. PMM parameters are adjusted on the dropout patterns while LCM parameters are adjusted on the latent classes. This difference is highlighted in our data set because the latent classes exhibit much more heterogeneity than dropout patterns. We suggest several complementary analyses to investigate the characteristics of latent classes in order to understand the meaning of the parameters when using LCM to deal with informative dropout.

The use of Gaussian quadrature for estimation in frailty proportional hazards models

In this paper, we propose a novel Gaussian quadrature estimation method in various frailty proportional hazards models. We approximate the unspecified baseline hazard by a piecewise constant one, resulting in a parametric model that can be fitted conveniently by Gaussian quadrature tools in standard software such as SAS Proc NLMIXED. We first apply our method to simple frailty models for correlated survival data (e.g. recurrent or clustered failure times), then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death. Simulation studies show that our method compares favorably with the well-received penalized partial likelihood method and the Monte Carlo EM (MCEM) method, for both normal and Gamma frailty models. We apply our method to three real data examples: (1) the time to blindness of both eyes in a diabetic retinopathy study, (2) the joint analysis of recurrent opportunistic diseases in the presence of death for HIV-infected patients, and (3) the joint modeling of local, distant tumor recurrences and patients survival in a soft tissue sarcoma study. The proposed method greatly simplifies the implementation of the (joint) frailty models and makes them much more accessible to general statistical practitioners.