Analysis of non-ignorable missing and left-censored longitudinal data using a weighted random effects tobit model

In a longitudinal study with response data collected during a hospital stay, observations may be missing because of the subject's discharge from the hospital prior to completion of the study or the death of the subject, resulting in non-ignorable missing data. In addition to non-ignorable missingness, there is left-censoring in the response measurements because of the inherent limit of detection. For analyzing non-ignorable missing and left-censored longitudinal data, we have proposed to extend the theory of random effects tobit regression model to weighted random effects tobit regression model. The weights are computed on the basis of inverse probability weighted augmented methodology. An extensive simulation study was performed to compare the performance of the proposed model with a number of competitive models. The simulation study shows that the estimates are consistent and that the root mean square errors of the estimates are minimal for the use of augmented inverse probability weights in the random effects tobit model. The proposed method is also applied to the non-ignorable missing and left-censored interleukin-6 biomarker data obtained from the Genetic and Inflammatory Markers of Sepsis study.     

Joint mixed‐effects models for causal inference with longitudinal data

Causal inference with observational longitudinal data and time‐varying exposures is complicated due to the potential for time‐dependent confounding and unmeasured confounding. Most causal inference methods that handle time‐dependent confounding rely on either the assumption of no unmeasured confounders or the availability of an unconfounded variable that is associated with the exposure (eg, an instrumental variable). Furthermore, when data are incomplete, validity of many methods often depends on the assumption of missing at random. We propose an approach that combines a parametric joint mixed‐effects model for the study outcome and the exposure with g‐computation to identify and estimate causal effects in the presence of time‐dependent confounding and unmeasured confounding. G‐computation can estimate participant‐specific or population‐average causal effects using parameters of the joint model. The joint model is a type of shared parameter model where the outcome and exposure‐selection models share common random effect(s). We also extend the joint model to handle missing data and truncation by death when missingness is possibly not at random. We evaluate the performance of the proposed method using simulation studies and compare the method to both linear mixed‐ and fixed‐effects models combined with g‐computation as well as to targeted maximum likelihood estimation. We apply the method to an epidemiologic study of vitamin D and depressive symptoms in older adults and include code using SAS PROC NLMIXED software to enhance the accessibility of the method to applied researchers.     

Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt‐PA stroke trial

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. Copyright © 2009 John Wiley & Sons, Ltd.     

Smooth random change point models

Change point models are used to describe processes over time that show a change in direction. An example of such a process is cognitive ability, where a decline a few years before death is sometimes observed. A broken-stick model consists of two linear parts and a breakpoint where the two lines intersect. Alternatively, models can be formulated that imply a smooth change between the two linear parts. Change point models can be extended by adding random effects to account for variability between subjects. A new smooth change point model is introduced and examples are presented that show how change point models can be estimated using functions in R for mixed-effects models. The Bayesian inference using WinBUGS is also discussed. The methods are illustrated using data from a population-based longitudinal study of ageing, the Cambridge City over 75 Cohort Study. The aim is to identify how many years before death individuals experience a change in the rate of decline of their cognitive ability.     

Analysis of crossover designs with nonignorable dropout

This article addresses the analysis of crossover designs with nonignorable dropout. We study nonreplicated crossover designs and replicated designs separately. With a primary objective of comparing the treatment mean effects, we jointly model the longitudinal measures and discrete time to dropout. We propose shared‐parameter models and mixed‐effects selection models. We adapt a linear‐mixed effects model as the conditional model for the longitudinal outcomes. We invoke a discrete‐time hazards model with a complementary log‐log link function for the conditional distribution of time to dropout. We apply maximum likelihood for parameter estimation. We perform simulation studies to investigate the robustness of our proposed approaches under various missing data mechanisms. We then apply the approaches to two examples with a continuous outcome and one example with a binary outcome using existing software. We also implement the controlled multiple imputation methods as a sensitivity analysis of the missing data assumption.     

Power analyses for longitudinal study designs with missing data

Existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns. Although the pattern of missing data can be assessed during data analysis, it is unknown during the design phase of a study. The random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis. In this paper, we model the occurrence of missing data with a two-state, first-order Markov process and integrate the modelling information into the power function to account for random missing data patterns. The Markov model is easily specified to accommodate different anticipated missing data processes. We develop this approach for the two most popular longitudinal models: the generalized estimating equations (GEE) and the linear mixed-effects model under the missing completely at random (MCAR) assumption. For GEE, we also limit our consideration to the working independence correlation model. The proposed methodology is illustrated with numerous examples that are motivated by real study designs.     

A shared parameter location scale mixed effect model for EMA data subject to informative missing

In this paper, we address the problem of accounting for informative missing in the context of ecological momentary assessment studies (sometimes referred to as intensive longitudinal studies), where each study unit gets measured intensively over time and intermittent missing is usually present. We present a shared parameter modeling approach that links the primary longitudinal outcome with potentially informative missingness by a common set of random effects that summarize a subjects’ specific traits in terms of their mean (location) and variability (scale). The primary outcome, conditional on the random effects, are allowed to exhibit heterogeneity in terms of both the mean and within subject variance. Unlike previous methods which largely rely on numerical integration or approximation, we estimate the model by a full Bayesian approach using Markov Chain Monte Carlo. An adolescent mood study example is illustrated together with a series of simulation studies. Results in comparison to more conventional approaches suggest that accounting for the common but unobserved random subject mean and variance effects, shared between the primary outcome and missingness models, can significantly improve the model fit, and also provide the benefit of understanding how missingness can affect the inference for the primary outcome.     

Analysis of longitudinal studies with death and drop-out: a case study

The analysis of longitudinal data has recently been an active area of biostatistical research. Two main approaches to analysis have emerged, the first concentrating on modelling evolution of marginal distributions of the main response variable of interest and the other on subject-specific trajectories. In epidemiology the analysis is usually complicated by missing data and by death of study participants. Motivated by a study of cognitive decline in the elderly, this paper argues that these two types of incomplete follow-up may need to be treated differently in the analysis, and proposes an extension to the marginal modelling approach. The problem of informative drop-out is also discussed. The methods are implemented in the 'Stata' statistical package.     

A Bayesian approach to joint analysis of multivariate longitudinal data and parametric accelerated failure time

Impairment caused by Parkinson's disease (PD) is multidimensional (e.g., sensoria, functions, and cognition) and progressive. Its multidimensional nature precludes a single outcome to measure disease progression. Clinical trials of PD use multiple categorical and continuous longitudinal outcomes to assess the treatment effects on overall improvement. A terminal event such as death or dropout can stop the follow‐up process. Moreover, the time to the terminal event may be dependent on the multivariate longitudinal measurements. In this article, we consider a joint random‐effects model for the correlated outcomes. A multilevel item response theory model is used for the multivariate longitudinal outcomes and a parametric accelerated failure time model is used for the failure time because of the violation of proportional hazard assumption. These two models are linked via random effects. The Bayesian inference via MCMC is implemented in ‘BUGS ’ language. Our proposed method is evaluated by a simulation study and is applied to DATATOP study, a motivating clinical trial to determine if deprenyl slows the progression of PD. © 2013 The authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Joint two‐part Tobit models for longitudinal and time‐to‐event data

In this article, we show how Tobit models can address problems of identifying characteristics of subjects having left‐censored outcomes in the context of developing a method for jointly analyzing time‐to‐event and longitudinal data. There are some methods for handling these types of data separately, but they may not be appropriate when time to event is dependent on the longitudinal outcome, and a substantial portion of values are reported to be below the limits of detection. An alternative approach is to develop a joint model for the time‐to‐event outcome and a two‐part longitudinal outcome, linking them through random effects. This proposed approach is implemented to assess the association between the risk of decline of CD4/CD8 ratio and rates of change in viral load, along with discriminating between patients who are potentially progressors to AIDS from patients who do not. We develop a fully Bayesian approach for fitting joint two‐part Tobit models and illustrate the proposed methods on simulated and real data from an AIDS clinical study.     

Bayesian semiparametric joint modeling of longitudinal explanatory variables of mixed types and a binary outcome

Many prospective biomedical studies collect longitudinal clinical and lifestyle data that are both continuous and discrete. In some studies, there is interest in the association between a binary outcome and the values of these longitudinal measurements at a specific time point. A common problem in these studies is inconsistency in timing of measurements and missing follow-ups which can lead to few measurements at the time of interest. Some methods have been developed to address this problem, but are only applicable to continuous measurements. To address this limitation, we propose a new class of joint models for a binary outcome and longitudinal explanatory variables of mixed types. The longitudinal model uses a latent normal random variable construction with regression splines to model time-dependent trends in mean with a Dirichlet Process prior assigned to random effects to relax distribution assumptions. We also standardize timing of the explanatory variables by relating the binary outcome to imputed longitudinal values at a set time point. The proposed model is evaluated through simulation studies and applied to data from a cancer survivor study of participants in the Women's Health Initiative.     

Analysis of longitudinal Gaussian data with missing data on the response variable

BACKGROUND: Using an application and a simulation study we show the bias induced by missing data in the outcome in longitudinal studies and discuss suitable statistical methods according to the type of missing responses when the variable under study is gaussian. Method: The model used for the analysis of gaussian longitudinal data is the mixed effects linear model. When the probability of response does not depend on the missing values of the outcome and on the parameters of the linear model, missing data are ignorable, and parameters of the mixed effects linear model may be estimated by the maximum likelihood method with classical softwares. When the missing data are non ignorable, several methods have been proposed. We describe the method proposed by Diggle and Kenward (1994) (DK method) for which a software is available. This model consists in the combination of a linear mixed effects model for the outcome variable and a logistic model for the probability of response which depends on the outcome variable. RESULTS: A simulation study shows the efficacy of this method and its limits when the data are not normal. In this case, estimators obtained by the DK approach may be more biased than estimators obtained under the hypothesis of ignorable missing data even if the data are non ignorable. Data of the Paquid cohort about the evolution of the scores to a neuropsychological test among elderly subjects show the bias of a naive analysis using all available data. Although missing responses are not ignorable in this study, estimates of the linear mixed effects model are not very different using the DK approach and the hypothesis of ignorable missing data. CONCLUSION: Statistical methods for longitudinal data including non ignorable missing responses are sensitive to hypotheses difficult to verify. Thus, it will be better in practical applications to perform an analysis under the hypothesis of ignorable missing responses and compare the results obtained with several approaches for non ignorable missing data. However, such a strategy requires development of new softwares.     

Joint modeling of longitudinal outcomes and survival using latent growth modeling approach in a mesothelioma trial

Joint modeling of longitudinal and survival data can provide more efficient and less biased estimates of treatment effects through accounting for the associations between these two data types. Sponsors of oncology clinical trials routinely and increasingly include patient-reported outcome (PRO) instruments to evaluate the effect of treatment on symptoms, functioning, and quality of life. Known publications of these trials typically do not include jointly modeled analyses and results. We formulated several joint models based on a latent growth model for longitudinal PRO data and a Cox proportional hazards model for survival data. The longitudinal and survival components were linked through either a latent growth trajectory or shared random effects. We applied these models to data from a randomized phase III oncology clinical trial in mesothelioma. We compared the results derived under different model specifications and showed that the use of joint modeling may result in improved estimates of the overall treatment effect.     

Bayesian modeling of the covariance structure for irregular longitudinal data using the partial autocorrelation function

In long‐term follow‐up studies, irregular longitudinal data are observed when individuals are assessed repeatedly over time but at uncommon and irregularly spaced time points. Modeling the covariance structure for this type of data is challenging, as it requires specification of a covariance function that is positive definite. Moreover, in certain settings, careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure. Two common settings where this occurs are studies with ‘outcome‐dependent follow‐up’ and studies with ‘ignorable missing data’. ‘Outcome‐dependent follow‐up’ occurs when individuals with a history of poor health outcomes had more follow‐up measurements, and the intervals between the repeated measurements were shorter. When the follow‐up time process only depends on previous outcomes, likelihood‐based methods can still provide consistent estimates of the regression parameters, given that both the mean and covariance structures of the irregular longitudinal data are correctly specified and no model for the follow‐up time process is required. For ‘ignorable missing data’, the missing data mechanism does not need to be specified, but valid likelihood‐based inference requires correct specification of the covariance structure. In both cases, flexible modeling approaches for the covariance structure are essential. In this paper, we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function and the variance function. In particular, we propose semiparametric non‐stationary partial autocorrelation function models, which do not suffer from complex positive definiteness restrictions like the autocorrelation function. We describe a Bayesian approach, discuss computational issues, and apply the proposed methods to CD4 count data from a pediatric AIDS clinical trial. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

A varying-coefficient model for the evaluation of time-varying concomitant intervention effects in longitudinal studies

Concomitant interventions are often introduced during a longitudinal clinical trial to patients who respond undesirably to the pre-specified treatments. In addition to the main objective of evaluating the pre-specified treatment effects, an important secondary objective in such a trial is to evaluate whether a concomitant intervention could change a patient's response over time. Because the initiation of a concomitant intervention may depend on the patient's general trend of pre-intervention outcomes, regression approaches that treat the presence of the intervention as a time-dependent covariate may lead to biased estimates for the intervention effects. Borrowing the techniques of Follmann and Wu (Biometrics 1995; 51:151-168) for modeling informative missing data, we propose a varying-coefficient mixed-effects model to evaluate the patient's longitudinal outcome trends before and after the patient's starting time of the intervention. By allowing the random coefficients to be correlated with the patient's starting time of the intervention, our model leads to less biased estimates of the intervention effects. Nonparametric estimation and inferences of the coefficient curves and intervention effects are developed using B-splines. Our methods are demonstrated through a longitudinal clinical trial in depression and heart disease and a simulation study.     

Correction of bias from non‐random missing longitudinal data using auxiliary information

Missing data are common in longitudinal studies due to drop‐out, loss to follow‐up, and death. Likelihood‐based mixed effects models for longitudinal data give valid estimates when the data are missing at random (MAR). These assumptions, however, are not testable without further information. In some studies, there is additional information available in the form of an auxiliary variable known to be correlated with the missing outcome of interest. Availability of such auxiliary information provides us with an opportunity to test the MAR assumption. If the MAR assumption is violated, such information can be utilized to reduce or eliminate bias when the missing data process depends on the unobserved outcome through the auxiliary information. We compare two methods of utilizing the auxiliary information: joint modeling of the outcome of interest and the auxiliary variable, and multiple imputation (MI). Simulation studies are performed to examine the two methods. The likelihood‐based joint modeling approach is consistent and most efficient when correctly specified. However, mis‐specification of the joint distribution can lead to biased results. MI is slightly less efficient than a correct joint modeling approach and can also be biased when the imputation model is mis‐specified, though it is more robust to mis‐specification of the imputation distribution when all the variables affecting the missing data mechanism and the missing outcome are included in the imputation model. An example is presented from a dementia screening study. Copyright © 2009 John Wiley & Sons, Ltd.     

A selection model for longitudinal binary responses subject to non‐ignorable attrition

Longitudinal studies collect information on a sample of individuals which is followed over time to analyze the effects of individual and time‐dependent characteristics on the observed response. These studies often suffer from attrition: individuals drop out of the study before its completion time and thus present incomplete data records. When the missing mechanism, once conditioned on other (observed) variables, does not depend on current (eventually unobserved) values of the response variable, the dropout mechanism is known to be ignorable. We propose a selection model extending semiparametric variance component models for longitudinal binary responses to allow for dependence between the missing data mechanism and the primary response process. The model is applied to a data set from a methadone maintenance treatment programme held in Sidney, 1986. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiple imputation of cognitive performance as a repeatedly measured outcome

Longitudinal studies of cognitive performance are sensitive to dropout, as participants experiencing cognitive deficits are less likely to attend study visits, which may bias estimated associations between exposures of interest and cognitive decline. Multiple imputation is a powerful tool for handling missing data, however its use for missing cognitive outcome measures in longitudinal analyses remains limited. We use multiple imputation by chained equations (MICE) to impute cognitive performance scores of participants who did not attend the 2011–2013 exam of the Atherosclerosis Risk in Communities Study. We examined the validity of imputed scores using observed and simulated data under varying assumptions. We examined differences in the estimated association between diabetes at baseline and 20-year cognitive decline with and without imputed values. Lastly, we discuss how different analytic methods (mixed models and models fit using generalized estimate equations) and choice of for whom to impute result in different estimands. Validation using observed data showed MICE produced unbiased imputations. Simulations showed a substantial reduction in the bias of the 20-year association between diabetes and cognitive decline comparing MICE (3–4 % bias) to analyses of available data only (16–23 % bias) in a construct where missingness was strongly informative but realistic. Associations between diabetes and 20-year cognitive decline were substantially stronger with MICE than in available-case analyses. Our study suggests when informative data are available for non-examined participants, MICE can be an effective tool for imputing cognitive performance and improving assessment of cognitive decline, though careful thought should be given to target imputation population and analytic model chosen, as they may yield different estimands.     

A joint modeling approach to data with informative cluster size: robustness to the cluster size model

In many biomedical and epidemiological studies, data are often clustered due to longitudinal follow up or repeated sampling. While in some clustered data the cluster size is pre-determined, in others it may be correlated with the outcome of subunits, resulting in informative cluster size. When the cluster size is informative, standard statistical procedures that ignore cluster size may produce biased estimates. One attractive framework for modeling data with informative cluster size is the joint modeling approach in which a common set of random effects are shared by both the outcome and cluster size models. In addition to making distributional assumptions on the shared random effects, the joint modeling approach needs to specify the cluster size model. Questions arise as to whether the joint modeling approach is robust to misspecification of the cluster size model. In this paper, we studied both asymptotic and finite-sample characteristics of the maximum likelihood estimators in joint models when the cluster size model is misspecified. We found that using an incorrect distribution for the cluster size may induce small to moderate biases, while using a misspecified functional form for the shared random parameter in the cluster size model results in nearly unbiased estimation of outcome model parameters. We also found that there is little efficiency loss under this model misspecification. A developmental toxicity study was used to motivate the research and to demonstrate the findings.     

Latent trait shared-parameter mixed models for missing ecological momentary assessment data

Latent trait shared-parameter mixed models for ecological momentary assessment (EMA) data containing missing values are developed in which data are collected in an intermittent manner. In such studies, data are often missing due to unanswered prompts. Using item response theory models, a latent trait is used to represent the missing prompts and modeled jointly with a mixed model for bivariate longitudinal outcomes. Both one- and two-parameter latent trait shared-parameter mixed models are presented. These new models offer a unique way to analyze missing EMA data with many response patterns. Here, the proposed models represent missingness via a latent trait that corresponds to the students' “ability” to respond to the prompting device. Data containing more than 10 300 observations from an EMA study involving high school students' positive and negative affects are presented. The latent trait representing missingness was a significant predictor of both positive affect and negative affect outcomes. The models are compared to a missing at random mixed model. A simulation study indicates that the proposed models can provide lower bias and increased efficiency compared to the standard missing at random approach commonly used with intermittent missing longitudinal data.