Impact of missing data due to drop-outs on estimators for rates of change in longitudinal studies: a simulation study.

Many cohort studies and clinical trials are designed to compare rates of change over time in one or more disease markers in several groups. One major problem in such longitudinal studies is missing data due to patient drop-out. The bias and efficiency of six different methods to estimate rates of changes in longitudinal studies with incomplete observations were compared: generalized estimating equation estimates (GEE) proposed by Liang and Zeger (1986); unweighted average of ordinary least squares (OLSE) of individual rates of change (UWLS); weighted average of OLSE (WLS); conditional linear model estimates (CLE), a covariate type estimates proposed by Wu and Bailey (1989); random effect (RE), and joint multivariate RE (JMRE) estimates. The latter method combines a linear RE model for the underlying pattern of the marker with a log-normal survival model for informative drop-out process. The performance of these methods in the presence of missing data completely at random (MCAR), at random (MAR) and non-ignorable (NIM) were compared in simulation studies. Data for the disease marker were generated under the linear random effects model with parameter values derived from realistic examples in HIV infection. Rates of drop-out, assumed to increase over time, were allowed to be independent of marker values or to depend either only on previous marker values or on both previous and current marker values. Under MACR all six methods yielded unbiased estimates of both group mean rates and between-group difference. However, the cross-sectional view of the data in the GEE method resulted in seriously biased estimates under MAR and NIM drop-out process. The bias in the estimates ranged from 30 per cent to 50 per cent. The degree of bias in the GEE estimates increases with the severity of non-randomness and with the proportion of MAR data. Under MCAR and MAR all the other five methods performed relatively well. RE and JMRE estimates were more efficient(that is, had smaller variance) than UWLS, WLS and CL estimates. Under NIM, WLS and particularly RE estimates tended to underestimate the average rate of marker change (bias approximately 10 per cent). Under NIM, UWLS, CL and JMRE performed better in terms of bias (3-5 per cent) with the JMRE giving the most efficient estimates. Given that markers are key variables related to disease progression, missing marker data are likely to be at least MAR. Thus, the GEE method may not be appropriate for analysing such longitudinal marker data. The potential biases due to incomplete data require greater recognition in reports of longitudinal studies. Sensitivity analyses to assess the effect of drop-outs on inferences about the target parameters are important.     

Bias adjustment in analysing longitudinal data with informative missingness.

The recent biostatistical literature contains a number of methods for handling the bias caused by 'informative censoring', which refers to drop-out from a longitudinal study after a number of visits scheduled at predetermined intervals. The same or related methods can be extended to situations where the missing pattern is intermittent. The pattern of missingness is often assumed to be related to the outcome through random effects which represent unmeasured individual characteristics such as health awareness. To date there is only limited experience with applying the methods for informative censoring in practice, mostly because of complicated modelling and difficult computations. In this paper, we propose an estimation method based on grouping the data. The proposed estimator is asymptotically unbiased in various situations under informative missingness. Several existing methods are reviewed and compared in simulation studies. We apply the methods to data from the Wisconsin Diabetes Registry Project, a longitudinal study tracking glycaemic control and acute and chronic complications from the diagnosis of type I diabetes.     

Missing data in longitudinal studies: cross-sectional multiple imputation provides similar estimates to full-information maximum likelihood

PurposeThe aim of this research was to examine, in an exploratory manner, whether cross-sectional multiple imputation generates valid parameter estimates for a latent growth curve model in a longitudinal data set with nonmonotone missingness.MethodsA simulated longitudinal data set of N = 5000 was generated and consisted of a continuous dependent variable, assessed at three measurement occasions and a categorical time-invariant independent variable. Missing data had a nonmonotone pattern and the proportion of missingness increased from the initial to the final measurement occasion (5%–20%). Three methods were considered to deal with missing data: listwise deletion, full-information maximum likelihood, and multiple imputation. A latent growth curve model was specified and analysis of variance was used to compare parameter estimates between the full data set and missing data approaches.ResultsMultiple imputation resulted in significantly lower slope variance compared with the full data set. There were no differences in any parameter estimates between the multiple imputation and full-information maximum likelihood approaches.ConclusionsThis study suggested that in longitudinal studies with nonmonotone missingness, cross-sectional imputation at each time point may be viable and produces estimates comparable with those obtained with full-information maximum likelihood. Future research pursuing the validity of this method is warranted.     

A comparison of two methods for the estimation of precision with incomplete longitudinal data, jointly modelled with a time-to-event outcome

Several methods for the estimation and comparison of rates of change in longitudinal studies with staggered entry and informative drop-outs have been recently proposed. For multivariate normal linear models, REML estimation is used. There are various approaches to maximizing the corresponding log-likelihood; in this paper we use a restricted iterative generalized least squares method (RIGLS) combined with a nested EM algorithm. An important statistical problem in such approaches is the estimation of the standard errors adjusted for the missing data (observed data information matrix). Louis has provided a general technique for computing the observed data information in terms of completed data quantities within the EM framework. The multiple imputation (MI) method for obtaining variances can be regarded as an alternative to this. The aim of this paper is to develop, apply and compare the Louis and a modified MI method in the setting of longitudinal studies where the source of missing data is either death or disease progression (informative) or end of the study (assumed non-informative). Longitudinal data are simultaneously modelled with the missingness process. The methods are illustrated by modelling CD4 count data from an HIV-1 clinical trial and evaluated through simulation studies. Both methods, Louis and MI, are used with Monte Carlo simulations of the missing data using the appropriate conditional distributions, the former with 100 simulations, the latter with 5 and 10. It is seen that naive SEs based on the completed data likelihood can be seriously biased. This bias was largely corrected by Louis and modified MI methods, which gave broadly similar estimates. Given the relative simplicity of the modified MI method, it may be preferable.     

Allowing for uncertainty due to missing continuous outcome data in pairwise and network meta‐analysis

Missing outcome data are commonly encountered in randomized controlled trials and hence may need to be addressed in a meta‐analysis of multiple trials. A common and simple approach to deal with missing data is to restrict analysis to individuals for whom the outcome was obtained (complete case analysis). However, estimated treatment effects from complete case analyses are potentially biased if informative missing data are ignored. We develop methods for estimating meta‐analytic summary treatment effects for continuous outcomes in the presence of missing data for some of the individuals within the trials. We build on a method previously developed for binary outcomes, which quantifies the degree of departure from a missing at random assumption via the informative missingness odds ratio. Our new model quantifies the degree of departure from missing at random using either an informative missingness difference of means or an informative missingness ratio of means, both of which relate the mean value of the missing outcome data to that of the observed data. We propose estimating the treatment effects, adjusted for informative missingness, and their standard errors by a Taylor series approximation and by a Monte Carlo method. We apply the methodology to examples of both pairwise and network meta‐analysis with multi‐arm trials. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

An index of local sensitivity to nonignorable drop-out in longitudinal modelling

In longitudinal studies with potentially nonignorable drop-out, one can assess the likely effect of the nonignorability in a sensitivity analysis. Troxel et al. proposed a general index of sensitivity to nonignorability, or ISNI, to measure sensitivity of key inferences in a neighbourhood of the ignorable, missing at random (MAR) model. They derived detailed formulas for ISNI in the special case of the generalized linear model with a potentially missing univariate outcome. In this paper, we extend the method to longitudinal modelling. We use a multivariate normal model for the outcomes and a regression model for the drop-out process, allowing missingness probabilities to depend on an unobserved response. The computation is straightforward, and merely involves estimating a mixed-effects model and a selection model for the drop-out, together with some simple arithmetic calculations. We illustrate the method with three examples.     

Pseudo-likelihood methods for longitudinal binary data with non-ignorable missing responses and covariates

In this paper we consider longitudinal studies in which the outcome to be measured over time is binary, and the covariates of interest are categorical. In longitudinal studies it is common for the outcomes and any time-varying covariates to be missing due to missed study visits, resulting in non-monotone patterns of missingness. Moreover, the reasons for missed visits may be related to the specific values of the response and/or covariates that should have been obtained, i.e. missingness is non-ignorable. With non-monotone non-ignorable missing response and covariate data, a full likelihood approach is quite complicated, and maximum likelihood estimation can be computationally prohibitive when there are many occasions of follow-up. Furthermore, the full likelihood must be correctly specified to obtain consistent parameter estimates. We propose a pseudo-likelihood method for jointly estimating the covariate effects on the marginal probabilities of the outcomes and the parameters of the missing data mechanism. The pseudo-likelihood requires specification of the marginal distributions of the missingness indicator, outcome, and possibly missing covariates at each occasions, but avoids making assumptions about the joint distribution of the data at two or more occasions. Thus, the proposed method can be considered semi-parametric. The proposed method is an extension of the pseudo-likelihood approach in Troxel et al. to handle binary responses and possibly missing time-varying covariates. The method is illustrated using data from the Six Cities study, a longitudinal study of the health effects of air pollution.     

The problem of attrition in a Finnish longitudinal survey on depression

A cohort of all school children aged 16 years in 1983 (n=2194, 96.7%) in Tampere, Finland were studied at 16, 22 and 32 years of age by self-reported questionnaires. The non-response pattern was considered by modelling the individual response probability by panel year and gender. Gender and school performance at age 16 years were the most important predictors of non-response. They explained away the effect of all other variables at 16 and 22 years, except for earlier non-response at age 22. However, the ability of the models to predict non-respondents was very poor. The effect of attrition for the estimation of depression prevalence was evaluated first by longitudinal weighting methods used commonly in survey studies and then by Markov chain Monte Carlo (MCMC) simulation of the missing depression status. Under the missing-at-random assumption (MAR), both applied correction methods gave estimates of roughly the same size and did not significantly differ from the observed prevalence of depression. No indication of informative missingness was found. We therefore conclude that attrition does not seriously bias the estimation of depression prevalence in the data. In general, non-response models, which are needed to correct for informative missingness, are likely to have poor ability to predict non-response. Therefore, the plausibility of the MAR assumption is important in the presence of attrition.     

Sensitivity analysis of longitudinal normal data with drop-outs

We propose to perform a sensitivity analysis to evaluate the extent to which results from a longitudinal study can be affected by informative drop-outs. The method is based on a selection model, where the parameter relating the dropout probability to the current observation is not estimated, but fixed to a set of values. This allows to evaluate several hypotheses for the degree of informativeness of the drop-out process. Expectation and variance of missing data, conditional on the drop-out time are computed, and a stochastic EM algorithm is used to obtain maximum likelihood estimates. Simulations show that when the drop-out parameter is correctly specified, unbiased estimates of the other parameters are obtained, and coverage percentages of their confidence intervals are close to their theoretical value. More interestingly, misspecification of the drop-out parameter does not considerably alter these results. This method was applied to a randomized clinical trial, designed to demonstrate non-inferiority of an inhaled corticosteroid in terms of bone density, compared with a reference treatment. Sensitivity analysis showed that the conclusion of non-inferiority was robust against different hypotheses for the drop-out process.     

A mixed-effects model for cognitive decline with non-monotone non-response from a two-phase longitudinal study of dementia

Most longitudinal studies of elderly are characterized by substantial drop-out due to death and many other factors beyond the control of the investigators. In a two-phase longitudinal study of dementia, subjects with cognitive impairment skip the first phase survey in the next follow-up, leading to intermittent missing variables measured in that phase. In the context of analysing pre-dementia cognitive decline in an elderly population, both of the two causes of non-response can potentially be informative in the sense that the missingness is dependent on the unobserved outcome. To take these factors into account, mixed-effects models are constructed to allow the outcome and the multiple causes of missing values to share the same 'random parameter' or random effect. The crucial assumption of our model is that the random effects of the model for the outcome and that of the model for the missing-data indicators are linked in a deterministic manner. It can be thought of as an approximation of a more general and realistic situation, in which the two models have distinct, yet dependent, random effects. We conduct a simulation study to investigate possible deviations of the estimates under such a scenario. A second simulation illustrates the magnitude of the bias in estimating the difference of decline rate between two groups when the random effects are linked in different manners for the two groups.     

Mixed models for bivariate response repeated measures data using Gibbs sampling

Repeated measures data are frequently incomplete, unbalanced and correlated. There has been a great deal of recent interest in mixed effects models for analysing such data. In this paper, we develop bivariate response mixed effects models that are a generalization of linear mixed effects models for a single response variable. We describe their estimation procedures using a Markov chain Monte Carlo method, the Gibbs sampler. We illustrate the methods with analyses of intravenous vitamin D₃ administration for secondary hyperparathyroidism in hemodialysis patients. In these data there were two response variables on each individual (PTH and calcium level). This study also suffered from attrition, like many longitudinal studies. While, considering the study design, it was reasonable to assume the drop-out mechanism for the calcium (Ca) level to be ‘missing at random’, the drop-out mechanism for the PTH level was likely to be non-ignorable. We found that the posterior treatment effects for the PTH level by the single response model were underestimated compared with those obtained by the bivariate response model, while there were little differences in the posterior features for the Ca level under both models. © 1997 John Wiley & Sons Ltd.     

Pattern mixture models for longitudinal quality of life studies in advanced stage disease

Analyses of longitudinal quality of life (QOL) for patients with advanced stage disease are frequently plagued by problems of non-random drop-out attributable to deteriorating health and/or death. As an example, Moinpour et al. cite specific challenges which limited their report and assessment of QOL for patients treated for advanced stage colorectal cancer in a clinical trial of several chemotherapeutic regimes performed by the Southwest Oncology Group. A particular source of confusion that arises in studies of advanced stage disease is whether or not to differentiate loss of follow-up due to death from drop-out where the patient is still alive but has dropped from the study. In this paper we examine exploratory data techniques for longitudinal QOL data with non-random missingness due to drop-out and censorship by death. We propose a pattern mixture model for longitudinal QOL, time of drop-out and survival, which allows for straightforward implementation of sensitivity analyses and explicit comparisons to the raw data. Our method is illustrated in the context of analysing the data and addressing the challenges posed by Moinpour et al.     

A transition model for quality‐of‐life data with non‐ignorable non‐monotone missing data

In this paper, we consider a full likelihood method to analyze continuous longitudinal responses with non‐ignorable non‐monotone missing data. We consider a transition probability model for the missingness mechanism. A first‐order Markov dependence structure is assumed for both the missingness mechanism and observed data. This process fits the natural data structure in the longitudinal framework. Our main interest is in estimating the parameters of the marginal model and evaluating the missing‐at‐random assumption in the Effects of Public Information Study, a cancer‐related study recently conducted at the University of Pennsylvania. We also present a simulation study to assess the performance of the model. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Stratification of summary statistic tests according to missing data patterns

Summary statistics, such as slope or area under the time-response curve, reduce the dimensionality of repeated measures data and can thereby simplify the comparison of groups in longitudinal studies. Since summary statistic distributions vary according to the amount, timing, and type of any missingness that occurs, one must choose between analysing the data unconditionally or conditionally on the missingness patterns. This paper uses simulations to compare such unstratified and stratified summary statistic analyses with respect to their size and power under models that allow for both non-informative and informative missingness mechanisms. Of particular interest is the robustness of these methods to violations of the assumptions that one must make if they are to have proper test size. It is found that stratification of the analysis tends to result in an increase of power, and improves the robustness to violations of missing data assumptions.     

A correlated random‐effects model for normal longitudinal data with nonignorable missingness

The missing data problem is common in longitudinal or hierarchical structure studies. In this paper, we propose a correlated random‐effects model to fit normal longitudinal or cluster data when the missingness mechanism is nonignorable. Computational challenges arise in the model fitting due to intractable numerical integrations. We obtain the estimates of the parameters based on an accurate approximation of the log likelihood, which has higher‐order accuracy but with less computational burden than the existing approximation. We apply the proposed method it to a real data set arising from an autism study. Copyright © 2009 John Wiley & Sons, Ltd.     

A tractable method to account for high-dimensional nonignorable missing data in intensive longitudinal data

Despite the need for sensitivity analysis to nonignorable missingness in intensive longitudinal data (ILD), such analysis is greatly hindered by novel ILD features, such as large data volume and complex nonmonotonic missing-data patterns. Likelihood of alternative models permitting nonignorable missingness often involves very high-dimensional integrals, causing curse of dimensionality and rendering solutions computationally prohibitive to obtain. We aim to overcome this challenge by developing a computationally feasible method, nonlinear indexes of local sensitivity to nonignorability (NISNI). We use linear mixed effects models for the incomplete outcome and covariates. We use Markov multinomial models to describe complex missing-data patterns and mechanisms in ILD, thereby permitting missingness probabilities to depend directly on missing data. Using a second-order Taylor series to approximate likelihood under nonignorability, we develop formulas and closed-form expressions for NISNI. Our approach permits the outcome and covariates to be missing simultaneously, as is often the case in ILD, and can capture U-shaped impact of nonignorability in the neighborhood of the missing at random model without fitting alternative models or evaluating integrals. We evaluate performance of this method using simulated data and real ILD collected by the ecological momentary assessment method.     

A random-effects Markov transition model for Poisson-distributed repeated measures with non-ignorable missing values

In biomedical research with longitudinal designs, missing values due to intermittent non-response or premature withdrawal are usually 'non-ignorable' in the sense that unobserved values are related to the patterns of missingness. By drawing the framework of a shared-parameter mechanism, the process yielding the repeated count measures and the process yielding missing values can be modelled separately, conditionally on a group of shared parameters. For chronic diseases, Markov transition models can be used to study the transitional features of the pathologic processes. In this paper, Markov Chain Monte Carlo algorithms are developed to fit a random-effects Markov transition model for incomplete count repeated measures, within which random effects are shared by the counting process and the missing-data mechanism. Assuming a Poisson distribution for the count measures, the transition probabilities are estimated using a Poisson regression model. The missingness mechanism is modelled with a multinomial-logit regression to calculate the transition probabilities of the missingness indicators. The method is demonstrated using both simulated data sets and a practical data set from a smoking cessation clinical trial.     

A new Bayesian joint model for longitudinal count data with many zeros,intermittent missingness,and dropout with applications to HIV prevention trials

In longitudinal clinical trials, it is common that subjects may permanently withdraw from the study (dropout), or return to the study after missing one or more visits (intermittent missingness). It is also routinely encountered in HIV prevention clinical trials that there is a large proportion of zeros in count response data. In this paper, a sequential multinomial model is adopted for dropout and subsequently a conditional model is constructed for intermittent missingness. The new model captures the complex structure of missingness and incorporates dropout and intermittent missingness simultaneously. The model also allows us to easily compute the predictive probabilities of different missing data patterns. A zero-inflated Poisson mixed-effects regression model is assumed for the longitudinal count response data. We also propose an approach to assess the overall treatment effects under the zero-inflated Poisson model. We further show that the joint posterior distribution is improper if uniform priors are specified for the regression coefficients under the proposed model. Variations of the g-prior, Jeffreys prior, and maximally dispersed normal prior are thus established as remedies for the improper posterior distribution. An efficient Gibbs sampling algorithm is developed using a hierarchical centering technique. A modified logarithm of the pseudomarginal likelihood and a concordance based area under the curve criterion are used to compare the models under different missing data mechanisms. We then conduct an extensive simulation study to investigate the empirical performance of the proposed methods and further illustrate the methods using real data from an HIV prevention clinical trial.     

Longitudinal latent variable models given incompletely observed biomarkers and covariates

In this paper, we analyze a two‐level latent variable model for longitudinal data from the National Growth and Health Study where surrogate outcomes or biomarkers and covariates are subject to missingness at any of the levels. A conventional method for efficient handling of missing data is to re‐express the desired model as a joint distribution of variables, including the biomarkers, that are subject to missingness conditional on all of the covariates that are completely observed, and estimate the joint model by maximum likelihood, which is then transformed to the desired model. The joint model, however, identifies more parameters than desired, in general. We show that the over‐identified joint model produces biased estimation of the latent variable model and describe how to impose constraints on the joint model so that it has a one‐to‐one correspondence with the desired model for unbiased estimation. The constrained joint model handles missing data efficiently under the assumption of ignorable missing data and is estimated by a modified application of the expectation‐maximization algorithm. Copyright © 2016 John Wiley & Sons, Ltd.