An index of local sensitivity to non‐ignorability for multivariate longitudinal mixed data with potential non‐random dropout

Multivariate longitudinal data with mixed continuous and discrete responses with the possibility of non‐ignorable missingness are often common in follow‐up medical studies and their analysis needs to be developed. Standard methods of analysis based on the strong and the unverifiable assumption of missing at random (MAR) mechanism could be highly misleading. A way out of this problem is to start with methods that simultaneously allow modelling non‐ignorable mechanism, which includes somehow troubling computations that are often time consuming, then we can use a sensitivity analysis, in which one estimates models under a range of assumptions about non‐ignorability parameters to study the impact of these parameters on key inferences. A general index of sensitivity to non‐ignorability (ISNI) to measure sensitivity of key inferences in a neighborhood of MAR model without fitting a complicated not MAR (NMAR) model for univariate generalized linear models and for models used for univariate longitudinal normal and non‐Gaussian data with potentially NMAR dropout are well presented in the literature. In this paper we extend ISNI methodology to analyze multivariate longitudinal mixed data subject to non‐ignorable dropout in which the non‐ignorable dropout model could be dependent on the mixed responses. The approach is illustrated by analyzing a longitudinal data set in which the general substantive goal of the study is to better understand the relations between parental assessment of child's antisocial behavior and child's reading recognition skill. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

Accounting for informatively missing data in logistic regression by means of reassessment sampling

We explore the ‘reassessment’ design in a logistic regression setting, where a second wave of sampling is applied to recover a portion of the missing data on a binary exposure and/or outcome variable. We construct a joint likelihood function based on the original model of interest and a model for the missing data mechanism, with emphasis on non‐ignorable missingness. The estimation is carried out by numerical maximization of the joint likelihood function with close approximation of the accompanying Hessian matrix, using sharable programs that take advantage of general optimization routines in standard software. We show how likelihood ratio tests can be used for model selection and how they facilitate direct hypothesis testing for whether missingness is at random. Examples and simulations are presented to demonstrate the performance of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of change in the presence of informative censoring: application to a longitudinal clinical trial of progressive renal disease

The rate of change in a continuous variable, measured serially over time, is often used as an outcome in longitudinal studies or clinical trials. When patients terminate the study before the scheduled end of the study, there is a potential for bias in estimation of rate of change using standard methods which ignore the missing data mechanism. These methods include the use of unweighted generalized estimating equations methods and likelihood-based methods assuming an ignorable missing data mechanism. We present a model for analysis of informatively censored data, based on an extension of the two-stage linear random effects model, where each subject's random intercept and slope are allowed to be associated with an underlying time to event. The joint distribution of the continuous responses and the time-to-event variable are then estimated via maximum likelihood using the EM algorithm, and using the bootstrap to calculate standard errors. We illustrate this methodology and compare it to simpler approaches and usual maximum likelihood using data from a multi-centre study of the effects of diet and blood pressure control on progression of renal disease, the Modification of Diet in Renal Disease (MDRD) Study. Sensitivity analyses and simulations are used to evaluate the performance of this methodology in the context of the MDRD data, under various scenarios where the drop-out mechanism is ignorable as well as non-ignorable.     

Maximum likelihood estimation with missing outcomes: From simplicity to complexity

Many clinical or prevention studies involve missing or censored outcomes. Maximum likelihood (ML) methods provide a conceptually straightforward approach to estimation when the outcome is partially missing. Methods of implementing ML methods range from the simple to the complex, depending on the type of data and the missing-data mechanism. Simple ML methods for ignorable missing-data mechanisms (when data are missing at random) include complete-case analysis, complete-case analysis with covariate adjustment, survival analysis with covariate adjustment, and analysis via propensity-to-be-missing scores. More complex ML methods for ignorable missing-data mechanisms include the analysis of longitudinal dropouts via a marginal model for continuous data or a conditional model for categorical data. A moderately complex ML method for categorical data with a saturated model and either ignorable or nonignorable missing-data mechanisms is a perfect fit analysis, an algebraic method involving closed-form estimates and variances. A complex and flexible ML method with categorical data and either ignorable or nonignorable missing-data mechanisms is the method of composite linear models, a matrix method requiring specialized software. Except for the method of composite linear models, which can involve challenging matrix specifications, the implementation of these ML methods ranges in difficulty from easy to moderate.     

Non‐ignorable loss to follow‐up: correcting mortality estimates based on additional outcome ascertainment

Loss to follow‐up (LTFU) is a common problem in many epidemiological studies. In antiretroviral treatment (ART) programs for patients with human immunodeficiency virus (HIV), mortality estimates can be biased if the LTFU mechanism is non‐ignorable, that is, mortality differs between lost and retained patients. In this setting, routine procedures for handling missing data may lead to biased estimates. To appropriately deal with non‐ignorable LTFU, explicit modeling of the missing data mechanism is needed. This can be based on additional outcome ascertainment for a sample of patients LTFU, for example, through linkage to national registries or through survey‐based methods. In this paper, we demonstrate how this additional information can be used to construct estimators based on inverse probability weights (IPW) or multiple imputation. We use simulations to contrast the performance of the proposed estimators with methods widely used in HIV cohort research for dealing with missing data. The practical implications of our approach are illustrated using South African ART data, which are partially linkable to South African national vital registration data. Our results demonstrate that while IPWs and proper imputation procedures can be easily constructed from additional outcome ascertainment to obtain valid overall estimates, neglecting non‐ignorable LTFU can result in substantial bias. We believe the proposed estimators are readily applicable to a growing number of studies where LTFU is appreciable, but additional outcome data are available through linkage or surveys of patients LTFU. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A multiple imputation strategy for incomplete longitudinal data

Longitudinal studies are commonly used to study processes of change. Because data are collected over time, missing data are pervasive in longitudinal studies, and complete ascertainment of all variables is rare. In this paper a new imputation strategy for completing longitudinal data sets is proposed. The proposed methodology makes use of shrinkage estimators for pooling information across geographic entities, and of model averaging for pooling predictions across different statistical models. Bayes factors are used to compute weights (probabilities) for a set of models considered to be reasonable for at least some of the units for which imputations must be produced, imputations are produced by draws from the predictive distributions of the missing data, and multiple imputations are used to better reflect selected sources of uncertainty in the imputation process. The imputation strategy is developed within the context of an application to completing incomplete longitudinal variables in the so-called Area Resource File. The proposed procedure is compared with several other imputation procedures in terms of inferences derived with the imputations, and the proposed methodology is demonstrated to provide valid estimates of model parameters when the completed data are analysed. Extensions to other missing data problems in longitudinal studies are straightforward so long as the missing data mechanism can be assumed to be ignorable.     

Bayesian inference from incomplete longitudinal data: A simple method to quantify sensitivity to nonignorable dropout

Bayesian approach has been increasingly used for analyzing longitudinal data. When dropout occurs in the study, analysis often relies on the assumption of ignorable dropout. Because ignorability is a critical and untestable assumption without obtaining additional data or making other unverifiable assumptions, it is important to assess the impact of departures from the ignorability assumption on the key Bayesian inferences. In this paper, we extend the Bayesian index of local sensitivity to non‐ignorability (ISNI) method proposed by Zhang and Heitjan to longitudinal data with dropout. We derive formulas for the Bayesian ISNI when the complete longitudinal data follow a linear mixed‐effect model. The calculation of the index only requires the posterior draws or summary statistics of these draws from the standard analysis of the ignorable model. Thus, our approach avoids fitting any complicated nonignorable model. One can use the method to evaluate which Bayesian parameter estimates or functions of these estimates in a linear mixed‐effect model are susceptible to nonignorable dropout and which ones are not. We illustrate the method using a simulation study and two real examples: rats data set and rheumatoid arthritis clinical trial data set. Copyright © 2009 John Wiley & Sons, Ltd.     

A weighted combination of pseudo‐likelihood estimators for longitudinal binary data subject to non‐ignorable non‐monotone missingness

For longitudinal binary data with non‐monotone non‐ignorably missing outcomes over time, a full likelihood approach is complicated algebraically, and with many follow‐up times, maximum likelihood estimation can be computationally prohibitive. As alternatives, two pseudo‐likelihood approaches have been proposed that use minimal parametric assumptions. One formulation requires specification of the marginal distributions of the outcome and missing data mechanism at each time point, but uses an ‘independence working assumption,’ i.e. an assumption that observations are independent over time. Another method avoids having to estimate the missing data mechanism by formulating a ‘protective estimator.’ In simulations, these two estimators can be very inefficient, both for estimating time trends in the first case and for estimating both time‐varying and time‐stationary effects in the second. In this paper, we propose the use of the optimal weighted combination of these two estimators, and in simulations we show that the optimal weighted combination can be much more efficient than either estimator alone. Finally, the proposed method is used to analyze data from two longitudinal clinical trials of HIV‐infected patients. Copyright © 2010 John Wiley & Sons, Ltd.     

Multivariate longitudinal data analysis with censored and intermittent missing responses

The multivariate linear mixed model (MLMM) has emerged as an important analytical tool for longitudinal data with multiple outcomes. However, the analysis of multivariate longitudinal data could be complicated by the presence of censored measurements because of a detection limit of the assay in combination with unavoidable missing values arising when subjects miss some of their scheduled visits intermittently. This paper presents a generalization of the MLMM approach, called the MLMM‐CM, for a joint analysis of the multivariate longitudinal data with censored and intermittent missing responses. A computationally feasible expectation maximization–based procedure is developed to carry out maximum likelihood estimation within the MLMM‐CM framework. Moreover, the asymptotic standard errors of fixed effects are explicitly obtained via the information‐based method. We illustrate our methodology by using simulated data and a case study from an AIDS clinical trial. Experimental results reveal that the proposed method is able to provide more satisfactory performance as compared with the traditional MLMM approach.     

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.     

Methods for the analysis of informatively censored longitudinal data.

This paper describes the problem of informative censoring in longitudinal studies where the primary outcome is rate of change in a continuous variable. Standard approaches based on the linear random effects model are valid only when the data are missing in a non-ignorable fashion. Informative censoring, which is a special type of non-ignorably missing data, occurs when the probability of early termination is related to an individual subject's true rate of change. When present, informative censoring causes bias in standard likelihood-based analyses, as well as in weighted averages of individual least-squares slopes. This paper reviews several methods proposed by others for analysis of informatively censored longitudinal data, and outlines a new approach based on a log-normal survival model. Maximum likelihood estimates may be obtained via the EM algorithm. Advantages of this approach are that it allows general unbalanced data caused by staggered entry and unequally-timed visits, it utilizes all available data, including data from patients with only a single measurement, and it provides a unified method for estimating all model parameters. Issues related to study design when informative censoring may occur are also discussed.     

A Bayesian analysis of mixture structural equation models with non‐ignorable missing responses and covariates

In behavioral, biomedical, and social‐psychological sciences, it is common to encounter latent variables and heterogeneous data. Mixture structural equation models (SEMs) are very useful methods to analyze these kinds of data. Moreover, the presence of missing data, including both missing responses and missing covariates, is an important issue in practical research. However, limited work has been done on the analysis of mixture SEMs with non‐ignorable missing responses and covariates. The main objective of this paper is to develop a Bayesian approach for analyzing mixture SEMs with an unknown number of components, in which a multinomial logit model is introduced to assess the influence of some covariates on the component probability. Results of our simulation study show that the Bayesian estimates obtained by the proposed method are accurate, and the model selection procedure via a modified DIC is useful in identifying the correct number of components and in selecting an appropriate missing mechanism in the proposed mixture SEMs. A real data set related to a longitudinal study of polydrug use is employed to illustrate the methodology. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Nonlinear multiple imputation for continuous covariate within semiparametric Cox model: application to HIV data in Senegal

Multiple imputation is commonly used to impute missing covariate in Cox semiparametric regression setting. It is to fill each missing data with more plausible values, via a Gibbs sampling procedure, specifying an imputation model for each missing variable. This imputation method is implemented in several softwares that offer imputation models steered by the shape of the variable to be imputed, but all these imputation models make an assumption of linearity on covariates effect. However, this assumption is not often verified in practice as the covariates can have a nonlinear effect. Such a linear assumption can lead to a misleading conclusion because imputation model should be constructed to reflect the true distributional relationship between the missing values and the observed values. To estimate nonlinear effects of continuous time invariant covariates in imputation model, we propose a method based on B‐splines function. To assess the performance of this method, we conducted a simulation study, where we compared the multiple imputation method using Bayesian splines imputation model with multiple imputation using Bayesian linear imputation model in survival analysis setting. We evaluated the proposed method on the motivated data set collected in HIV‐infected patients enrolled in an observational cohort study in Senegal, which contains several incomplete variables. We found that our method performs well to estimate hazard ratio compared with the linear imputation methods, when data are missing completely at random, or missing at random. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint modeling of missing data due to non‐participation and death in longitudinal aging studies

Specific age‐related hypotheses are tested in population‐based longitudinal studies. At specific time intervals, both the outcomes of interest and the time‐varying covariates are measured. When participants are approached for follow‐up, some participants do not provide data. Investigations may show that many have died before the time of follow‐up whereas others refused to participate. Some of these non‐participants do not provide data at later follow‐ups. Few statistical methods for missing data distinguish between ‘non‐participation’ and ‘death’ among study participants. The augmented inverse probability‐weighted estimators are most commonly used in marginal structure models when data are missing at random. Treating non‐participation and death as the same, however, may lead to biased estimates and invalid inferences. To overcome this limitation, a multiple inverse probability‐weighted approach is presented to account for two types of missing data, non‐participation and death, when using a marginal mean model. Under certain conditions, the multiple weighted estimators are consistent and asymptotically normal. Simulation studies will be used to study the finite sample efficiency of the multiple weighted estimators. The proposed method will be applied to study the risk factors associated with the cognitive decline among the aging adults, using data from the Chicago Health and Aging Project (CHAP). Copyright © 2010 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.