Covariate Decomposition Methods for Longitudinal Missing‐at‐Random Data and Predictors Associated with Subject‐Specific Effects

Investigators often gather longitudinal data to assess changes in responses over time within subjects and to relate these changes to within‐subject changes in predictors. Missing data are common in such studies and predictors can be correlated with subject‐specific effects. Maximum likelihood methods for generalized linear mixed models provide consistent estimates when the data are ‘missing at random’ (MAR) but can produce inconsistent estimates in settings where the random effects are correlated with one of the predictors. On the other hand, conditional maximum likelihood methods (and closely related maximum likelihood methods that partition covariates into between‐ and within‐cluster components) provide consistent estimation when random effects are correlated with predictors but can produce inconsistent covariate effect estimates when data are MAR. Using theory, simulation studies, and fits to example data this paper shows that decomposition methods using complete covariate information produce consistent estimates. In some practical cases these methods, that ostensibly require complete covariate information, actually only involve the observed covariates. These results offer an easy‐to‐use approach to simultaneously protect against bias from both cluster‐level confounding and MAR missingness in assessments of change.     

Different methods for handling incomplete longitudinal binary outcome due to missing at random dropout

This paper compares the performance of weighted generalized estimating equations (WGEEs), multiple imputation based on generalized estimating equations (MI-GEEs) and generalized linear mixed models (GLMMs) for analyzing incomplete longitudinal binary data when the underlying study is subject to dropout. The paper aims to explore the performance of the above methods in terms of handling dropouts that are missing at random (MAR). The methods are compared on simulated data. The longitudinal binary data are generated from a logistic regression model, under different sample sizes. The incomplete data are created for three different dropout rates. The methods are evaluated in terms of bias, precision and mean square error in case where data are subject to MAR dropout. In conclusion, across the simulations performed, the MI-GEE method performed better in both small and large sample sizes. Evidently, this should not be seen as formal and definitive proof, but adds to the body of knowledge about the methods’ relative performance. In addition, the methods are compared using data from a randomized clinical trial.     

Application of sensitivity analysis to incomplete longitudinal CD4 count data

In this paper, we investigate the effect of tuberculosis pericarditis (TBP) treatment on CD4 count changes over time and draw inferences in the presence of missing data. We accounted for missing data and conducted sensitivity analyses to assess whether inferences under missing at random (MAR) assumption are sensitive to not missing at random (NMAR) assumptions using the selection model (SeM) framework. We conducted sensitivity analysis using the local influence approach and stress-testing analysis. Our analyses showed that the inferences from the MAR are robust to the NMAR assumption and influential subjects do not overturn the study conclusions about treatment effects and the dropout mechanism. Therefore, the missing CD4 count measurements are likely to be MAR. The results also revealed that TBP treatment does not interact with HIV/AIDS treatment and that TBP treatment has no significant effect on CD4 count changes over time. Although the methods considered were applied to data in the IMPI trial setting, the methods can also be applied to clinical trials with similar settings.     

Statistical methods of risk assessment for continuous variables

Adverse health effects for continuous responses are not as easily defined as adverse health effects for binary responses. Kodell and West (1993) developed methods for defining adverse effects for continuous responses and the associated risk. Procedures were developed for finding point estimates and upper confidence limits for additional risk under the assumption of a normal distribution and quadratic mean response curve with equal variances at each dose level. In this paper, methods are developed for point estimates and upper confidence limits for additional risk at experimental doses when the equal variance assumption is relaxed. An interpolation procedure is discussed for obtaining information at doses other than the experimental doses. A small simulation study is presented to test the performance of the methods discussed.     

Joint generalized estimating equations for multivariate longitudinal binary outcomes with missing data: an application to acquired immune deficiency syndrome data

Summary.  In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to women who are infected with the human immunodeficiency virus, instead of a single outcome variable, there are multiple binary outcomes (e.g. abnormal heart rate, abnormal blood pressure and abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated by using generalized estimating equations (GEEs), and consistent estimates can be obtained under the assumption of a missingness completely at random mechanism. When the missing data mechanism is missingness at random, i.e. the probability of missing a particular outcome at a time point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models by using a single modified GEE based on an EM-type algorithm. The method proposed is motivated by the longitudinal study of cardiac abnormalities in children who were born to women infected with the human immunodeficiency virus, and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that, under a missingness at random mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided that the correlation model has been correctly specified, whereas estimates from standard GEEs can lead to substantial bias.     

Type I error rates from likelihood‐based repeated measures analyses of incomplete longitudinal data

The last observation carried forward (LOCF) approach is commonly utilized to handle missing values in the primary analysis of clinical trials. However, recent evidence suggests that likelihood‐based analyses developed under the missing at random (MAR) framework are sensible alternatives. The objective of this study was to assess the Type I error rates from a likelihood‐based MAR approach – mixed‐model repeated measures (MMRM) – compared with LOCF when estimating treatment contrasts for mean change from baseline to endpoint (Δ). Data emulating neuropsychiatric clinical trials were simulated in a 4 × 4 factorial arrangement of scenarios, using four patterns of mean changes over time and four strategies for deleting data to generate subject dropout via an MAR mechanism. In data with no dropout, estimates of Δ and SE_Δ from MMRM and LOCF were identical. In data with dropout, the Type I error rates (averaged across all scenarios) for MMRM and LOCF were 5.49% and 16.76%, respectively. In 11 of the 16 scenarios, the Type I error rate from MMRM was at least 1.00% closer to the expected rate of 5.00% than the corresponding rate from LOCF. In no scenario did LOCF yield a Type I error rate that was at least 1.00% closer to the expected rate than the corresponding rate from MMRM. The average estimate of SE_Δ from MMRM was greater in data with dropout than in complete data, whereas the average estimate of SE_Δ from LOCF was smaller in data with dropout than in complete data, suggesting that standard errors from MMRM better reflected the uncertainty in the data. The results from this investigation support those from previous studies, which found that MMRM provided reasonable control of Type I error even in the presence of MNAR missingness. No universally best approach to analysis of longitudinal data exists. However, likelihood‐based MAR approaches have been shown to perform well in a variety of situations and are a sensible alternative to the LOCF approach. MNAR methods can be used within a sensitivity analysis framework to test the potential presence and impact of MNAR data, thereby assessing robustness of results from an MAR method. Copyright © 2004 John Wiley & Sons, Ltd.     

Latent class models for longitudinal studies of the elderly with data missing at random

The elderly population in the USA is expected to double in size by the year 2025, making longitudinal health studies of this population of increasing importance. The degree of loss to follow-up in studies of the elderly, which is often because elderly people cannot remain in the study, enter a nursing home or die, make longitudinal studies of this population problematic. We propose a latent class model for analysing multiple longitudinal binary health outcomes with multiple-cause non-response when the data are missing at random and a non-likelihood-based analysis is performed. We extend the estimating equations approach of Robins and co-workers to latent class models by reweighting the multiple binary longitudinal outcomes by the inverse probability of being observed. This results in consistent parameter estimates when the probability of non-response depends on observed outcomes and covariates (missing at random) assuming that the model for non-response is correctly specified. We extend the non-response model so that institutionalization, death and missingness due to failure to locate, refusal or incomplete data each have their own set of non-response probabilities. Robust variance estimates are derived which account for the use of a possibly misspecified covariance matrix, estimation of missing data weights and estimation of latent class measurement parameters. This approach is then applied to a study of lower body function among a subsample of the elderly participating in the 6-year Longitudinal Study of Aging.     

Mean estimation with data missing at random for functional covariables

In a missing-data setting, we want to estimate the mean of a scalar outcome, based on a sample in which an explanatory variable is observed for every subject while responses are missing by happenstance for some of them. We consider two kinds of estimates of the mean response when the explanatory variable is functional. One is based on the average of the predicted values and the second one is a functional adaptation of the Horvitz–Thompson estimator. We show that the infinite dimensionality of the problem does not affect the rates of convergence by stating that the estimates are root-n consistent, under missing at random (MAR) assumption. These asymptotic features are completed by simulated experiments illustrating the easiness of implementation and the good behaviour on finite sample sizes of the method. This is the first paper emphasizing that the insensitiveness of averaged estimates, well known in multivariate non-parametric statistics, remains true for an infinite-dimensional covariable. In this sense, this work opens the way for various other results of this kind in functional data analysis.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Influence of human immunodeficiency virus infection on neurological impairment: an analysis of longitudinal binary data with informative drop-out

A study to investigate the human immunodeficiency virus (HIV) status on the course of neurological impairment, conducted by the HIV Center at Columbia University, followed a cohort of HIV positive and negative gay men for 5 years and assessed the presence or absence of neurological impairment every 6 months. Almost half of the subjects dropped out before the end of the study for reasons that might have been related to the missing neurological data. We propose likelihood-based methods for analysing such binary longitudinal data under informative and non-informative drop-out. A transition model is assumed for the binary response, and several models for the drop-out processes are considered which are functions of the response variable (neurological impairment). The likelihood ratio test is used to compare models with informative and non-informative drop-out mechanisms. Using simulations, we investigate the percentage bias and mean-squared error (MSE) of the parameter estimates in the transition model under various assumptions for the drop-out. We find evidence for informative drop-out in the study, and we illustrate that the bias and MSE for the parameters of the transition model are not directly related to the observed drop-out or missing data rates. The effect of HIV status on the neurological impairment is found to be statistically significant under each of the models considered for the drop-out, although the regression coefficient may be biased in certain cases. The presence and relative magnitude of the bias depend on factors such as the probability of drop-out conditional on the presence of neurological impairment and the prevalence of neurological impairment in the population under study.     

A protective estimator for longitudinal binary data subject to non-ignorable non-monotone missingness

Summary.  In longitudinal studies missing data are the rule not the exception. We consider the analysis of longitudinal binary data with non-monotone missingness that is thought to be non-ignorable. In this setting a full likelihood approach is complicated algebraically and can be computationally prohibitive when there are many measurement occasions. We propose a 'protective' estimator that assumes that the probability that a response is missing at any occasion depends, in a completely unspecified way, on the value of that variable alone. Relying on this 'protectiveness' assumption, we describe a pseudolikelihood estimator of the regression parameters under non-ignorable missingness, without having to model the missing data mechanism directly. The method proposed is applied to CD4 cell count data from two longitudinal clinical trials of patients infected with the human immunodeficiency virus.     

The impact of dichotomization in longitudinal data analysis: a simulation study

In this paper, a simulation study is conducted to systematically investigate the impact of dichotomizing longitudinal continuous outcome variables under various types of missing data mechanisms. Generalized linear models (GLM) with standard generalized estimating equations (GEE) are widely used for longitudinal outcome analysis, but these semi‐parametric approaches are only valid under missing data completely at random (MCAR). Alternatively, weighted GEE (WGEE) and multiple imputation GEE (MI‐GEE) were developed to ensure validity under missing at random (MAR). Using a simulation study, the performance of standard GEE, WGEE and MI‐GEE on incomplete longitudinal dichotomized outcome analysis is evaluated. For comparisons, likelihood‐based linear mixed effects models (LMM) are used for incomplete longitudinal original continuous outcome analysis. Focusing on dichotomized outcome analysis, MI‐GEE with original continuous missing data imputation procedure provides well controlled test sizes and more stable power estimates compared with any other GEE‐based approaches. It is also shown that dichotomizing longitudinal continuous outcome will result in substantial loss of power compared with LMM. Copyright © 2009 John Wiley & Sons, Ltd.     

Handling of missing data in long‐term clinical trials: a case study

Missing data in clinical trials is a well‐known problem, and the classical statistical methods used can be overly simple. This case study shows how well‐established missing data theory can be applied to efficacy data collected in a long‐term open‐label trial with a discontinuation rate of almost 50%. Satisfaction with treatment in chronically constipated patients was the efficacy measure assessed at baseline and every 3 months postbaseline. The improvement in treatment satisfaction from baseline was originally analyzed with a paired t‐test ignoring missing data and discarding the correlation structure of the longitudinal data. As the original analysis started from missing completely at random assumptions regarding the missing data process, the satisfaction data were re‐examined, and several missing at random (MAR) and missing not at random (MNAR) techniques resulted in adjusted estimate for the improvement in satisfaction over 12 months. Throughout the different sensitivity analyses, the effect sizes remained significant and clinically relevant. Thus, even for an open‐label trial design, sensitivity analysis, with different assumptions for the nature of dropouts (MAR or MNAR) and with different classes of models (selection, pattern‐mixture, or multiple imputation models), has been found useful and provides evidence towards the robustness of the original analyses; additional sensitivity analyses could be undertaken to further qualify robustness. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonparametric Estimation of the Number of Drug Users in Hong Kong Using Repeated Multiple Lists

We update a previous approach to the estimation of the size of an open population when there are multiple lists at each time point. Our motivation is 35 years of longitudinal data on the detection of drug users by the Central Registry of Drug Abuse in Hong Kong. We develop a two‐stage smoothing spline approach. This gives a flexible and easily implemented alternative to the previous method which was based on kernel smoothing. The new method retains the property of reducing the variability of the individual estimates at each time point. We evaluate the new method by means of a simulation study that includes an examination of the effects of variable selection. The new method is then applied to data collected by the Central Registry of Drug Abuse. The parameter estimates obtained are compared with the well known Jolly–Seber estimates based on single capture methods.     

Extending the Mann–Whitney–Wilcoxon rank sum test to longitudinal regression analysis

Outliers are commonly observed in psychosocial research, generally resulting in biased estimates when comparing group differences using popular mean-based models such as the analysis of variance model. Rank-based methods such as the popular Mann–Whitney–Wilcoxon (MWW) rank sum test are more effective to address such outliers. However, available methods for inference are limited to cross-sectional data and cannot be applied to longitudinal studies under missing data. In this paper, we propose a generalized MWW test for comparing multiple groups with covariates within a longitudinal data setting, by utilizing the functional response models. Inference is based on a class of U-statistics-based weighted generalized estimating equations, providing consistent and asymptotically normal estimates not only under complete but missing data as well. The proposed approach is illustrated with both real and simulated study data.     

Linear Increments with Non‐monotone Missing Data and Measurement Error

Linear increments (LI) are used to analyse repeated outcome data with missing values. Previously, two LI methods have been proposed, one allowing non‐monotone missingness but not independent measurement error and one allowing independent measurement error but only monotone missingness. In both, it was suggested that the expected increment could depend on current outcome. We show that LI can allow non‐monotone missingness and either independent measurement error of unknown variance or dependence of expected increment on current outcome but not both. A popular alternative to LI is a multivariate normal model ignoring the missingness pattern. This gives consistent estimation when data are normally distributed and missing at random (MAR). We clarify the relation between MAR and the assumptions of LI and show that for continuous outcomes multivariate normal estimators are also consistent under (non‐MAR and non‐normal) assumptions not much stronger than those of LI. Moreover, when missingness is non‐monotone, they are typically more efficient.     

Estimation Bias in Complete-Case Analysis in Crossover Studies with Missing Data

Crossover designs are used often in clinical trials. It is not uncommon that subjects discontinue before completing all treatment periods in a crossover study. Despite availability of statistical methodologies utilizing all available data and software for obtaining valid inferences under the assumption of missing at random (MAR), naïve approaches, such as the complete case (CC) analysis, which is only valid with a strong assumption of missing completely at random are still widely used in practice. In this article, we obtain the analytical form of the estimation bias of treatment effects with CC for linear mixed models. We use simulation studies to examine the inflation of Type I error and efficiency loss in the inferences with CC under MAR. Invalidity and inefficiency of two other commonly used approaches for defining analyzed data in the presence of missing data, including data from at least two periods in three period crossover and available cases for a specific comparison of interest, are also demonstrated through simulation studies.     

Missing values: sparse inverse covariance estimation and?an?extension to sparse regression

We propose an ℓ ₁-regularized likelihood method for estimating the inverse covariance matrix in the high-dimensional multivariate normal model in presence of missing data. Our method is based on the assumption that the data are missing at random (MAR) which entails also the completely missing at random case. The implementation of the method is non-trivial as the observed negative log-likelihood generally is a complicated and non-convex function. We propose an efficient EM algorithm for optimization with provable numerical convergence properties. Furthermore, we extend the methodology to handle missing values in a sparse regression context. We demonstrate both methods on simulated and real data.     

A Bayesian shared parameter model for joint modeling of longitudinal continuous and binary outcomes

Joint modeling of associated mixed biomarkers in longitudinal studies leads to a better clinical decision by improving the efficiency of parameter estimates. In many clinical studies, the observed time for two biomarkers may not be equivalent and one of the longitudinal responses may have recorded in a longer time than the other one. In addition, the response variables may have different missing patterns. In this paper, we propose a new joint model of associated continuous and binary responses by accounting different missing patterns for two longitudinal outcomes. A conditional model for joint modeling of the two responses is used and two shared random effects models are considered for intermittent missingness of two responses. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is adopted for parameter estimation and model implementation. The validation and performance of the proposed model are investigated using some simulation studies. The proposed model is also applied for analyzing a real data set of bariatric surgery.     

A Bayesian Approach in Estimating Transition Probabilities of a Discrete-time Markov Chain for Ignorable Intermittent Missing Data

This article focuses on data analyses under the scenario of missing at random within discrete-time Markov chain models. The naive method, nonlinear (NL) method, and Expectation-Maximization (EM) algorithm are discussed. We extend the NL method into a Bayesian framework, using an adjusted rejection algorithm to sample the posterior distribution, and estimating the transition probabilities with a Monte Carlo algorithm. We compare the Bayesian nonlinear (BNL) method with the naive method and the EM algorithm with various missing rates, and comprehensively evaluate estimators in terms of biases, variances, mean square errors, and coverage probabilities (CPs). Our simulation results show that the EM algorithm usually offers smallest variances but with poorest CP, while the BNL method has smaller variances and better/similar CP as compared to the naive method. When the missing rate is low (about 9%, MAR), the three methods are comparable. Whereas when the missing rate is high (about 25%, MAR), overall, the BNL method performs slightly but consistently better than the naive method regarding variances and CP. Data from a longitudinal study of stress level among caregivers of individuals with Alzheimer’s disease is used to illustrate these methods.