Performance of weighted estimating equations for longitudinal binary data with drop-outs missing at random

The generalized estimating equations (GEE) approach is commonly used to model incomplete longitudinal binary data. When drop-outs are missing at random through dependence on observed responses (MAR), GEE may give biased parameter estimates in the model for the marginal means. A weighted estimating equations approach gives consistent estimation under MAR when the drop-out mechanism is correctly specified. In this approach, observations or person-visits are weighted inversely proportional to their probability of being observed. Using a simulation study, we compare the performance of unweighted and weighted GEE in models for time-specific means of a repeated binary response with MAR drop-outs. Weighted GEE resulted in smaller finite sample bias than GEE. However, when the drop-out model was misspecified, weighted GEE sometimes performed worse than GEE. Weighted GEE with observation-level weights gave more efficient estimates than a weighted GEE procedure with cluster-level weights.     

Simple generalized estimating equations (GEEs) and weighted generalized estimating equations (WGEEs) in longitudinal studies with dropouts: guidelines and implementation in R

Missing data are a common problem in clinical and epidemiological research, especially in longitudinal studies. Despite many methodological advances in recent decades, many papers on clinical trials and epidemiological studies do not report using principled statistical methods to accommodate missing data or use ineffective or inappropriate techniques. Two refined techniques are presented here: generalized estimating equations (GEEs) and weighted generalized estimating equations (WGEEs). These techniques are an extension of generalized linear models to longitudinal or clustered data, where observations are no longer independent. They can appropriately handle missing data when the missingness is completely at random (GEE and WGEE) or at random (WGEE) and do not require the outcome to be normally distributed. Our aim is to describe and illustrate with a real example, in a simple and accessible way to researchers, these techniques for handling missing data in the context of longitudinal studies subject to dropout and show how to implement them in R. We apply them to assess the evolution of health‐related quality of life in coronary patients in a data set subject to dropout. Copyright © 2016 John Wiley & Sons, Ltd.     

Attrition in longitudinal studies. How to deal with missing data

The purpose of this paper was to illustrate the influence of missing data on the results of longitudinal statistical analyses [i.e., MANOVA for repeated measurements and Generalised Estimating Equations (GEE)] and to illustrate the influence of using different imputation methods to replace missing data. Besides a complete dataset, four incomplete datasets were considered: two datasets with 10% missing data and two datasets with 25% missing data. In both situations missingness was considered independent and dependent on observed data. Imputation methods were divided into cross-sectional methods (i.e., mean of series, hot deck, and cross-sectional regression) and longitudinal methods (i.e., last value carried forward, longitudinal interpolation, and longitudinal regression). Besides these, also the multiple imputation method was applied and discussed. The analyses were performed on a particular (observational) longitudinal dataset, with particular missing data patterns and imputation methods. The results of this illustration shows that when MANOVA for repeated measurements is used, imputation methods are highly recommendable (because MANOVA as implemented in the software used, uses listwise deletion of cases with a missing value). Applying GEE analysis, imputation methods were not necessary. When imputation methods were used, longitudinal imputation methods were often preferable above cross-sectional imputation methods, in a way that the point estimates and standard errors were closer to the estimates derived from the complete dataset. Furthermore, this study showed that the theoretically more valid multiple imputation method did not lead to different point estimates than the more simple (longitudinal) imputation methods. However, the estimated standard errors appeared to be theoretically more adequate, because they reflect the uncertainty in estimation caused by missing values.     

Methods to Account for Attrition in Longitudinal Data: Do They Work? A Simulation Study

Attrition threatens the internal validity of cohort studies. Epidemiologists use various imputation and weighting methods to limit bias due to attrition. However, the ability of these methods to correct for attrition bias has not been tested. We simulated a cohort of 300 subjects using 500 computer replications to determine whether regression imputation, individual weighting, or multiple imputation is useful to reduce attrition bias. We compared these results to a complete subject analysis. Our logistic regression model included a binary exposure and two confounders. We generated 10, 25, and 40% attrition through three missing data mechanisms: missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR), and used four covariance matrices to vary attrition. We compared true and estimated mean odds ratios (ORs), standard deviations (SDs), and coverage. With data MCAR and MAR for all attrition rates, the complete subject analysis produced results at least as valid as those from the imputation and weighting methods. With data MNAR, no method provided unbiased estimates of the OR at attrition rates of 25 or 40%. When observations are not MAR or MCAR, imputation and weighting methods may not effectively reduce attrition bias.     

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.     

Joint Longitudinal Models for Dealing With Missing at Random Data in Trial-Based Economic Evaluations

ObjectivesIn trial-based economic evaluation, some individuals are typically associated with missing data at some time point, so that their corresponding aggregated outcomes (eg, quality-adjusted life-years) cannot be evaluated. Restricting the analysis to the complete cases is inefficient and can result in biased estimates, while imputation methods are often implemented under a missing at random (MAR) assumption. We propose the use of joint longitudinal models to extend standard approaches by taking into account the longitudinal structure to improve the estimation of the targeted quantities under MAR.MethodsWe compare the results from methods that handle missingness at an aggregated (case deletion, baseline imputation, and joint aggregated models) and disaggregated (joint longitudinal models) level under MAR. The methods are compared using a simulation study and applied to data from 2 real case studies.ResultsSimulations show that, according to which data affect the missingness process, aggregated methods may lead to biased results, while joint longitudinal models lead to valid inferences under MAR. The analysis of the 2 case studies support these results as both parameter estimates and cost-effectiveness results vary based on the amount of data incorporated into the model.ConclusionsOur analyses suggest that methods implemented at the aggregated level are potentially biased under MAR as they ignore the information from the partially observed follow-up data. This limitation can be overcome by extending the analysis to a longitudinal framework using joint models, which can incorporate all the available evidence.     

Meta‐analysis with missing study‐level sample variance data

We consider a study‐level meta‐analysis with a normally distributed outcome variable and possibly unequal study‐level variances, where the object of inference is the difference in means between a treatment and control group. A common complication in such an analysis is missing sample variances for some studies. A frequently used approach is to impute the weighted (by sample size) mean of the observed variances (mean imputation). Another approach is to include only those studies with variances reported (complete case analysis). Both mean imputation and complete case analysis are only valid under the missing‐completely‐at‐random assumption, and even then the inverse variance weights produced are not necessarily optimal. We propose a multiple imputation method employing gamma meta‐regression to impute the missing sample variances. Our method takes advantage of study‐level covariates that may be used to provide information about the missing data. Through simulation studies, we show that multiple imputation, when the imputation model is correctly specified, is superior to competing methods in terms of confidence interval coverage probability and type I error probability when testing a specified group difference. Finally, we describe a similar approach to handling missing variances in cross‐over studies. Copyright © 2016 John Wiley & Sons, Ltd.     

Recovery of information from multiple imputation: a simulation study

ABSTRACT: BACKGROUND: Multiple imputation is becoming increasingly popular for handling missing data. However, it is often implemented without adequate consideration of whether it offers any advantage over complete case analysis for the research question of interest, or whether potential gains may be offset by bias from a poorly fitting imputation model, particularly as the amount of missing data increases. METHODS: Simulated datasets (n = 1000) drawn from a synthetic population were used to explore information recovery from multiple imputation in estimating the coefficient of a binary exposure variable when various proportions of data (10-90%) were set missing at random in a highly-skewed continuous covariate or in the binary exposure. Imputation was performed using multivariate normal imputation (MVNI), with a simple or zero-skewness log transformation to manage non-normality. Bias, precision, mean-squared error and coverage for a set of regression parameter estimates were compared between multiple imputation and complete case analyses. RESULTS: For missingness in the continuous covariate, multiple imputation produced less bias and greater precision for the effect of the binary exposure variable, compared with complete case analysis, with larger gains in precision with more missing data. However, even with only moderate missingness, large bias and substantial under-coverage were apparent in estimating the continuous covariate's effect when skewness was not adequately addressed. For missingness in the binary covariate, all estimates had negligible bias but gains in precision from multiple imputation were minimal, particularly for the coefficient of the binary exposure. CONCLUSIONS: Although multiple imputation can be useful if covariates required for confounding adjustment are missing, benefits are likely to be minimal when data are missing in the exposure variable of interest. Furthermore, when there are large amounts of missingness, multiple imputation can become unreliable and introduce bias not present in a complete case analysis if the imputation model is not appropriate. Epidemiologists dealing with missing data should keep in mind the potential limitations as well as the potential benefits of multiple imputation. Further work is needed to provide clearer guidelines on effective application of this method.     

Sensitivity analysis for the estimation of rates of change with non-ignorable drop-out: an application to a randomized clinical trial of the vitamin D3

The vitamin D(3) trial was a repeated measures randomized clinical trial for secondary hyperparathyroidism in haemodialysis patients where the efficacy of the vitamin D(3) infusions for suppressing the secretion of parathyroid hormone (PTH) was compared among four dose groups over 12 weeks. In this trial, patients terminated the study before the scheduled end of the study due to their elevated serum calcium (Ca) level, that is, the administration of the vitamin D(3) was expected to cause hypercalcaemia as an adverse event. In this setting of monotone missingness, there is a potential for bias in estimation of mean rates of decline in PTH for each treatment group using the standard methods such as the generalized estimating equations (GEE) which ignore the observed past Ca histories. We estimated the treatment-group-specific mean rates of decline in PTH by the inverse probability of censoring weighted (IPCW) methods which account for the observed past histories of time-dependent factors that are both a predictor of drop-out and are correlated with the outcomes. The IPCW estimator can be viewed as an extension of the GEE estimator that allows for the data to be MAR but not MCAR. With missing data, it is rarely appropriate to analyse the data solely under the assumption that the missing data process is ignorable, because the assumption of ignorable missingness cannot be guaranteed to hold and is untestable from the observed data. We proposed a sensitivity analysis that examines how inference about the IPCW estimates of the treatment-group-specific mean rates of decline in PTH changes as we vary the non-ignorable selection bias parameter over a range of plausible values.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

L’imputation multiple des données manquantes aléatoirement : concepts généraux et présentation d’une méthode Monte-Carlo

BackgroundStatistical analysis of a data set with missing data is a frequent problem to deal with in epidemiology. Methods are available to manage incomplete observations, avoiding biased estimates and improving their precision, compared to more traditional methods, such as the analysis of the sub-sample of complete observations.MethodsOne of these approaches is multiple imputation, which consists in imputing successively several values for each missing data item. Several completed data sets having the same distribution characteristics as the observed data (variability and correlations) are thus generated. Standard analyses are done separately on each completed dataset then combined to obtain a global result. In this paper, we discuss the various assumptions made on the origin of missing data (at random or not), and we present in a pragmatic way the process of multiple imputation. A recent method, Multiple Imputation by Chained Equations (MICE), based on a Monte-Carlo Markov Chain algorithm under missing at random data (MAR) hypothesis, is described. An illustrative example of the MICE method is detailed for the analysis of the relation between a dichotomous variable and two covariates presenting MAR data with no particular structure, through multivariate logistic regression.ResultsCompared with the original dataset without missing data, the results show a substantial improvement of the regression coefficient estimates with the MICE method, relatively to those obtained on the dataset with complete observations.ConclusionThis method does not require any direct assumption on joint distribution of the variables and it is presently implemented in standard statistical software (Splus, Stata). It can be used for multiple imputation of missing data of several variables with no particular structure.     

An autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes

We are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient, and many patients drop out with a reason related to the response. In this paper, we focus on a model that assumes a dropout process is missing at random (MAR). In this dropout process, we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified. However, parsimonious covariance structures for the profiles approaching asymptotes are unclear. An autoregressive linear mixed effects model can express the profile with random individual asymptotes. We show that this model provides a new parsimonious covariance structure. The covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points. In simulation studies, the estimate of the asymptote is unbiased in MAR dropouts, but biased in non-ignorable dropouts. We also applied this model to actual schizophrenia trial data.     

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.     

A new estimation with minimum trace of asymptotic covariance matrix for incomplete longitudinal data with a surrogate process

Missing data is a very common problem in medical and social studies, especially when data are collected longitudinally. It is a challenging problem to utilize observed data effectively. Many papers on missing data problems can be found in statistical literature. It is well known that the inverse weighted estimation is neither efficient nor robust. On the other hand, the doubly robust (DR) method can improve the efficiency and robustness. As is known, the DR estimation requires a missing data model (i.e., a model for the probability that data are observed) and a working regression model (i.e., a model for the outcome variable given covariates and surrogate variables). Because the DR estimating function has mean zero for any parameters in the working regression model when the missing data model is correctly specified, in this paper, we derive a formula for the estimator of the parameters of the working regression model that yields the optimally efficient estimator of the marginal mean model (the parameters of interest) when the missing data model is correctly specified. Furthermore, the proposed method also inherits the DR property. Simulation studies demonstrate the greater efficiency of the proposed method compared with the standard DR method. A longitudinal dementia data set is used for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Review: a gentle introduction to imputation of missing values

In most situations, simple techniques for handling missing data (such as complete case analysis, overall mean imputation, and the missing-indicator method) produce biased results, whereas imputation techniques yield valid results without complicating the analysis once the imputations are carried out. Imputation techniques are based on the idea that any subject in a study sample can be replaced by a new randomly chosen subject from the same source population. Imputation of missing data on a variable is replacing that missing by a value that is drawn from an estimate of the distribution of this variable. In single imputation, only one estimate is used. In multiple imputation, various estimates are used, reflecting the uncertainty in the estimation of this distribution. Under the general conditions of so-called missing at random and missing completely at random, both single and multiple imputations result in unbiased estimates of study associations. But single imputation results in too small estimated standard errors, whereas multiple imputation results in correctly estimated standard errors and confidence intervals. In this article we explain why all this is the case, and use a simple simulation study to demonstrate our explanations. We also explain and illustrate why two frequently used methods to handle missing data, i.e., overall mean imputation and the missing-indicator method, almost always result in biased estimates.     

Explicating the Conditions Under Which Multilevel Multiple Imputation Mitigates Bias Resulting from Random Coefficient-Dependent Missing Longitudinal Data

Random coefficient-dependent (RCD) missingness is a non-ignorable mechanism through which missing data can arise in longitudinal designs. RCD, for which we cannot test, is a problematic form of missingness that occurs if subject-specific random effects correlate with propensity for missingness or dropout. Particularly when covariate missingness is a problem, investigators typically handle missing longitudinal data by using single-level multiple imputation procedures implemented with long-format data, which ignores within-person dependency entirely, or implemented with wide-format (i.e., multivariate) data, which ignores some aspects of within-person dependency. When either of these standard approaches to handling missing longitudinal data is used, RCD missingness leads to parameter bias and incorrect inference. We explain why multilevel multiple imputation (MMI) should alleviate bias induced by a RCD missing data mechanism under conditions that contribute to stronger determinacy of random coefficients. We evaluate our hypothesis with a simulation study. Three design factors are considered: intraclass correlation (ICC; ranging from .25 to .75), number of waves (ranging from 4 to 8), and percent of missing data (ranging from 20 to 50%). We find that MMI greatly outperforms the single-level wide-format (multivariate) method for imputation under a RCD mechanism. For the MMI analyses, bias was most alleviated when the ICC is high, there were more waves of data, and when there was less missing data. Practical recommendations for handling longitudinal missing data are suggested.     

Out of sight, not out of mind: strategies for handling missing data

OBJECTIVE: To describe and illustrate missing data mechanisms (MCAR, MAR, NMAR) and missing data techniques (MDTs) and offer recommended best practices for addressing missingness. METHOD: We simulated data sets and employed ad hoc MDTs (deletion techniques, mean substitution) and sophisticated MDTs (full information maximum likelihood, Bayesian estimation, multiple imputation) in linear regression analyses. RESULTS: MCAR data yielded unbiased parameter estimates across all MDTs, but loss of power with deletion methods. NMAR results were biased towards larger values and greater significance. Under MAR the sophisticated MDTs returned estimates closer to their original values. CONCLUSION: State-of-the-art, readily available MDTs outperform ad hoc techniques.     

Multinomial logistic regression with missing outcome data: An application to cancer subtypes

Many diseases such as cancer and heart diseases are heterogeneous and it is of great interest to study the disease risk specific to the subtypes in relation to genetic and environmental risk factors. However, due to logistic and cost reasons, the subtype information for the disease is missing for some subjects. In this article, we investigate methods for multinomial logistic regression with missing outcome data, including a bootstrap hot deck multiple imputation (BHMI), simple inverse probability weighted (SIPW), augmented inverse probability weighted (AIPW), and expected estimating equation (EEE) estimators. These methods are important approaches for missing data regression. The BHMI modifies the standard hot deck multiple imputation method such that it can provide valid confidence interval estimation. Under the situation when the covariates are discrete, the SIPW, AIPW, and EEE estimators are numerically identical. When the covariates are continuous, nonparametric smoothers can be applied to estimate the selection probabilities and the estimating scores. These methods perform similarly. Extensive simulations show that all of these methods yield unbiased estimators while the complete-case (CC) analysis can be biased if the missingness depends on the observed data. Our simulations also demonstrate that these methods can gain substantial efficiency compared with the CC analysis. The methods are applied to a colorectal cancer study in which cancer subtype data are missing among some study individuals.     

An alternative empirical likelihood method in missing response problems and causal inference

Missing responses are common problems in medical, social, and economic studies. When responses are missing at random, a complete case data analysis may result in biases. A popular debias method is inverse probability weighting proposed by Horvitz and Thompson. To improve efficiency, Robins et al. proposed an augmented inverse probability weighting method. The augmented inverse probability weighting estimator has a double‐robustness property and achieves the semiparametric efficiency lower bound when the regression model and propensity score model are both correctly specified. In this paper, we introduce an empirical likelihood‐based estimator as an alternative to Qin and Zhang (2007). Our proposed estimator is also doubly robust and locally efficient. Simulation results show that the proposed estimator has better performance when the propensity score is correctly modeled. Moreover, the proposed method can be applied in the estimation of average treatment effect in observational causal inferences. Finally, we apply our method to an observational study of smoking, using data from the Cardiovascular Outcomes in Renal Atherosclerotic Lesions clinical trial. Copyright © 2016 John Wiley & Sons, Ltd.     

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.