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.     

The handling of missing data in trial-based economic evaluations: should data be multiply imputed prior to longitudinal linear mixed-model analyses?

Introduction

For the analysis of clinical effects, multiple imputation (MI) of missing data were shown to be unnecessary when using longitudinal linear mixed-models (LLM). It remains unclear whether this also applies to trial-based economic evaluations. Therefore, this study aimed to assess whether MI is required prior to LLM when analyzing longitudinal cost and effect data.

Methods

Two-thousand complete datasets were simulated containing five time points. Incomplete datasets were generated with 10, 25, and 50% missing data in follow-up costs and effects, assuming a Missing At Random (MAR) mechanism. Six different strategies were compared using empirical bias (EB), root-mean-squared error (RMSE), and coverage rate (CR). These strategies were: LLM alone (LLM) and MI with LLM (MI-LLM), and, as reference strategies, mean imputation with LLM (M-LLM), seemingly unrelated regression alone (SUR-CCA), MI with SUR (MI-SUR), and mean imputation with SUR (M-SUR).

Results

For costs and effects, LLM, MI-LLM, and MI-SUR performed better than M-LLM, SUR-CCA, and M-SUR, with smaller EBs and RMSEs as well as CRs closers to nominal levels. However, even though LLM, MI-LLM and MI-SUR performed equally well for effects, MI-LLM and MI-SUR were found to perform better than LLM for costs at 10 and 25% missing data. At 50% missing data, all strategies resulted in relatively high EBs and RMSEs for costs.

Conclusion

LLM should be combined with MI when analyzing trial-based economic evaluation data. MI-SUR is more efficient and can also be used, but then an average intervention effect over time cannot be estimated.

     

Diagnosing imputation models by applying target analyses to posterior replicates of completed data

Multiple imputation fills in missing data with posterior predictive draws from imputation models. To assess the adequacy of imputation models, we can compare completed data with their replicates simulated under the imputation model. We apply analyses of substantive interest to both datasets and use posterior predictive checks of the differences of these estimates to quantify the evidence of model inadequacy. We can further integrate out the imputed missing data and their replicates over the completed-data analyses to reduce variance in the comparison. In many cases, the checking procedure can be easily implemented using standard imputation software by treating re-imputations under the model as posterior predictive replicates. Thus, it can be applied for non-Bayesian imputation methods. We also sketch several strategies for applying the method in the context of practical imputation analyses. We illustrate the method using two real data applications and study its property using a simulation.     

Combining fractional polynomial model building with multiple imputation

Multivariable fractional polynomial (MFP) models are commonly used in medical research. The datasets in which MFP models are applied often contain covariates with missing values. To handle the missing values, we describe methods for combining multiple imputation with MFP modelling, considering in turn three issues: first, how to impute so that the imputation model does not favour certain fractional polynomial (FP) models over others; second, how to estimate the FP exponents in multiply imputed data; and third, how to choose between models of differing complexity. Two imputation methods are outlined for different settings. For model selection, methods based on Wald‐type statistics and weighted likelihood‐ratio tests are proposed and evaluated in simulation studies. The Wald‐based method is very slightly better at estimating FP exponents. Type I error rates are very similar for both methods, although slightly less well controlled than analysis of complete records; however, there is potential for substantial gains in power over the analysis of complete records. We illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published, contrasting the selected models with that obtained by analysing the complete records only. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

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.     

A bias‐corrected estimator in multiple imputation for missing data

Multiple imputation (MI) is one of the most popular methods to deal with missing data, and its use has been rapidly increasing in medical studies. Although MI is rather appealing in practice since it is possible to use ordinary statistical methods for a complete data set once the missing values are fully imputed, the method of imputation is still problematic. If the missing values are imputed from some parametric model, the validity of imputation is not necessarily ensured, and the final estimate for a parameter of interest can be biased unless the parametric model is correctly specified. Nonparametric methods have been also proposed for MI, but it is not so straightforward as to produce imputation values from nonparametrically estimated distributions. In this paper, we propose a new method for MI to obtain a consistent (or asymptotically unbiased) final estimate even if the imputation model is misspecified. The key idea is to use an imputation model from which the imputation values are easily produced and to make a proper correction in the likelihood function after the imputation by using the density ratio between the imputation model and the true conditional density function for the missing variable as a weight. Although the conditional density must be nonparametrically estimated, it is not used for the imputation. The performance of our method is evaluated by both theory and simulation studies. A real data analysis is also conducted to illustrate our method by using the Duke Cardiac Catheterization Coronary Artery Disease Diagnostic Dataset.     

Joint multiple imputation for longitudinal outcomes and clinical events that truncate longitudinal follow‐up

Longitudinal cohort studies often collect both repeated measurements of longitudinal outcomes and times to clinical events whose occurrence precludes further longitudinal measurements. Although joint modeling of the clinical events and the longitudinal data can be used to provide valid statistical inference for target estimands in certain contexts, the application of joint models in medical literature is currently rather restricted because of the complexity of the joint models and the intensive computation involved. We propose a multiple imputation approach to jointly impute missing data of both the longitudinal and clinical event outcomes. With complete imputed datasets, analysts are then able to use simple and transparent statistical methods and standard statistical software to perform various analyses without dealing with the complications of missing data and joint modeling. We show that the proposed multiple imputation approach is flexible and easy to implement in practice. Numerical results are also provided to demonstrate its performance. Copyright © 2015 John Wiley & Sons, Ltd.     

Combining multiple imputation and meta‐analysis with individual participant data

Multiple imputation is a strategy for the analysis of incomplete data such that the impact of the missingness on the power and bias of estimates is mitigated. When data from multiple studies are collated, we can propose both within‐study and multilevel imputation models to impute missing data on covariates. It is not clear how to choose between imputation models or how to combine imputation and inverse‐variance weighted meta‐analysis methods. This is especially important as often different studies measure data on different variables, meaning that we may need to impute data on a variable which is systematically missing in a particular study. In this paper, we consider a simulation analysis of sporadically missing data in a single covariate with a linear analysis model and discuss how the results would be applicable to the case of systematically missing data. We find in this context that ensuring the congeniality of the imputation and analysis models is important to give correct standard errors and confidence intervals. For example, if the analysis model allows between‐study heterogeneity of a parameter, then we should incorporate this heterogeneity into the imputation model to maintain the congeniality of the two models. In an inverse‐variance weighted meta‐analysis, we should impute missing data and apply Rubin's rules at the study level prior to meta‐analysis, rather than meta‐analyzing each of the multiple imputations and then combining the meta‐analysis estimates using Rubin's rules. We illustrate the results using data from the Emerging Risk Factors Collaboration. © 2013 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Missing data in a multi-item instrument were best handled by multiple imputation at the item score level

ObjectivesRegardless of the proportion of missing values, complete-case analysis is most frequently applied, although advanced techniques such as multiple imputation (MI) are available. The objective of this study was to explore the performance of simple and more advanced methods for handling missing data in cases when some, many, or all item scores are missing in a multi-item instrument.Study Design and SettingReal-life missing data situations were simulated in a multi-item variable used as a covariate in a linear regression model. Various missing data mechanisms were simulated with an increasing percentage of missing data. Subsequently, several techniques to handle missing data were applied to decide on the most optimal technique for each scenario. Fitted regression coefficients were compared using the bias and coverage as performance parameters.ResultsMean imputation caused biased estimates in every missing data scenario when data are missing for more than 10% of the subjects. Furthermore, when a large percentage of subjects had missing items (>25%), MI methods applied to the items outperformed methods applied to the total score.ConclusionWe recommend applying MI to the item scores to get the most accurate regression model estimates. Moreover, we advise not to use any form of mean imputation to handle missing data.     

Imputation-based strategies for clinical trial longitudinal data with nonignorable missing values

Biomedical research is plagued with problems of missing data, especially in clinical trials of medical and behavioral therapies adopting longitudinal design. After a literature review on modeling incomplete longitudinal data based on full-likelihood functions, this paper proposes a set of imputation-based strategies for implementing selection, pattern-mixture, and shared-parameter models for handling intermittent missing values and dropouts that are potentially nonignorable according to various criteria. Within the framework of multiple partial imputation, intermittent missing values are first imputed several times; then, each partially imputed data set is analyzed to deal with dropouts with or without further imputation. Depending on the choice of imputation model or measurement model, there exist various strategies that can be jointly applied to the same set of data to study the effect of treatment or intervention from multi-faceted perspectives. For illustration, the strategies were applied to a data set with continuous repeated measures from a smoking cessation clinical trial.     

Addressing missing data in confounders when estimating propensity scores for continuous exposures

Propensity score models are frequently used to estimate causal effects in observational studies. One unresolved issue in fitting these models is handling missing values in the propensity score model covariates. As these models usually contain a large set of covariates, using only individuals with complete data significantly decreases the sample size and statistical power. Several missing data imputation approaches have been proposed, including multiple imputation (MI), MI with missingness pattern (MIMP), and treatment mean imputation. Generalized boosted modeling (GBM), which is a nonparametric approach to estimate propensity scores, can automatically handle missingness in the covariates. Although the performance of MI, MIMP, and treatment mean imputation have previously been compared for binary treatments, they have not been compared for continuous exposures or with single imputation and GBM. We compared these approaches in estimating the generalized propensity score (GPS) for a continuous exposure in both a simulation study and in empirical data. Using GBM with the incomplete data to estimate the GPS did not perform well in the simulation. Missing values should be imputed before estimating propensity scores using GBM or any other approach for estimating the GPS.     

Properties of the full random-effect modeling approach with missing covariate data

During drug development, a key step is the identification of relevant covariates predicting between-subject variations in drug response. The full random effects model (FREM) is one of the full-covariate approaches used to identify relevant covariates in nonlinear mixed effects models. Here we explore the ability of FREM to handle missing (both missing completely at random (MCAR) and missing at random (MAR)) covariate data and compare it to the full fixed-effects model (FFEM) approach, applied either with complete case analysis or mean imputation. A global health dataset (20 421 children) was used to develop a FREM describing the changes of height for age Z-score (HAZ) over time. Simulated datasets (n = 1000) were generated with variable rates of missing (MCAR) covariate data (0%-90%) and different proportions of missing (MAR) data condition on either observed covariates or predicted HAZ. The three methods were used to re-estimate model and compared in terms of bias and precision which showed that FREM had only minor increases in bias and minor loss of precision at increasing percentages of missing (MCAR) covariate data and performed similarly in the MAR scenarios. Conversely, the FFEM approaches either collapsed at $$70% of missing (MCAR) covariate data (FFEM complete case analysis) or had large bias increases and loss of precision (FFEM with mean imputation). Our results suggest that FREM is an appropriate approach to covariate modeling for datasets with missing (MCAR and MAR) covariate data, such as in global health studies.     

多种缺失机制共存的定量纵向缺失数据处理方法的模拟比较研究

目的 比较在处理多种缺失机制共存的定量纵向缺失数据时,基于对照的模式混合模型（PMM）、重复测量的混合效应模型（MMRM）以及多重填补法(MI)的统计性能。方法 采用Monte Carlo技术模拟产生包含完全随机缺失、随机缺失和非随机缺失中两种或三种缺失机制的定量纵向缺失数据集,评价三类处理方法的统计性能。结果 基于对照的PMM控制Ⅰ类错误率在较低水平,检验效能最低。MMRM和MI的Ⅰ类错误率可控,检验效能高于基于对照的PMM。两组疗效无差异的情况下,所有方法的估计误差相当,基于对照的PMM方法的95%置信区间覆盖率最高;有差异的情况下,各方法受符合其缺失机制假设的缺失比例大小影响。含有非随机缺失数据时,基于对照的PMM基本不高估疗效差异,95%置信区间覆盖率最高,MMRM和MI高估疗效差异,95%置信区间覆盖率较低。所有方法的95%置信区间宽度相当。结论 分析多种缺失机制共存,特别是含有非随机缺失的纵向缺失数据时,MMRM和MI的统计性能有所降低,可采用基于对照的PMM进行敏感性分析,但需要注意其具体假设,防止估计过于保守。     

Binary variable multiple‐model multiple imputation to address missing data mechanism uncertainty: application to a smoking cessation trial

The true missing data mechanism is never known in practice. We present a method for generating multiple imputations for binary variables, which formally incorporates missing data mechanism uncertainty. Imputations are generated from a distribution of imputation models rather than a single model, with the distribution reflecting subjective notions of missing data mechanism uncertainty. Parameter estimates and standard errors are obtained using rules for nested multiple imputation. Using simulation, we investigate the impact of missing data mechanism uncertainty on post‐imputation inferences and show that incorporating this uncertainty can increase the coverage of parameter estimates. We apply our method to a longitudinal smoking cessation trial where nonignorably missing data were a concern. Our method provides a simple approach for formalizing subjective notions regarding nonresponse and can be implemented using existing imputation software. Copyright © 2014 John Wiley & Sons, Ltd.     

Missing phenotype data imputation in pedigree data analysis

Mapping complex traits or phenotypes with small genetic effects, whose phenotypes may be modulated by temporal trends in families are challenging. Detailed and accurate data must be available on families, whether or not the data were collected over time. Missing data complicate matters in pedigree analysis, especially in the case of a longitudinal pedigree analysis. Because most analytical methods developed for the analysis of longitudinal pedigree data require no missing data, the researcher is left with the option of dropping those cases (individuals) with missing data from the analysis or imputing values for the missing data. We present the use of data augmentation within Bayesian polygenic and longitudinal polygenic models to produce k complete datasets. The data augmentation, or imputation step of the Markov chain Monte Carlo, takes into account the observed familial information and the observed subject information available at other time points. These k complete datasets can then be used to fit single time point or longitudinal pedigree models. By producing a set of k complete datasets and thus k sets of parameter estimates, the total variance associated with an estimate can be partitioned into a within-imputation and a between-imputation component. The method is illustrated using the Genetic Analysis Workshop simulated data.     

Missing data strategies for time-varying confounders in comparative effectiveness studies of non-missing time-varying exposures and right-censored outcomes

The treatment of missing data in comparative effectiveness studies with right-censored outcomes and time-varying covariates is challenging because of the multilevel structure of the data. In particular, the performance of an accessible method like multiple imputation (MI) under an imputation model that ignores the multilevel structure is unknown and has not been compared to complete-case (CC) and single imputation methods that are most commonly applied in this context. Through an extensive simulation study, we compared statistical properties among CC analysis, last value carried forward, mean imputation, the use of missing indicators, and MI-based approaches with and without auxiliary variables under an extended Cox model when the interest lies in characterizing relationships between non-missing time-varying exposures and right-censored outcomes. MI demonstrated favorable properties under a moderate missing-at-random condition (absolute bias <0.1) and outperformed CC and single imputation methods, even when the MI method did not account for correlated observations in the imputation model. The performance of MI decreased with increasing complexity such as when the missing data mechanism involved the exposure of interest, but was still preferred over other methods considered and performed well in the presence of strong auxiliary variables. We recommend considering MI that ignores the multilevel structure in the imputation model when data are missing in a time-varying confounder, incorporating variables associated with missingness in the MI models as well as conducting sensitivity analyses across plausible assumptions.     

Multiple imputation for handling systematically missing confounders in meta‐analysis of individual participant data

A variable is ‘systematically missing’ if it is missing for all individuals within particular studies in an individual participant data meta‐analysis. When a systematically missing variable is a potential confounder in observational epidemiology, standard methods either fail to adjust the exposure–disease association for the potential confounder or exclude studies where it is missing. We propose a new approach to adjust for systematically missing confounders based on multiple imputation by chained equations. Systematically missing data are imputed via multilevel regression models that allow for heterogeneity between studies. A simulation study compares various choices of imputation model. An illustration is given using data from eight studies estimating the association between carotid intima media thickness and subsequent risk of cardiovascular events. Results are compared with standard methods and also with an extension of a published method that exploits the relationship between fully adjusted and partially adjusted estimated effects through a multivariate random effects meta‐analysis model. We conclude that multiple imputation provides a practicable approach that can handle arbitrary patterns of systematic missingness. Bias is reduced by including sufficient between‐study random effects in the imputation model. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple imputation for an incomplete covariate that is a ratio

We are concerned with multiple imputation of the ratio of two variables, which is to be used as a covariate in a regression analysis. If the numerator and denominator are not missing simultaneously, it seems sensible to make use of the observed variable in the imputation model. One such strategy is to impute missing values for the numerator and denominator, or the log‐transformed numerator and denominator, and then calculate the ratio of interest; we call this ‘passive’ imputation. Alternatively, missing ratio values might be imputed directly, with or without the numerator and/or the denominator in the imputation model; we call this ‘active’ imputation. In two motivating datasets, one involving body mass index as a covariate and the other involving the ratio of total to high‐density lipoprotein cholesterol, we assess the sensitivity of results to the choice of imputation model and, as an alternative, explore fully Bayesian joint models for the outcome and incomplete ratio. Fully Bayesian approaches using Winbugs were unusable in both datasets because of computational problems. In our first dataset, multiple imputation results are similar regardless of the imputation model; in the second, results are sensitive to the choice of imputation model. Sensitivity depends strongly on the coefficient of variation of the ratio's denominator. A simulation study demonstrates that passive imputation without transformation is risky because it can lead to downward bias when the coefficient of variation of the ratio's denominator is larger than about 0.1. Active imputation or passive imputation after log‐transformation is preferable. © 2013 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Variable selection for multiply‐imputed data with application to dioxin exposure study

Multiple imputation (MI) is a commonly used technique for handling missing data in large‐scale medical and public health studies. However, variable selection on multiply‐imputed data remains an important and longstanding statistical problem. If a variable selection method is applied to each imputed dataset separately, it may select different variables for different imputed datasets, which makes it difficult to interpret the final model or draw scientific conclusions. In this paper, we propose a novel multiple imputation‐least absolute shrinkage and selection operator (MI‐LASSO) variable selection method as an extension of the least absolute shrinkage and selection operator (LASSO) method to multiply‐imputed data. The MI‐LASSO method treats the estimated regression coefficients of the same variable across all imputed datasets as a group and applies the group LASSO penalty to yield a consistent variable selection across multiple‐imputed datasets. We use a simulation study to demonstrate the advantage of the MI‐LASSO method compared with the alternatives. We also apply the MI‐LASSO method to the University of Michigan Dioxin Exposure Study to identify important circumstances and exposure factors that are associated with human serum dioxin concentration in Midland, Michigan. Copyright © 2013 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.