Multiple imputation methods for handling incomplete longitudinal and clustered data where the target analysis is a linear mixed effects model

Multiple imputation (MI) is increasingly popular for handling multivariate missing data. Two general approaches are available in standard computer packages: MI based on the posterior distribution of incomplete variables under a multivariate (joint) model, and fully conditional specification (FCS), which imputes missing values using univariate conditional distributions for each incomplete variable given all the others, cycling iteratively through the univariate imputation models. In the context of longitudinal or clustered data, it is not clear whether these approaches result in consistent estimates of regression coefficient and variance component parameters when the analysis model of interest is a linear mixed effects model (LMM) that includes both random intercepts and slopes with either covariates or both covariates and outcome contain missing information. In the current paper, we compared the performance of seven different MI methods for handling missing values in longitudinal and clustered data in the context of fitting LMMs with both random intercepts and slopes. We study the theoretical compatibility between specific imputation models fitted under each of these approaches and the LMM, and also conduct simulation studies in both the longitudinal and clustered data settings. Simulations were motivated by analyses of the association between body mass index (BMI) and quality of life (QoL) in the Longitudinal Study of Australian Children (LSAC). Our findings showed that the relative performance of MI methods vary according to whether the incomplete covariate has fixed or random effects and whether there is missingnesss in the outcome variable. We showed that compatible imputation and analysis models resulted in consistent estimation of both regression parameters and variance components via simulation. We illustrate our findings with the analysis of LSAC data.     

Multiple Imputation Approaches for the Analysis of Dichotomized Responses in Longitudinal Studies with Missing Data

Summary Often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value. In the event that the underlying continuous measurements are from a longitudinal study, one can use the repeated‐measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status. Standard Bayesian multiple imputation techniques ( Rubin, 1987 , in Multiple Imputation for Nonresponse in Surveys) that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within‐ and between‐imputation variances are found to be conservative. The frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of Robins and Wang (2000, Biometrika 87, 113–124) is shown to be more efficient. We propose to apply ( Kenward and Roger, 1997 , Biometrics 53, 983–997) degrees of freedom to account for the uncertainty associated with variance–covariance parameter estimates for the repeated measures model.     

Multiple imputation for discrete data: Evaluation of the joint latent normal model

Missing data are ubiquitous in clinical and social research, and multiple imputation (MI) is increasingly the methodology of choice for practitioners. Two principal strategies for imputation have been proposed in the literature: joint modelling multiple imputation (JM‐MI) and full conditional specification multiple imputation (FCS‐MI). While JM‐MI is arguably a preferable approach, because it involves specification of an explicit imputation model, FCS‐MI is pragmatically appealing, because of its flexibility in handling different types of variables. JM‐MI has developed from the multivariate normal model, and latent normal variables have been proposed as a natural way to extend this model to handle categorical variables. In this article, we evaluate the latent normal model through an extensive simulation study and an application on data from the German Breast Cancer Study Group, comparing the results with FCS‐MI. We divide our investigation in four sections, focusing on (i) binary, (ii) categorical, (iii) ordinal, and (iv) count data. Using data simulated from both the latent normal model and the general location model, we find that in all but one extreme general location model setting JM‐MI works very well, and sometimes outperforms FCS‐MI. We conclude the latent normal model, implemented in the R package jomo , can be used with confidence by researchers, both for single and multilevel multiple imputation.     

Approaches for missing covariate data in logistic regression with MNAR sensitivity analyses

Data with missing covariate values but fully observed binary outcomes are an important subset of the missing data challenge. Common approaches are complete case analysis (CCA) and multiple imputation (MI). While CCA relies on missing completely at random (MCAR), MI usually relies on a missing at random (MAR) assumption to produce unbiased results. For MI involving logistic regression models, it is also important to consider several missing not at random (MNAR) conditions under which CCA is asymptotically unbiased and, as we show, MI is also valid in some cases. We use a data application and simulation study to compare the performance of several machine learning and parametric MI methods under a fully conditional specification framework (MI-FCS). Our simulation includes five scenarios involving MCAR, MAR, and MNAR under predictable and nonpredictable conditions, where “predictable” indicates missingness is not associated with the outcome. We build on previous results in the literature to show MI and CCA can both produce unbiased results under more conditions than some analysts may realize. When both approaches were valid, we found that MI-FCS was at least as good as CCA in terms of estimated bias and coverage, and was superior when missingness involved a categorical covariate. We also demonstrate how MNAR sensitivity analysis can build confidence that unbiased results were obtained, including under MNAR-predictable, when CCA and MI are both valid. Since the missingness mechanism cannot be identified from observed data, investigators should compare results from MI and CCA when both are plausibly valid, followed by MNAR sensitivity analysis.     

Rounding Strategies for Multiply Imputed Binary Data

Multiple imputation (MI) has emerged in the last two decades as a frequently used approach in dealing with incomplete data. Gaussian and log‐linear imputation models are fairly straightforward to implement for continuous and discrete data, respectively. However, in missing data settings that include a mix of continuous and discrete variables, the lack of flexible models for the joint distribution of different types of variables can make the specification of the imputation model a daunting task. The widespread availability of software packages that are capable of carrying out MI under the assumption of joint multivariate normality allows applied researchers to address this complication pragmatically by treating the discrete variables as continuous for imputation purposes and subsequently rounding the imputed values to the nearest observed category. In this article, we compare several rounding rules for binary variables based on simulated longitudinal data sets that have been used to illustrate other missing‐data techniques. Using a combination of conditional and marginal data generation mechanisms and imputation models, we study the statistical properties of multiple‐imputation‐based estimates for various population quantities under different rounding rules from bias and coverage standpoints. We conclude that a good rule should be driven by borrowing information from other variables in the system rather than relying on the marginal characteristics and should be relatively insensitive to imputation model specifications that may potentially be incompatible with the observed data. We also urge researchers to consider the applied context and specific nature of the problem, to avoid uncritical and possibly inappropriate use of rounding in imputation models.     

Imputation and variable selection in linear regression models with missing covariates

Across multiply imputed data sets, variable selection methods such as stepwise regression and other criterion-based strategies that include or exclude particular variables typically result in models with different selected predictors, thus presenting a problem for combining the results from separate complete-data analyses. Here, drawing on a Bayesian framework, we propose two alternative strategies to address the problem of choosing among linear regression models when there are missing covariates. One approach, which we call "impute, then select" (ITS) involves initially performing multiple imputation and then applying Bayesian variable selection to the multiply imputed data sets. A second strategy is to conduct Bayesian variable selection and missing data imputation simultaneously within one Gibbs sampling process, which we call "simultaneously impute and select" (SIAS). The methods are implemented and evaluated using the Bayesian procedure known as stochastic search variable selection for multivariate normal data sets, but both strategies offer general frameworks within which different Bayesian variable selection algorithms could be used for other types of data sets. A study of mental health services utilization among children in foster care programs is used to illustrate the techniques. Simulation studies show that both ITS and SIAS outperform complete-case analysis with stepwise variable selection and that SIAS slightly outperforms ITS.     

Model selection of generalized estimating equations with multiply imputed longitudinal data

Longitudinal data often encounter missingness with monotone and/or intermittent missing patterns. Multiple imputation (MI) has been popularly employed for analysis of missing longitudinal data. In particular, the MI‐GEE method has been proposed for inference of generalized estimating equations (GEE) when missing data are imputed via MI. However, little is known about how to perform model selection with multiply imputed longitudinal data. In this work, we extend the existing GEE model selection criteria, including the “quasi‐likelihood under the independence model criterion” (QIC) and the “missing longitudinal information criterion” (MLIC), to accommodate multiple imputed datasets for selection of the MI‐GEE mean model. According to real data analyses from a schizophrenia study and an AIDS study, as well as simulations under nonmonotone missingness with moderate proportion of missing observations, we conclude that: (i) more than a few imputed datasets are required for stable and reliable model selection in MI‐GEE analysis; (ii) the MI‐based GEE model selection methods with a suitable number of imputations generally perform well, while the naive application of existing model selection methods by simply ignoring missing observations may lead to very poor performance; (iii) the model selection criteria based on improper (frequentist) multiple imputation generally performs better than their analogies based on proper (Bayesian) multiple imputation.     

Local average treatment effects estimation via substantive model compatible multiple imputation

Nonadherence to assigned treatment is common in randomized controlled trials (RCTs). Recently, there has been increased interest in estimating causal effects of treatment received, for example, the so‐called local average treatment effect (LATE). Instrumental variables (IV) methods can be used for identification, with estimation proceeding either via fully parametric mixture models or two‐stage least squares (TSLS). TSLS is popular but can be problematic for binary outcomes where the estimand of interest is a causal odds ratio. Mixture models are rarely used in practice, perhaps because of their perceived complexity and need for specialist software. Here, we propose using multiple imputation (MI) to impute the latent compliance class appearing in the mixture models. Since such models include an interaction term between the latent compliance class and randomized treatment, we use “substantive model compatible” MI (SMC MIC), which can additionally handle missing data in outcomes and other variables in the model, before fitting the mixture models via maximum likelihood to the MI data sets and combining results via Rubin's rules. We use simulations to compare the performance of SMC MIC to existing approaches and also illustrate the methods by reanalyzing an RCT in UK primary health. We show that SMC MIC can be more efficient than full Bayesian estimation when auxiliary variables are incorporated, and is superior to two‐stage methods, especially for binary outcomes.     

Analysis of Longitudinal Clinical Trials with Missing Data Using Multiple Imputation in Conjunction with Robust Regression

Summary In a typical randomized clinical trial, a continuous variable of interest (e.g., bone density) is measured at baseline and fixed postbaseline time points. The resulting longitudinal data, often incomplete due to dropouts and other reasons, are commonly analyzed using parametric likelihood‐based methods that assume multivariate normality of the response vector. If the normality assumption is deemed untenable, then semiparametric methods such as (weighted) generalized estimating equations are considered. We propose an alternate approach in which the missing data problem is tackled using multiple imputation, and each imputed dataset is analyzed using robust regression (M‐estimation; Huber, 1973 , Annals of Statistics 1, 799–821.) to protect against potential non‐normality/outliers in the original or imputed dataset. The robust analysis results from each imputed dataset are combined for overall estimation and inference using either the simple Rubin (1987 , Multiple Imputation for Nonresponse in Surveys, New York: Wiley) method, or the more complex but potentially more accurate Robins and Wang (2000 , Biometrika 87, 113–124.) method. We use simulations to show that our proposed approach performs at least as well as the standard methods under normality, but is notably better under both elliptically symmetric and asymmetric non‐normal distributions. A clinical trial example is used for illustration.     

Evaluation of measures of correctness of genotype imputation in the context of genomic prediction: a review of livestock applications

In livestock, many studies have reported the results of imputation to 50k single nucleotide polymorphism (SNP) genotypes for animals that are genotyped with low-density SNP panels. The objective of this paper is to review different measures of correctness of imputation, and to evaluate their utility depending on the purpose of the imputed genotypes. Across studies, imputation accuracy, computed as the correlation between true and imputed genotypes, and imputation error rates, that counts the number of incorrectly imputed alleles, are commonly used measures of imputation correctness. Based on the nature of both measures and results reported in the literature, imputation accuracy appears to be a more useful measure of the correctness of imputation than imputation error rates, because imputation accuracy does not depend on minor allele frequency (MAF), whereas imputation error rate depends on MAF. Therefore imputation accuracy can be better compared across loci with different MAF. Imputation accuracy depends on the ability of identifying the correct haplotype of a SNP, but many other factors have been identified as well, including the number of genotyped immediate ancestors, the number of animals with genotypes at the high-density panel, the SNP density on the low- and high-density panel, the MAF of the imputed SNP and whether imputed SNP are located at the end of a chromosome or not. Some of these factors directly contribute to the linkage disequilibrium between imputed SNP and SNP on the low-density panel. When imputation accuracy is assessed as a predictor for the accuracy of subsequent genomic prediction, we recommend that: (1) individual-specific imputation accuracies should be used that are computed after centring and scaling both true and imputed genotypes; and (2) imputation of gene dosage is preferred over imputation of the most likely genotype, as this increases accuracy and reduces bias of the imputed genotypes and the subsequent genomic predictions.     

Imputation of missing covariate values in epigenome-wide analysis of DNA methylation data

DNA methylation is a widely studied epigenetic mechanism and alterations in methylation patterns may be involved in the development of common diseases. Unlike inherited changes in genetic sequence, variation in site-specific methylation varies by tissue, developmental stage, and disease status, and may be impacted by aging and exposure to environmental factors, such as diet or smoking. These non-genetic factors are typically included in epigenome-wide association studies (EWAS) because they may be confounding factors to the association between methylation and disease. However, missing values in these variables can lead to reduced sample size and decrease the statistical power of EWAS. We propose a site selection and multiple imputation (MI) method to impute missing covariate values and to perform association tests in EWAS. Then, we compare this method to an alternative projection-based method. Through simulations, we show that the MI-based method is slightly conservative, but provides consistent estimates for effect size. We also illustrate these methods with data from the Atherosclerosis Risk in Communities (ARIC) study to carry out an EWAS between methylation levels and smoking status, in which missing cell type compositions and white blood cell counts are imputed.     

Quantifying the impact of fixed effects modeling of clusters in multiple imputation for cluster randomized trials

In cluster randomized trials (CRTs), identifiable clusters rather than individuals are randomized to study groups. Resulting data often consist of a small number of clusters with correlated observations within a treatment group. Missing data often present a problem in the analysis of such trials, and multiple imputation (MI) has been used to create complete data sets, enabling subsequent analysis with well-established analysis methods for CRTs. We discuss strategies for accounting for clustering when multiply imputing a missing continuous outcome, focusing on estimation of the variance of group means as used in an adjusted t-test or ANOVA. These analysis procedures are congenial to (can be derived from) a mixed effects imputation model; however, this imputation procedure is not yet available in commercial statistical software. An alternative approach that is readily available and has been used in recent studies is to include fixed effects for cluster, but the impact of using this convenient method has not been studied. We show that under this imputation model the MI variance estimator is positively biased and that smaller intraclass correlations (ICCs) lead to larger overestimation of the MI variance. Analytical expressions for the bias of the variance estimator are derived in the case of data missing completely at random, and cases in which data are missing at random are illustrated through simulation. Finally, various imputation methods are applied to data from the Detroit Middle School Asthma Project, a recent school-based CRT, and differences in inference are compared.     

Imputation of missing prostate cancer stage in English cancer registry data based on clinical assumptions

BackgroundCancer stage can be missing in national cancer registry records. We explored whether missing prostate cancer stage can be imputed using specific clinical assumptions.MethodsProstate cancer patients diagnosed between 2010 and 2013 were identified in English cancer registry data and linked to administrative hospital and mortality data (n = 139,807). Missing staging items were imputed based on specific assumptions: men with recorded N-stage but missing M-stage have no distant metastases (M0); low/intermediate-risk men with missing N- and/or M-stage have no nodal disease (N0) or metastases; and high-risk men with missing M-stage have no metastases. We tested these clinical assumptions by comparing 4-year survival in men with the same recorded and imputed cancer stage. Multi-variable Cox regression was used to test the validity of the clinical assumptions and multiple imputation.ResultsSurvival was similar for men with recorded N-stage but missing M-stage and corresponding men with M0 (89.5% vs 89.6%); for low/intermediate-risk men with missing M-stage and corresponding men with M0 (92.0% vs 93.1%); and for low/intermediate-risk men with missing N-stage and corresponding men with N0 (90.9% vs 93.7%). However, survival was different for high-risk men with missing M-stage and corresponding men with M0. Imputation based on clinical imputation performs as well as statistical multiple imputation.ConclusionSpecific clinical assumptions can be used to impute missing information on nodal involvement and distant metastases in some patients with prostate cancer.     

A comparison of model selection methods for prediction in the presence of multiply imputed data

Many approaches for variable selection with multiply imputed data in the development of a prognostic model have been proposed. However, no method prevails as uniformly best. We conducted a simulation study with a binary outcome and a logistic regression model to compare two classes of variable selection methods in the presence of MI data: (I) Model selection on bootstrap data, using backward elimination based on AIC or lasso, and fit the final model based on the most frequently (e.g. ) selected variables over all MI and bootstrap data sets; (II) Model selection on original MI data, using lasso. The final model is obtained by (i) averaging estimates of variables that were selected in any MI data set or (ii) in 50% of the MI data; (iii) performing lasso on the stacked MI data, and (iv) as in (iii) but using individual weights as determined by the fraction of missingness. In all lasso models, we used both the optimal penalty and the 1‐se rule. We considered recalibrating models to correct for overshrinkage due to the suboptimal penalty by refitting the linear predictor or all individual variables. We applied the methods on a real dataset of 951 adult patients with tuberculous meningitis to predict mortality within nine months. Overall, applying lasso selection with the 1‐se penalty shows the best performance, both in approach I and II. Stacking MI data is an attractive approach because it does not require choosing a selection threshold when combining results from separate MI data sets     

Multiple imputation to minimise bias from missing stage information in estimates of early cancer diagnosis in England: a population-based study

IntroductionMonitoring early diagnosis is a priority of cancer policy in England. Information on stage has not always been available for a large proportion of patients, however, which may bias temporal comparisons. We previously estimated that early-stage diagnosis of colorectal cancer rose from 32% to 44% during 2008–2013, using multiple imputation. Here we examine the underlying assumptions of multiple imputation for missing stage using the same dataset.MethodsIndividually-linked cancer registration, Hospital Episode Statistics (HES), and audit data were examined. Six imputation models including different interaction terms, post-diagnosis treatment, and survival information were assessed, and comparisons drawn with the a priori optimal model. Models were further tested by setting stage values to missing for some patients under one plausible mechanism, then comparing actual and imputed stage distributions for these patients. Finally, a pattern-mixture sensitivity analysis was conducted.ResultsData from 196,511 colorectal patients were analysed, with 39.2% missing stage. Inclusion of survival time increased the accuracy of imputation: the odds ratio for change in early-stage diagnosis during 2008–2013 was 1.7 (95% CI: 1.6, 1.7) with survival to 1 year included, compared to 1.9 (95% CI 1.9–2.0) with no survival information. Imputation estimates of stage were accurate in one plausible simulation. Pattern-mixture analyses indicated our previous analysis conclusions would only change materially if stage were misclassified for 20% of the patients who had it categorised as late.InterpretationMultiple imputation models can substantially reduce bias from missing stage, but data on patient’s one-year survival should be included for highest accuracy.     

Effect of Genome-Wide Genotyping and Reference Panels on Rare Variants Imputation

Common variants explain little of the variance of most common disease,prompting large-scale sequencing studies to understand the contribution of rare variants to these diseases.Imputation of rare variants from genome-wide genotypic arrays offers a cost-efficient strategy to achieve necessary sample sizes required for adequate statistical power.To estimate the performance of imputation of rare variants,we imputed 153 individuals,each of whom was genotyped on 3 different genotype arrays including 317k,610k and 1 million single nucleotide polymorphisms(SNPs),to two different reference panels:HapMap2 and 1000 Genomes pilot March 2010 release (lKGpilot) by using IMPUTE version 2.We found that more than 94%and 84%of all SNPs yield acceptable accuracy(info > 0.4) in HapMap2 and lKGpilot-based imputation,respectively.For rare variants(minor allele frequency(MAF) <5%),the proportion of wellimputed SNPs increased as the MAF increased from 0.3%to 5%across all 3 genome-wide association study(GWAS) datasets.The proportion of well-imputed SNPs was 69%,60%and 49%for SNPs with a MAF from 0.3%to 5%for 1M,610k and 317k,respectively. None of the very rare variants(MAF < 0.3%) were well imputed.We conclude that the imputation accuracy of rare variants increases with higher density of genome-wide genotyping arrays when the size of the reference panel is small.Variants with lower MAF are more difficult to impute.These findings have important implications in the design and replication of large-scale sequencing studies.     

Comparison of Results from Different Imputation Techniques for Missing Data from an Anti-Obesity Drug Trial

Background

In randomised trials of medical interventions, the most reliable analysis follows the intention-to-treat (ITT) principle. However, the ITT analysis requires that missing outcome data have to be imputed. Different imputation techniques may give different results and some may lead to bias. In anti-obesity drug trials, many data are usually missing, and the most used imputation method is last observation carried forward (LOCF). LOCF is generally considered conservative, but there are more reliable methods such as multiple imputation (MI).

Objectives

To compare four different methods of handling missing data in a 60-week placebo controlled anti-obesity drug trial on topiramate.

Methods

We compared an analysis of complete cases with datasets where missing body weight measurements had been replaced using three different imputation methods: LOCF, baseline carried forward (BOCF) and MI.

Results

561 participants were randomised. Compared to placebo, there was a significantly greater weight loss with topiramate in all analyses: 9.5 kg (SE 1.17) in the complete case analysis (N = 86), 6.8 kg (SE 0.66) using LOCF (N = 561), 6.4 kg (SE 0.90) using MI (N = 561) and 1.5 kg (SE 0.28) using BOCF (N = 561).

Conclusions

The different imputation methods gave very different results. Contrary to widely stated claims, LOCF did not produce a conservative (i.e., lower) efficacy estimate compared to MI. Also, LOCF had a lower SE than MI.     

Survival analysis using auxiliary variables via multiple imputation, with application to AIDS clinical trial data

We develop an approach, based on multiple imputation, to using auxiliary variables to recover information from censored observations in survival analysis. We apply the approach to data from an AIDS clinical trial comparing ZDV and placebo, in which CD4 count is the time-dependent auxiliary variable. To facilitate imputation, a joint model is developed for the data, which includes a hierarchical change-point model for CD4 counts and a time-dependent proportional hazards model for the time to AIDS. Markov chain Monte Carlo methods are used to multiply impute event times for censored cases. The augmented data are then analyzed and the results combined using standard multiple-imputation techniques. A comparison of our multiple-imputation approach to simply analyzing the observed data indicates that multiple imputation leads to a small change in the estimated effect of ZDV and smaller estimated standard errors. A sensitivity analysis suggests that the qualitative findings are reproducible under a variety of imputation models. A simulation study indicates that improved efficiency over standard analyses and partial corrections for dependent censoring can result. An issue that arises with our approach, however, is whether the analysis of primary interest and the imputation model are compatible.     

Imputation accuracy is robust to cattle reference genome updates

Genotype imputation is routinely applied in a large number of cattle breeds. Imputation has become a need due to the large number of SNP arrays with variable density (currently, from 2900 to 777 962 SNPs). Although many authors have studied the effect of different statistical methods on imputation accuracy, the impact of a (likely) change in the reference genome assembly on imputation from lower to higher density has not been determined so far. In this work, 1021 Italian Simmental SNP genotypes were remapped on the three most recent reference genome assemblies. Four imputation methods were used to assess the impact of an update in the reference genome. As expected, the four methods behaved differently, with large differences in terms of accuracy. Updating SNP coordinates on the three tested cattle reference genome assemblies determined only a slight variation on imputation results within method.     

Doubly robust multiple imputation using kernel‐based techniques

We consider the problem of estimating the marginal mean of an incompletely observed variable and develop a multiple imputation approach. Using fully observed predictors, we first establish two working models: one predicts the missing outcome variable, and the other predicts the probability of missingness. The predictive scores from the two models are used to measure the similarity between the incomplete and observed cases. Based on the predictive scores, we construct a set of kernel weights for the observed cases, with higher weights indicating more similarity. Missing data are imputed by sampling from the observed cases with probability proportional to their kernel weights. The proposed approach can produce reasonable estimates for the marginal mean and has a double robustness property, provided that one of the two working models is correctly specified. It also shows some robustness against misspecification of both models. We demonstrate these patterns in a simulation study. In a real‐data example, we analyze the total helicopter response time from injury in the Arizona emergency medical service data.