Using the outcome for imputation of missing predictor values was preferred

BACKGROUND AND OBJECTIVE: Epidemiologic studies commonly estimate associations between predictors (risk factors) and outcome. Most software automatically exclude subjects with missing values. This commonly causes bias because missing values seldom occur completely at random (MCAR) but rather selectively based on other (observed) variables, missing at random (MAR). Multiple imputation (MI) of missing predictor values using all observed information including outcome is advocated to deal with selective missing values. This seems a self-fulfilling prophecy. METHODS: We tested this hypothesis using data from a study on diagnosis of pulmonary embolism. We selected five predictors of pulmonary embolism without missing values. Their regression coefficients and standard errors (SEs) estimated from the original sample were considered as "true" values. We assigned missing values to these predictors--both MCAR and MAR--and repeated this 1,000 times using simulations. Per simulation we multiple imputed the missing values without and with the outcome, and compared the regression coefficients and SEs to the truth. RESULTS: Regression coefficients based on MI including outcome were close to the truth. MI without outcome yielded very biased--underestimated--coefficients. SEs and coverage of the 90% confidence intervals were not different between MI with and without outcome. Results were the same for MCAR and MAR. CONCLUSION: For all types of missing values, imputation of missing predictor values using the outcome is preferred over imputation without outcome and is no self-fulfilling prophecy.     

Multiple imputation of missing values was not necessary before performing a longitudinal mixed-model analysis

Background and ObjectivesAs a result of the development of sophisticated techniques, such as multiple imputation, the interest in handling missing data in longitudinal studies has increased enormously in past years. Within the field of longitudinal data analysis, there is a current debate on whether it is necessary to use multiple imputations before performing a mixed-model analysis to analyze the longitudinal data. In the current study this necessity is evaluated.Study Design and SettingThe results of mixed-model analyses with and without multiple imputation were compared with each other. Four data sets with missing values were created—one data set with missing completely at random, two data sets with missing at random, and one data set with missing not at random). In all data sets, the relationship between a continuous outcome variable and two different covariates were analyzed: a time-independent dichotomous covariate and a time-dependent continuous covariate.ResultsAlthough for all types of missing data, the results of the mixed-model analysis with or without multiple imputations were slightly different, they were not in favor of one of the two approaches. In addition, repeating the multiple imputations 100 times showed that the results of the mixed-model analysis with multiple imputation were quite unstable.ConclusionIt is not necessary to handle missing data using multiple imputations before performing a mixed-model analysis on longitudinal data.     

Propensity score estimation with missing values using a multiple imputation missingness pattern (MIMP) approach

Propensity scores have been used widely as a bias reduction method to estimate the treatment effect in nonrandomized studies. Since many covariates are generally included in the model for estimating the propensity scores, the proportion of subjects with at least one missing covariate could be large. While many methods have been proposed for propensity score‐based estimation in the presence of missing covariates, little has been published comparing the performance of these methods. In this article we propose a novel method called multiple imputation missingness pattern (MIMP) and compare it with the naive estimator (ignoring propensity score) and three commonly used methods of handling missing covariates in propensity score‐based estimation (separate estimation of propensity scores within each pattern of missing data, multiple imputation and discarding missing data) under different mechanisms of missing data and degree of correlation among covariates. Simulation shows that all adjusted estimators are much less biased than the naive estimator. Under certain conditions MIMP provides benefits (smaller bias and mean‐squared error) compared with existing alternatives. Copyright © 2009 John Wiley & Sons, Ltd.     

Survival estimates of a prognostic classification depended more on year of treatment than on imputation of missing values

BACKGROUND AND OBJECTIVE: The International Germ Cell Consensus (IGCC) classification defines good, intermediate, and poor prognosis groups among patients with nonseminomatous germ cell cancer. In the database used to develop the IGCC classification (n = 5,202), >40% of patients were excluded because of missing values (n = 2,154). We looked for effects of this exclusion on survival estimates in the three IGCC prognosis groups. STUDY DESIGN AND SETTING: We imputed missing values using a multiple imputation procedure. The IGCC classification was applied to patients with complete data (n = 3,048) and with imputed data (n = 2,154), and 5-year survival was calculated for each prognosis group. RESULTS: Patients with missing values had a lower 5-year survival than those without missing values: 76% vs. 82%. Five-year survival in the complete and imputed data samples was 92% and 87% for the good prognosis groups and 80% and 70% for the intermediate prognosis groups, whereas 5-year survival for the poor prognosis groups in both samples was similar (50% and 47%, respectively). This difference in survival was largely explained by a higher proportion of missing values among patients treated before 1985, who had a worse survival than patients treated after 1985. CONCLUSION: Multiple imputation of the missing values led to lower survival estimates across the IGCC prognosis groups, compared with estimates based on the complete data. Although imputation of missing values gives statistically better survival estimates, adjustments for year of treatment are necessary to make the estimates applicable to currently diagnosed patients with testicular cancer.     

Guided multiple imputation of missing data: using a subsample to strengthen the missing-at-random assumption

Multiple imputation can be a good solution to handling missing data if data are missing at random. However, this assumption is often difficult to verify. We describe an application of multiple imputation that makes this assumption plausible. This procedure requires contacting a random sample of subjects with incomplete data to fill in the missing information, and then adjusting the imputation model to incorporate the new data. Simulations with missing data that were decidedly not missing at random showed, as expected, that the method restored the original beta coefficients, whereas other methods of dealing with missing data failed. Using a dataset with real missing data, we found that different approaches to imputation produced moderately different results. Simulations suggest that filling in 10% of data that was initially missing is sufficient for imputation in many epidemiologic applications, and should produce approximately unbiased results, provided there is a high response on follow-up from the subsample of those with some originally missing data. This response can probably be achieved if this data collection is planned as an initial approach to dealing with the missing data, rather than at later stages, after further attempts that leave only data that is very difficult to complete.     

A model-based approach to the imputation of missing data: home injury incidences

Missing or incomplete data cases are a problem in all types of statistical analyses. In disease surveillance, this problem inhibits determining the actual incidence of a disease event and monitoring the disease occurrence. Several statistical techniques have been developed to impute values for incomplete data cases. We present a model-based approach to the imputation of missing data elements as applied to determining the incidence of home injury deaths.     

Multiple imputation to account for missing data in a survey: estimating the prevalence of osteoporosis

BACKGROUND: Nonresponse bias is a concern in any epidemiologic survey in which a subset of selected individuals declines to participate. METHODS: We reviewed multiple imputation, a widely applicable and easy to implement Bayesian methodology to adjust for nonresponse bias. To illustrate the method, we used data from the Canadian Multicentre Osteoporosis Study, a large cohort study of 9423 randomly selected Canadians, designed in part to estimate the prevalence of osteoporosis. Although subjects were randomly selected, only 42% of individuals who were contacted agreed to participate fully in the study. The study design included a brief questionnaire for those invitees who declined further participation in order to collect information on the major risk factors for osteoporosis. These risk factors (which included age, sex, previous fractures, family history of osteoporosis, and current smoking status) were then used to estimate the missing osteoporosis status for nonparticipants using multiple imputation. Both ignorable and nonignorable imputation models are considered. RESULTS: Our results suggest that selection bias in the study is of concern, but only slightly, in very elderly (age 80+ years), both women and men. CONCLUSIONS: Epidemiologists should consider using multiple imputation more often than is current practice.     

Nonparametric multiple imputation for receiver operating characteristics analysis when some biomarker values are missing at random

The receiver operating characteristics (ROC) curve is a widely used tool for evaluating discriminative and diagnostic power of a biomarker. When the biomarker value is missing for some observations, the ROC analysis based solely on complete cases loses efficiency because of the reduced sample size, and more importantly, it is subject to potential bias. In this paper, we investigate nonparametric multiple imputation methods for ROC analysis when some biomarker values are missing at random and there are auxiliary variables that are fully observed and predictive of biomarker values and/or missingness of biomarker values. Although a direct application of standard nonparametric imputation is robust to model misspecification, its finite sample performance suffers from curse of dimensionality as the number of auxiliary variables increases. To address this problem, we propose new nonparametric imputation methods, which achieve dimension reduction through the use of one or two working models, namely, models for prediction and propensity scores. The proposed imputation methods provide a platform for a full range of ROC analysis and hence are more flexible than existing methods that primarily focus on estimating the area under the ROC curve. We conduct simulation studies to evaluate the finite sample performance of the proposed methods and find that the proposed methods are robust to various types of model misidentification and outperform the standard nonparametric approach even when the number of auxiliary variables is moderate. We further illustrate the proposed methods by using an observational study of maternal depression during pregnancy.     

Bias and efficiency of multiple imputation compared with complete‐case analysis for missing covariate values

When missing data occur in one or more covariates in a regression model, multiple imputation (MI) is widely advocated as an improvement over complete‐case analysis (CC). We use theoretical arguments and simulation studies to compare these methods with MI implemented under a missing at random assumption. When data are missing completely at random, both methods have negligible bias, and MI is more efficient than CC across a wide range of scenarios. For other missing data mechanisms, bias arises in one or both methods. In our simulation setting, CC is biased towards the null when data are missing at random. However, when missingness is independent of the outcome given the covariates, CC has negligible bias and MI is biased away from the null. With more general missing data mechanisms, bias tends to be smaller for MI than for CC. Since MI is not always better than CC for missing covariate problems, the choice of method should take into account what is known about the missing data mechanism in a particular substantive application. Importantly, the choice of method should not be based on comparison of standard errors. We propose new ways to understand empirical differences between MI and CC, which may provide insights into the appropriateness of the assumptions underlying each method, and we propose a new index for assessing the likely gain in precision from MI: the fraction of incomplete cases among the observed values of a covariate (FICO). Copyright © 2010 John Wiley & Sons, Ltd.     

Complete imputation of missing repeated categorical data: one-sample applications.

Longitudinal studies with repeated measures are often subject to non-response. Methods currently employed to alleviate the difficulties caused by missing data are typically unsatisfactory, especially when the cause of the missingness is related to the outcomes. We present an approach for incomplete categorical data in the repeated measures setting that allows missing data to depend on other observed outcomes for a study subject. The proposed methodology also allows a broader examination of study findings through interpretation of results in the framework of the set of all possible test statistics that might have been observed had no data been missing. The proposed approach consists of the following general steps. First, we generate all possible sets of missing values and form a set of possible complete data sets. We then weight each data set according to clearly defined assumptions and apply an appropriate statistical test procedure to each data set, combining the results to give an overall indication of significance. We make use of the EM algorithm and a Bayesian prior in this approach. While not restricted to the one-sample case, the proposed methodology is illustrated for one-sample data and compared to the common complete-case and available-case analysis methods.     

Cumulative sojourn time in longitudinal studies: a sequential imputation method to handle missing health state data due to dropout

Missing data are ubiquitous in longitudinal studies. In this paper, we propose an imputation procedure to handle dropouts in longitudinal studies. By taking advantage of the monotone missing pattern resulting from dropouts, our imputation procedure can be carried out sequentially, which substantially reduces the computation complexity. In addition, at each step of the sequential imputation, we set up a model selection mechanism that chooses between a parametric model and a nonparametric model to impute each missing observation. Unlike usual model selection procedures that aim at finding a single model fitting the entire data set well, our model selection procedure is customized to find a suitable model for the prediction of each missing observation. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple imputation of missing blood pressure covariates in survival analysis

This paper studies a non-response problem in survival analysis where the occurrence of missing data in the risk factor is related to mortality. In a study to determine the influence of blood pressure on survival in the very old (85+ years), blood pressure measurements are missing in about 12.5 per cent of the sample. The available data suggest that the process that created the missing data depends jointly on survival and the unknown blood pressure, thereby distorting the relation of interest. Multiple imputation is used to impute missing blood pressure and then analyse the data under a variety of non-response models. One special modelling problem is treated in detail; the construction of a predictive model for drawing imputations if the number of variables is large. Risk estimates for these data appear robust to even large departures from the simplest non-response model, and are similar to those derived under deletion of the incomplete records.     

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.     

Multiple imputation of missing repeated outcome measurements did not add to linear mixed-effects models

ObjectiveTo assess the added value of multiple imputation (MI) of missing repeated outcomes measures in longitudinal data sets analyzed with linear mixed-effects (LME) models.Study Design and SettingData were used from a trial on the effects of Rosuvastatin on rate of change in carotid intima-media thickness (CIMT). The reference treatment effect was derived from a complete data set. Scenarios and proportions of missing values in CIMT measurements were applied and LME analyses were used before and after MI. The added value of MI, in terms of bias and precision, was assessed using the mean-squared error (MSE) of the treatment effects and coverage of the 95% confidence interval.ResultsThe reference treatment effect was ?0.0177 mm/y. The MSEs for LME analysis without and with MI were similar in scenarios with up to 40% missing values. Coverage was large in all scenarios and was similar for LME with and without MI.ConclusionOur study empirically shows that MI of missing end point data before LME analyses does not increase precision in the estimated rate of change in the end point. Hence, MI had no added value in this setting and standard LME modeling remains the method of choice.     

Dealing with missing covariates in epidemiologic studies: a comparison between multiple imputation and a full Bayesian approach

Incomplete data are generally a challenge to the analysis of most large studies. The current gold standard to account for missing data is multiple imputation, and more specifically multiple imputation with chained equations (MICE). Numerous studies have been conducted to illustrate the performance of MICE for missing covariate data. The results show that the method works well in various situations. However, less is known about its performance in more complex models, specifically when the outcome is multivariate as in longitudinal studies. In current practice, the multivariate nature of the longitudinal outcome is often neglected in the imputation procedure, or only the baseline outcome is used to impute missing covariates. In this work, we evaluate the performance of MICE using different strategies to include a longitudinal outcome into the imputation models and compare it with a fully Bayesian approach that jointly imputes missing values and estimates the parameters of the longitudinal model. Results from simulation and a real data example show that MICE requires the analyst to correctly specify which components of the longitudinal process need to be included in the imputation models in order to obtain unbiased results. The full Bayesian approach, on the other hand, does not require the analyst to explicitly specify how the longitudinal outcome enters the imputation models. It performed well under different scenarios. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple imputation of missing covariates for the Cox proportional hazards cure model

We explore several approaches for imputing partially observed covariates when the outcome of interest is a censored event time and when there is an underlying subset of the population that will never experience the event of interest. We call these subjects ‘cured’, and we consider the case where the data are modeled using a Cox proportional hazards (CPH) mixture cure model. We study covariate imputation approaches using fully conditional specification. We derive the exact conditional distribution and suggest a sampling scheme for imputing partially observed covariates in the CPH cure model setting. We also propose several approximations to the exact distribution that are simpler and more convenient to use for imputation. A simulation study demonstrates that the proposed imputation approaches outperform existing imputation approaches for survival data without a cure fraction in terms of bias in estimating CPH cure model parameters. We apply our multiple imputation techniques to a study of patients with head and neck cancer. Copyright © 2016 John Wiley & Sons, Ltd.     

An imputation method for non-ignorable missing data in studies of blood pressure

In studies with repeated measures of blood pressure (BP), particularly in trials of hypertension prevention, BP measurements often become censored once a participant commences antihypertensive medication. When prescribed by non-study physicians under uncontrolled conditions, the missing data mechanism is non-ignorable and may bias the BP effects of interest. I propose a method that models the distribution of BPs measured by non-study physicians and their relation to study BPs using random effects models. If treated for hypertension, I assume that BP measured outside the study is greater than a clinical cutpoint, such as diastolic BP⩾90 mmHg. I then compute estimates for the missing study BPs conditional on previously observed study BPs and treatment for hypertension. Multiple imputation is used to model the variability of the BP values and adjust the standard error estimates of the parameters. Examples are given using simulated data and data from the weight loss intervention of phase I of the Trials of Hypertension Prevention. © 1997 John Wiley & Sons, Ltd.