多重填补的方法及其统计推断原理

目的 描述数据缺失的特征和数据缺失模式，对Rubin最早提出的多重填补(multiple imputation,MI)的基本概念、填补和分析缺失数据的方法、综合统计推断进行了探讨，分析了MI的特点、局限性以及应用MI方法处理不完整数据集时需要注意的地方。方法 通过计算机模拟，用MI方法将每一个缺失值用一系列可能的值填补，然后使用常规的、针对完全数据集的统计方法对多重填补后得到的若干数据集进行分析，并把所得的结果进行综合。结果 多重填补值显示出了缺失数据的不确定性，使得已有数据得到了充分利用，从而对总体参数做出了更为准确的估计。结论 MI方法为处理存在缺失值的数据集提供了有用的策略，并且适用于多种数据缺失的场合。     

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.     

Imputation of missing values is superior to complete case analysis and the missing-indicator method in multivariable diagnostic research: a clinical example

BACKGROUND AND OBJECTIVES: To illustrate the effects of different methods for handling missing data--complete case analysis, missing-indicator method, single imputation of unconditional and conditional mean, and multiple imputation (MI)--in the context of multivariable diagnostic research aiming to identify potential predictors (test results) that independently contribute to the prediction of disease presence or absence. METHODS: We used data from 398 subjects from a prospective study on the diagnosis of pulmonary embolism. Various diagnostic predictors or tests had (varying percentages of) missing values. Per method of handling these missing values, we fitted a diagnostic prediction model using multivariable logistic regression analysis. RESULTS: The receiver operating characteristic curve area for all diagnostic models was above 0.75. The predictors in the final models based on the complete case analysis, and after using the missing-indicator method, were very different compared to the other models. The models based on MI did not differ much from the models derived after using single conditional and unconditional mean imputation. CONCLUSION: In multivariable diagnostic research complete case analysis and the use of the missing-indicator method should be avoided, even when data are missing completely at random. MI methods are known to be superior to single imputation methods. For our example study, the single imputation methods performed equally well, but this was most likely because of the low overall number of missing values.     

Development and validation of a prediction model with missing predictor data: a practical approach

ObjectiveTo illustrate the sequence of steps needed to develop and validate a clinical prediction model, when missing predictor values have been multiply imputed.Study Design and SettingWe used data from consecutive primary care patients suspected of deep venous thrombosis (DVT) to develop and validate a diagnostic model for the presence of DVT. Missing values were imputed 10 times with the MICE conditional imputation method. After the selection of predictors and transformations for continuous predictors according to three different methods, we estimated regression coefficients and performance measures.ResultsThe three methods to select predictors and transformations of continuous predictors showed similar results. Rubin's rules could easily be applied to estimate regression coefficients and performance measures, once predictors and transformations were selected.ConclusionWe provide a practical approach for model development and validation with multiply imputed data.     

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.     

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.     

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.     

Multiple imputation in the presence of non‐normal data

Multiple imputation (MI) is becoming increasingly popular for handling missing data. Standard approaches for MI assume normality for continuous variables (conditionally on the other variables in the imputation model). However, it is unclear how to impute non‐normally distributed continuous variables. Using simulation and a case study, we compared various transformations applied prior to imputation, including a novel non‐parametric transformation, to imputation on the raw scale and using predictive mean matching (PMM) when imputing non‐normal data. We generated data from a range of non‐normal distributions, and set 50% to missing completely at random or missing at random. We then imputed missing values on the raw scale, following a zero‐skewness log, Box–Cox or non‐parametric transformation and using PMM with both type 1 and 2 matching. We compared inferences regarding the marginal mean of the incomplete variable and the association with a fully observed outcome. We also compared results from these approaches in the analysis of depression and anxiety symptoms in parents of very preterm compared with term‐born infants. The results provide novel empirical evidence that the decision regarding how to impute a non‐normal variable should be based on the nature of the relationship between the variables of interest. If the relationship is linear in the untransformed scale, transformation can introduce bias irrespective of the transformation used. However, if the relationship is non‐linear, it may be important to transform the variable to accurately capture this relationship. A useful alternative is to impute the variable using PMM with type 1 matching. Copyright © 2016 John Wiley & Sons, Ltd.     

含有周期性的时间序列中连续型缺失数据的填补方法

目的用模拟研究的方法,对含周期性的时间序列数据中的连续型缺失数据进行填补,比较基于周期信息的时间序列缺失值填补法(简称周期性填补法)和spline插值填补法对连续型缺失数据的填补效果。方法分别应用模拟时间序列数据和实际时间序列数据模拟连续型缺失,比较两种方法在不同连续缺失个数下的缺失值填补效果。采用NRMSE和RMSE量化填补的误差。结果除连续型缺失长度为10和平,随着连续缺失个数的增加,周期性填补法的填补误均小于spline插值填补法。周期性填补方法的填补误差在5～30的连续缺失范围内无明显波动,始终保持在一个较低的水平;而spline填补值的误差随着缺失个数的增加明显增高。结论对于含有确切周期性的时间序列,周期性填补方法对连续型缺失数据的填补效果相对于spline填补更好,填补误差稳定,并且不随连续缺失长度的增加而有较大的变化。     

A distribution-based multiple imputation method for handling bivariate pesticide data with values below the limit of detection

Background

Environmental and biomedical researchers frequently encounter laboratory data constrained by a lower limit of detection (LOD). Commonly used methods to address these left-censored data, such as simple substitution of a constant for all values < LOD, may bias parameter estimation. In contrast, multiple imputation (MI) methods yield valid and robust parameter estimates and explicit imputed values for variables that can be analyzed as outcomes or predictors.

Objective

In this article we expand distribution-based MI methods for left-censored data to a bivariate setting, specifically, a longitudinal study with biological measures at two points in time.

Methods

We have presented the likelihood function for a bivariate normal distribution taking into account values < LOD as well as missing data assumed missing at random, and we use the estimated distributional parameters to impute values < LOD and to generate multiple plausible data sets for analysis by standard statistical methods. We conducted a simulation study to evaluate the sampling properties of the estimators, and we illustrate a practical application using data from the Community Participatory Approach to Measuring Farmworker Pesticide Exposure (PACE3) study to estimate associations between urinary acephate (APE) concentrations (indicating pesticide exposure) at two points in time and self-reported symptoms.

Results

Simulation study results demonstrated that imputed and observed values together were consistent with the assumed and estimated underlying distribution. Our analysis of PACE3 data using MI to impute APE values < LOD showed that urinary APE concentration was significantly associated with potential pesticide poisoning symptoms. Results based on simple substitution methods were substantially different from those based on the MI method.

Conclusions

The distribution-based MI method is a valid and feasible approach to analyze bivariate data with values < LOD, especially when explicit values for the nondetections are needed. We recommend the use of this approach in environmental and biomedical research.     

Multiple imputation of completely missing repeated measures data within person from a complex sample: application to accelerometer data in the National Health and Nutrition Examination Survey

The Physical Activity Monitor component was introduced into the 2003–2004 National Health and Nutrition Examination Survey (NHANES) to collect objective information on physical activity including both movement intensity counts and ambulatory steps. Because of an error in the accelerometer device initialization process, the steps data were missing for all participants in several primary sampling units, typically a single county or group of contiguous counties, who had intensity count data from their accelerometers. To avoid potential bias and loss in efficiency in estimation and inference involving the steps data, we considered methods to accurately impute the missing values for steps collected in the 2003–2004 NHANES. The objective was to come up with an efficient imputation method that minimized model‐based assumptions. We adopted a multiple imputation approach based on additive regression, bootstrapping and predictive mean matching methods. This method fits alternative conditional expectation (ace) models, which use an automated procedure to estimate optimal transformations for both the predictor and response variables. This paper describes the approaches used in this imputation and evaluates the methods by comparing the distributions of the original and the imputed data. A simulation study using the observed data is also conducted as part of the model diagnostics. Finally, some real data analyses are performed to compare the before and after imputation results. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.     

多重填补在随机干预试验研究中的应用

目的 利用多重填补方法实现对含缺失值的随机干预试验进行分析.方法 结合心理干预试验研究数据,利用SAS程序PROC MI和PROC MIANALYZE实现缺失数据的填补,应用稳健协方差分析评价心理健康干预效果.结果 填补与未填补分析结果一致,心理健康指标在干预组和对照组差别均无统计学意义,但CBO结局与干预有交互作用.结论 干预对学生心理健康起到一定的作用,但差别无统计学意义.     

A Scoping Review of Item-Level Missing Data in Within-Trial Cost-Effectiveness Analysis

ObjectivesCost-effectiveness analysis (CEA) alongside randomized controlled trials often relies on self-reported multi-item questionnaires that are invariably prone to missing item-level data. The purpose of this study is to review how missing multi-item questionnaire data are handled in trial-based CEAs.MethodsWe searched the National Institute for Health Research journals to identify within-trial CEAs published between January 2016 and April 2021 using multi-item instruments to collect costs and quality of life (QOL) data. Information on missing data handling and methods, with a focus on the level and type of imputation, was extracted.ResultsA total of 87 trial-based CEAs were included in the review. Complete case analysis or available case analysis and multiple imputation (MI) were the most popular methods, selected by similar numbers of studies, to handle missing costs and QOL in base-case analysis. Nevertheless, complete case analysis or available case analysis dominated sensitivity analysis. Once imputation was chosen, missing costs were widely imputed at item-level via MI, whereas missing QOL was usually imputed at the more aggregated time point level during the follow-up via MI.ConclusionsMissing costs and QOL tend to be imputed at different levels of missingness in current CEAs alongside randomized controlled trials. Given the limited information provided by included studies, the impact of applying different imputation methods at different levels of aggregation on CEA decision making remains unclear.     

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.     

Model validation and influence diagnostics for regression models with missing covariates

Missing covariate values are prevalent in regression applications. While an array of methods have been developed for estimating parameters in regression models with missing covariate data for a variety of response types, minimal focus has been given to validation of the response model and influence diagnostics. Previous research has mainly focused on estimating residuals for observations with missing covariates using expected values, after which specialized techniques are needed to conduct proper inference. We suggest a multiple imputation strategy that allows for the use of standard methods for residual analyses on the imputed data sets or a stacked data set. We demonstrate the suggested multiple imputation method by analyzing the Sleep in Mammals data in the context of a linear regression model and the New York Social Indicators Status data with a logistic regression model.     

含有周期性的时间序列中随机型缺失数据的填补方法

目的用模拟研究的方法,对含周期性的时间序列数据中随机型缺失数据进行填补,比较基于周期信息的时间序列缺失值填补法(简称周期性填补法)和spline插值填补法对缺失数据的填补效果。方法利用SAS模拟产生平稳、有周期性的时间序列数据并构造随机型缺失。分别比较相同序列长度不同缺失比例和相同缺失比例不同序列长度下,两种方法的缺失值填补效果。采用NRMSE和RMSE量化填补的误差。结果相同序列长度下,随着缺失比例的增加,两种填补方法的填补误差均增加,除缺失比例为30%的RMSE在两种方法间的差异无统计学意义外,周期性填补法的NRMSE和RMSE均小于spline填补法(P<0.05)。相同缺失比例下,序列长度较短时,两种填补方法的差异无统计学意义;序列长度较长时,周期性填补法的填补效果优于spline填补法。结论总体上,周期性填补法对含有确切周期性的时间序列中缺失数据的填补效果较好。     

Multiple imputation of missing dual-energy X-ray absorptiometry data in the National Health and Nutrition Examination Survey

In 1999, dual-energy x-ray absorptiometry (DXA) scans were added to the National Health and Nutrition Examination Survey (NHANES) to provide information on soft tissue composition and bone mineral content. However, in 1999-2004, DXA data were missing in whole or in part for about 21 per cent of the NHANES participants eligible for the DXA examination; and the missingness is associated with important characteristics such as body mass index and age. To handle this missing-data problem, multiple imputation of the missing DXA data was performed. Several features made the project interesting and challenging statistically, including the relationship between missingness on the DXA measures and the values of other variables; the highly multivariate nature of the variables being imputed; the need to transform the DXA variables during the imputation process; the desire to use a large number of non-DXA predictors, many of which had small amounts of missing data themselves, in the imputation models; the use of lower bounds in the imputation procedure; and relationships between the DXA variables and other variables, which helped both in creating and evaluating the imputations. This paper describes the imputation models, methods, and evaluations for this publicly available data resource and demonstrates properties of the imputations via examples of analyses of the data. The analyses suggest that imputation helps to correct biases that occur in estimates based on the data without imputation, and that it helps to increase the precision of estimates as well. Moreover, multiple imputation usually yields larger estimated standard errors than those obtained with single imputation.     

Multiple imputation of cognitive performance as a repeatedly measured outcome

Longitudinal studies of cognitive performance are sensitive to dropout, as participants experiencing cognitive deficits are less likely to attend study visits, which may bias estimated associations between exposures of interest and cognitive decline. Multiple imputation is a powerful tool for handling missing data, however its use for missing cognitive outcome measures in longitudinal analyses remains limited. We use multiple imputation by chained equations (MICE) to impute cognitive performance scores of participants who did not attend the 2011–2013 exam of the Atherosclerosis Risk in Communities Study. We examined the validity of imputed scores using observed and simulated data under varying assumptions. We examined differences in the estimated association between diabetes at baseline and 20-year cognitive decline with and without imputed values. Lastly, we discuss how different analytic methods (mixed models and models fit using generalized estimate equations) and choice of for whom to impute result in different estimands. Validation using observed data showed MICE produced unbiased imputations. Simulations showed a substantial reduction in the bias of the 20-year association between diabetes and cognitive decline comparing MICE (3–4 % bias) to analyses of available data only (16–23 % bias) in a construct where missingness was strongly informative but realistic. Associations between diabetes and 20-year cognitive decline were substantially stronger with MICE than in available-case analyses. Our study suggests when informative data are available for non-examined participants, MICE can be an effective tool for imputing cognitive performance and improving assessment of cognitive decline, though careful thought should be given to target imputation population and analytic model chosen, as they may yield different estimands.     

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.     

Sensitivity analysis for clinical trials with missing continuous outcome data using controlled multiple imputation: A practical guide

Missing data due to loss to follow-up or intercurrent events are unintended, but unfortunately inevitable in clinical trials. Since the true values of missing data are never known, it is necessary to assess the impact of untestable and unavoidable assumptions about any unobserved data in sensitivity analysis. This tutorial provides an overview of controlled multiple imputation (MI) techniques and a practical guide to their use for sensitivity analysis of trials with missing continuous outcome data. These include δ- and reference-based MI procedures. In δ-based imputation, an offset term, δ, is typically added to the expected value of the missing data to assess the impact of unobserved participants having a worse or better response than those observed. Reference-based imputation draws imputed values with some reference to observed data in other groups of the trial, typically in other treatment arms. We illustrate the accessibility of these methods using data from a pediatric eczema trial and a chronic headache trial and provide Stata code to facilitate adoption. We discuss issues surrounding the choice of δ in δ-based sensitivity analysis. We also review the debate on variance estimation within reference-based analysis and justify the use of Rubin's variance estimator in this setting, since as we further elaborate on within, it provides information anchored inference.