Comparison of several imputation methods for missing baseline data in propensity scores analysis of binary outcome

We performed a simulation study comparing the statistical properties of the estimated log odds ratio from propensity scores analyses of a binary response variable, in which missing baseline data had been imputed using a simple imputation scheme (Treatment Mean Imputation), compared with three ways of performing multiple imputation (MI) and with a Complete Case analysis. MI that included treatment (treated/untreated) and outcome (for our analyses, outcome was adverse event [yes/no]) in the imputer's model had the best statistical properties of the imputation schemes we studied. MI is feasible to use in situations where one has just a few outcomes to analyze. We also found that Treatment Mean Imputation performed quite well and is a reasonable alternative to MI in situations where it is not feasible to use MI. Treatment Mean Imputation performed better than MI methods that did not include both the treatment and outcome in the imputer's model. Copyright © 2009 John Wiley & Sons, Ltd.     

Variable selection models based on multiple imputation with an application for predicting median effective dose and maximum effect

The statistical methods for variable selection and prediction could be challenging when missing covariates exist. Although multiple imputation (MI) is a universally accepted technique for solving missing data problem, how to combine the MI results for variable selection is not quite clear, because different imputations may result in different selections. The widely applied variable selection methods include the sparse partial least-squares (SPLS) method and the penalized least-squares method, e.g. the elastic net (ENet) method. In this paper, we propose an MI-based weighted elastic net (MI-WENet) method that is based on stacked MI data and a weighting scheme for each observation in the stacked data set. In the MI-WENet method, MI accounts for sampling and imputation uncertainty for missing values, and the weight accounts for the observed information. Extensive numerical simulations are carried out to compare the proposed MI-WENet method with the other competing alternatives, such as the SPLS and ENet. In addition, we applied the MI-WENet method to examine the predictor variables for the endothelial function that can be characterized by median effective dose (ED50) and maximum effect (Emax) in an ex-vivo phenylephrine-induced extension and acetylcholine-induced relaxation experiment.     

Inverse Probability Weighting with Missing Predictors of Treatment Assignment or Missingness

Inverse probability weighting (IPW) can deal with confounding in non randomized studies. The inverse weights are probabilities of treatment assignment (propensity scores), estimated by regressing assignment on predictors. Problems arise if predictors can be missing. Solutions previously proposed include assuming assignment depends only on observed predictors and multiple imputation (MI) of missing predictors. For the MI approach, it was recommended that missingness indicators be used with the other predictors. We determine when the two MI approaches, (with/without missingness indicators) yield consistent estimators and compare their efficiencies.We find that, although including indicators can reduce bias when predictors are missing not at random, it can induce bias when they are missing at random. We propose a consistent variance estimator and investigate performance of the simpler Rubin’s Rules variance estimator. In simulations we find both estimators perform well. IPW is also used to correct bias when an analysis model is fitted to incomplete data by restricting to complete cases. Here, weights are inverse probabilities of being a complete case. We explain how the same MI methods can be used in this situation to deal with missing predictors in the weight model, and illustrate this approach using data from the National Child Development Survey.     

On the Performance of Sequential Regression Multiple Imputation Methods with Non Normal Error Distributions

Sequential regression multiple imputation has emerged as a popular approach for handling incomplete data with complex features. In this approach, imputations for each missing variable are produced based on a regression model using other variables as predictors in a cyclic manner. Normality assumption is frequently imposed for the error distributions in the conditional regression models for continuous variables, despite that it rarely holds in real scenarios. We use a simulation study to investigate the performance of several sequential regression imputation methods when the error distribution is flat or heavy tailed. The methods evaluated include the sequential normal imputation and its several extensions which adjust for non normal error terms. The results show that all methods perform well for estimating the marginal mean and proportion, as well as the regression coefficient when the error distribution is flat or moderately heavy tailed. When the error distribution is strongly heavy tailed, all methods retain their good performances for the mean and the adjusted methods have robust performances for the proportion; but all methods can have poor performances for the regression coefficient because they cannot accommodate the extreme values well. We caution against the mechanical use of sequential regression imputation without model checking and diagnostics.     

Efficient estimation in a regression model with missing responses

This article examines methods to efficiently estimate the mean response in a linear model with an unknown error distribution under the assumption that the responses are missing at random. We show how the asymptotic variance is affected by the estimator of the regression parameter, and by the imputation method. To estimate the regression parameter, the ordinary least squares is efficient only if the error distribution happens to be normal. If the errors are not normal, then we propose a one step improvement estimator or a maximum empirical likelihood estimator to efficiently estimate the parameter.To investigate the imputation’s impact on the estimation of the mean response, we compare the listwise deletion method and the propensity score method (which do not use imputation at all), and two imputation methods. We demonstrate that listwise deletion and the propensity score method are inefficient. Partial imputation, where only the missing responses are imputed, is compared to full imputation, where both missing and non-missing responses are imputed. Our results reveal that, in general, full imputation is better than partial imputation. However, when the regression parameter is estimated very poorly, the partial imputation will outperform full imputation. The efficient estimator for the mean response is the full imputation estimator that utilizes an efficient estimator of the parameter.     

Multiple Imputation of Predictor Variables Using Generalized Additive Models

The sensitivity of multiple imputation methods to deviations from their distributional assumptions is investigated using simulations, where the parameters of scientific interest are the coefficients of a linear regression model, and values in predictor variables are missing at random. The performance of a newly proposed imputation method based on generalized additive models for location, scale, and shape (GAMLSS) is investigated. Although imputation methods based on predictive mean matching are virtually unbiased, they suffer from mild to moderate under-coverage, even in the experiment where all variables are jointly normal distributed. The GAMLSS method features better coverage than currently available methods.     

Navigating choices when applying multiple imputation in the presence of multi-level categorical interaction effects

Multiple imputation (MI) is an appealing option for handling missing data. When implementing MI, however, users need to make important decisions to obtain estimates with good statistical properties. One such decision involves the choice of imputation model–the joint modeling (JM) versus fully conditional specification (FCS) approach. Another involves the choice of method to handle interactions. These include imputing the interaction term as any other variable (active imputation), or imputing the main effects and then deriving the interaction (passive imputation). Our study investigates the best approach to perform MI in the presence of interaction effects involving two categorical variables. Such effects warrant special attention as they involve multiple correlated parameters that are handled differently under JM and FCS modeling. Through an extensive simulation study, we compared active, passive and an improved passive approach under FCS, as JM precludes passive imputation. We additionally compared JM and FCS techniques using active imputation. Performance between active and passive imputation was comparable. The improved passive approach proved superior to the other two particularly when the number of parameters corresponding to the interaction was large. JM without rounding and FCS using active imputation were also mostly comparable, with JM outperforming FCS when the number of parameters was large. In a direct comparison of JM active and FCS improved passive, the latter was the clear winner. We recommend improved passive imputation under FCS along with sensitivity analyses to handle multi-level interaction terms.     

Alternative imputation techniques for complex metric variables

This paper deals with imputation techniques and strategies. Usually, imputation truly commences after the first data editing, but many preceding operations are needed before that. In this editing step, the missing or deficient items are to be recognized and coded, and then it is decided which of these, if any, should be substituted by imputing. There are a number of imputation methods and their specifications. Consequently, it is not clear what method finally should be chosen, especially when an imputation method may be best in one respect, and another method in the other. In this paper, we consider these questions through the following four imputation methods: (i) random hot decking, (ii) logistic regression imputation, (iii) linear regression imputation, and (iv) regression-based nearest neighbour hot decking. The last two methods are applied with the two different specifications. The two metric variables have been used in empirical tests. The first is very complex, but the second is more ordinary, and thus easier to handle. The empirical examples are based on simulations, which clearly show the biases of the various methods and their specifications. In general, it seems that method (iv) is recommendable although the results from it are not perfect either.     

Joint GEEs for multivariate correlated data with incomplete binary outcomes

This study considers a fully-parametric but uncongenial multiple imputation (MI) inference to jointly analyze incomplete binary response variables observed in a correlated data settings. Multiple imputation model is specified as a fully-parametric model based on a multivariate extension of mixed-effects models. Dichotomized imputed datasets are then analyzed using joint GEE models where covariates are associated with the marginal mean of responses with response-specific regression coefficients and a Kronecker product is accommodated for cluster-specific correlation structure for a given response variable and correlation structure between multiple response variables. The validity of the proposed MI-based JGEE (MI-JGEE) approach is assessed through a Monte Carlo simulation study under different scenarios. The simulation results, which are evaluated in terms of bias, mean-squared error, and coverage rate, show that MI-JGEE has promising inferential properties even when the underlying multiple imputation is misspecified. Finally, Adolescent Alcohol Prevention Trial data are used for illustration.     

A numerical study of multiple imputation methods using nonparametric multivariate outlier identifiers and depth-based performance criteria with clinical laboratory data

It is well known that if a multivariate outlier has one or more missing component values, then multiple imputation (MI) methods tend to impute nonextreme values and make the outlier become less extreme and less likely to be detected. In this paper, nonparametric depth-based multivariate outlier identifiers are used as criteria in a numerical study comparing several established methods of MI as well as a new proposed one, nine in all, in a setting of several actual clinical laboratory data sets of different dimensions. Two criteria, an ‘outlier recovery probability’ and a ‘relative accuracy measure’, are developed, based on depth functions. Three outlier identifiers, based on Mahalanobis distance, robust Mahalanobis distance, and generalized principle component analysis are also included in the study. Consequently, not only the comparison of imputation methods but also the comparison of outlier detection methods is accomplished in this study. Our findings show that the performance of an MI method depends on the choice of depth-based outlier detection criterion, as well as the size and dimension of the data and the fraction of missing components. By taking these features into account, an MI method for a given data set can be selected more optimally.     

Imputation techniques for incomplete data in quadratic discriminant analysis

We have compared the efficacy of five imputation algorithms readily available in SAS for the quadratic discriminant function. Here, we have generated several different parametric-configuration training data with missing data, including monotone missing-at-random observations, and used a Monte Carlo simulation to examine the expected probabilities of misclassification for the two-class quadratic statistical discrimination problem under five different imputation methods. Specifically, we have compared the efficacy of the complete observation-only method and the mean substitution, regression, predictive mean matching, propensity score, and Markov Chain Monte Carlo (MCMC) imputation methods. We found that the MCMC and propensity score multiple imputation approaches are, in general, superior to the other imputation methods for the configurations and training-sample sizes we considered.     

MIMCA: multiple imputation for categorical variables with multiple correspondence analysis

We propose a multiple imputation method to deal with incomplete categorical data. This method imputes the missing entries using the principal component method dedicated to categorical data: multiple correspondence analysis (MCA). The uncertainty concerning the parameters of the imputation model is reflected using a non-parametric bootstrap. Multiple imputation using MCA (MIMCA) requires estimating a small number of parameters due to the dimensionality reduction property of MCA. It allows the user to impute a large range of data sets. In particular, a high number of categories per variable, a high number of variables or a small number of individuals are not an issue for MIMCA. Through a simulation study based on real data sets, the method is assessed and compared to the reference methods (multiple imputation using the loglinear model, multiple imputation by logistic regressions) as well to the latest works on the topic (multiple imputation by random forests or by the Dirichlet process mixture of products of multinomial distributions model). The proposed method provides a good point estimate of the parameters of the analysis model considered, such as the coefficients of a main effects logistic regression model, and a reliable estimate of the variability of the estimators. In addition, MIMCA has the great advantage that it is substantially less time consuming on data sets of high dimensions than the other multiple imputation methods.     

An Examination of Discrepancies in Multiple Imputation Procedures Between SAS® and SPSS®

Multiple imputation (MI) has become a feasible method to replace missing data due to the rapid development of computer technology over the past three decades. Nonetheless, a unique issue with MI hinges on the fact that different software packages can give different results. Even when one begins with the same random number seed, conflicting findings can be obtained from the same data under an identical imputation model between SAS® and SPSS®. Consequently, as illustrated in this article, a predictor variable can be claimed both significant and not significant depending on the software being used. Based on the considerations of multiple imputation steps, including result pooling, default selection, and different numbers of imputations, practical suggestions are provided to minimize the discrepancies in the results obtained when using MI. Features of Stata® are briefly reviewed in the Discussion section to broaden the comparison of MI computing across widely used software packages.     

Comparisons of imputation methods with application to assess factors associated with self efficacy of physical activity in breast cancer survivors

ABSTRACT

Missing data are commonly encountered in self-reported measurements and questionnaires. It is crucial to treat missing values using appropriate method to avoid bias and reduction of power. Various types of imputation methods exist, but it is not always clear which method is preferred for imputation of data with non-normal variables. In this paper, we compared four imputation methods: mean imputation, quantile imputation, multiple imputation, and quantile regression multiple imputation (QRMI), using both simulated and real data investigating factors affecting self-efficacy in breast cancer survivors. The results displayed an advantage of using multiple imputation, especially QRMI when data are not normal.     

Balanced k-nearest neighbour imputation

Random imputation is an interesting class of imputation methods to handle item nonresponse because it tends to preserve the distribution of the imputed variable. However, such methods amplify the total variance of the estimators because values are imputed at random. This increase in variance is called imputation variance. In this paper, we propose a new random hot-deck imputation method that is based on the k-nearest neighbour methodology. It replaces the missing value of a unit with the observed value of a similar unit. Calibration and balanced sampling are applied to minimize the imputation variance. Moreover, our proposed method provides triple protection against nonresponse bias. This means that if at least one out of three specified models holds, then the resulting total estimator is unbiased. Finally, our approach allows the user to perform consistency edits and to impute simultaneously.     

Sensitivity to imputation models and assumptions in receiver operating characteristic analysis with incomplete data

Modern statistical methods using incomplete data have been increasingly applied in a wide variety of substantive problems. Similarly, receiver operating characteristic (ROC) analysis, a method used in evaluating diagnostic tests or biomarkers in medical research, has also been increasingly popular problem in both its development and application. While missing-data methods have been applied in ROC analysis, the impact of model mis-specification and/or assumptions (e.g. missing at random) underlying the missing data has not been thoroughly studied. In this work, we study the performance of multiple imputation (MI) inference in ROC analysis. Particularly, we investigate parametric and non-parametric techniques for MI inference under common missingness mechanisms. Depending on the coherency of the imputation model with the underlying data generation mechanism, our results show that MI generally leads to well-calibrated inferences under ignorable missingness mechanisms.     

Bias of factor loadings from questionnaire data with imputed scores

This study investigated the bias of factor loadings obtained from incomplete questionnaire data with imputed scores. Three models were used to generate discrete ordered rating scale data typical of questionnaires, also known as Likert data. These methods were the multidimensional polytomous latent trait model, a normal ogive item response theory model, and the discretized normal model. Incomplete data due to nonresponse were simulated using either missing completely at random or not missing at random mechanisms. Subsequently, for each incomplete data matrix, four imputation methods were applied for imputing item scores. Based on a completely crossed six-factor design, it was concluded that in general, bias was small for all data simulation methods and all imputation methods, and under all nonresponse mechanisms. Imputation method, two-way-plus-error, had the smallest bias in the factor loadings. Bias based on the discretized normal model was greater than that based on the other two models.     

Multiple imputation of censored survival data in the presence of missing covariates using restricted mean survival time

Missing covariates data with censored outcomes put a challenge in the analysis of clinical data especially in small sample settings. Multiple imputation (MI) techniques are popularly used to impute missing covariates and the data are then analyzed through methods that can handle censoring. However, techniques based on MI are available to impute censored data also but they are not much in practice. In the present study, we applied a method based on multiple imputation by chained equations to impute missing values of covariates and also to impute censored outcomes using restricted survival time in small sample settings. The complete data were then analyzed using linear regression models. Simulation studies and a real example of CHD data show that the present method produced better estimates and lower standard errors when applied on the data having missing covariate values and censored outcomes than the analysis of the data having censored outcome but excluding cases with missing covariates or the analysis when cases with missing covariate values and censored outcomes were excluded from the data (complete case analysis).     

A Simulation Study Comparing Multiple Imputation Methods for Incomplete Longitudinal Ordinal Data

Multiple imputation (MI) is now a reference solution for handling missing data. The default method for MI is the Multivariate Normal Imputation (MNI) algorithm that is based on the multivariate normal distribution. In the presence of longitudinal ordinal missing data, where the Gaussian assumption is no longer valid, application of the MNI method is questionable. This simulation study compares the performance of the MNI and ordinal imputation regression model for incomplete longitudinal ordinal data for situations covering various numbers of categories of the ordinal outcome, time occasions, sample sizes, rates of missingness, well-balanced, and skewed data.     

基于随机森林模型的分类数据缺失值插补

缺失数据是影响调查问卷数据质量的重要因素,对调查问卷中的缺失值进行插补可以显著提高调查数据的质量。调查问卷的数据类型多以分类型数据为主,数据挖掘技术中的分类算法是处理属性分类问题的常用方法,随机森林模型是众多分类算法中精度较高的方法之一。将随机森林模型引入调查问卷缺失数据的插补研究中,提出了基于随机森林模型的分类数据缺失值插补方法,并根据不同的缺失模式探讨了相应的插补步骤。通过与其它方法的实证模拟比较,表明随机森林插补法得到的插补值准确度更优、可信度更高。