Missing data: our view of the state of the art

Statistical procedures for missing data have vastly improved, yet misconception and unsound practice still abound. The authors frame the missing-data problem, review methods, offer advice, and raise issues that remain unresolved. They clear up common misunderstandings regarding the missing at random (MAR) concept. They summarize the evidence against older procedures and, with few exceptions, discourage their use. They present, in both technical and practical language, 2 general approaches that come highly recommended: maximum likelihood (ML) and Bayesian multiple imputation (MI). Newer developments are discussed, including some for dealing with missing data that are not MAR. Although not yet in the mainstream, these procedures may eventually extend the ML and MI methods that currently represent the state of the art.     

How to perform multiblock component analysis in practice

To explore structural differences and similarities in multivariate multiblock data (e.g., a number of variables have been measured for different groups of subjects, where the data for each group constitute a different data block), researchers have a variety of multiblock component analysis and factor analysis strategies at their disposal. In this article, we focus on three types of multiblock component methods—namely, principal component analysis on each data block separately, simultaneous component analysis, and the recently proposed clusterwise simultaneous component analysis, which is a generic and flexible approach that has no counterpart in the factor analysis tradition. We describe the steps to take when applying those methods in practice. Whereas plenty of software is available for fitting factor analysis solutions, up to now no easy-to-use software has existed for fitting these multiblock component analysis methods. Therefore, this article presents the MultiBlock Component Analysis program, which also includes procedures for missing data imputation and model selection.     

Multiple Imputation of Item Scores in Test and Questionnaire Data,and Influence on Psychometric Results

The performance of five simple multiple imputation methods for dealing with missing data were compared. In addition, random imputation and multivariate normal imputation were used as lower and upper benchmark, respectively. Test data were simulated and item scores were deleted such that they were either missing completely at random, missing at random, or not missing at random. Cronbach's alpha, Loevinger's scalability coefficient H, and the item cluster solution from Mokken scale analysis of the complete data were compared with the corresponding results based on the data including imputed scores. The multiple-imputation methods, two-way with normally distributed errors, corrected item-mean substitution with normally distributed errors, and response function, produced discrepancies in Cronbach's coefficient alpha, Loevinger's coefficient H, and the cluster solution from Mokken scale analysis, that were smaller than the discrepancies in upper benchmark multivariate normal imputation.     

Tests of Homoscedasticity,Normality, and Missing Completely at Random for Incomplete Multivariate Data

Test of homogeneity of covariances (or homoscedasticity) among several groups has many applications in statistical analysis. In the context of incomplete data analysis, tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random (MCAR). These tests of MCAR require large sample sizes n and/or large group sample sizes n _i , and they usually fail when applied to nonnormal data. Hawkins (Technometrics 23:105–110, 1981) proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when n _i are small. This paper proposes a modification of this test for complete data to improve its performance, and extends its application to test of homoscedasticity and MCAR when data are multivariate normal and incomplete. Moreover, it is shown that the statistic used in the Hawkins test in conjunction with a nonparametric k-sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and nonnormal data. It is explained how a combination of the proposed normal-theory Hawkins test and the nonparametric test can be employed to test for homoscedasticity, MCAR, and multivariate normality. Simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of Type I error rejection rates. Also, a power study of the proposed tests indicates good power. The proposed methods use appropriate missing data imputations to impute missing data. Methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods. Examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups.     

Estimating age change in list recall in asset and health dynamics of the oldest-old: the effects of attrition bias and missing data treatment

Average change in list recall was evaluated as a function of missing data treatment (Study 1) and dropout status (Study 2) over ages 70 to 105 in Asset and Health Dynamics of the Oldest-Old data. In Study 1 the authors compared results of full-information maximum likelihood (FIML) and the multiple imputation (MI) missing-data treatments with and without independent predictors of missingness. Results showed declines in all treatments, but declines were larger for FIML and MI treatments when predictors were included in the treatment of missing data, indicating that attrition bias was reduced. In Study 2, models that included dropout status had better fits and reduced random variance compared with models without dropout status. The authors conclude that change estimates are most accurate when independent predictors of missingness are included in the treatment of missing data with either MI or FIML and when dropout effects are modeled.     

2PLM下缺失数据处理方法及其比较

项目反应理论(IRT)是用于客观测量的现代教育与心理测量理论之一,广泛用于缺失数据十分常见的大尺度测验分析。IRT中两参数逻辑斯蒂克模型(2PLM)下仅有完全随机缺失机制下缺失反应和缺失能力处理的EM算法。本研究推导2PLM下缺失反应忽略的EM 算法,并提出随机缺失机制下缺失反应和缺失能力处理的EM算法和考虑能力估计和作答反应不确定性的多重借补法。研究显示：在各种缺失机制、缺失比例和测验设计下,缺失反应忽略的EM算法和多重借补法表现理想。     

Establishing normative data for repeated cognitive assessment: A comparison of different statistical methods

Serial cognitive assessment is conducted to monitor changes in the cognitive abilities of patients over time. At present, mainly the regression-based change and the ANCOVA approaches are used to establish normative data for serial cognitive assessment. These methods are straightforward, but they have some severe drawbacks. For example, they can only consider the data of two measurement occasions. In this article, we propose three alternative normative methods that are not hampered by these problems—that is, multivariate regression, the standard linear mixed model (LMM), and the linear mixed model combined with multiple imputation (LMM with MI) approaches. The multivariate regression method is primarily useful when a small number of repeated measurements are taken at fixed time points. When the data are more unbalanced, the standard LMM and the LMM with MI methods are more appropriate because they allow for a more adequate modeling of the covariance structure. The standard LMM has the advantage that it is easier to conduct and that it does not require a Monte Carlo component. The LMM with MI, on the other hand, has the advantage that it can flexibly deal with missing responses and missing covariate values at the same time. The different normative methods are illustrated on the basis of the data of a large longitudinal study in which a cognitive test (the Stroop Color Word Test) was administered at four measurement occasions (i.e., at baseline and 3, 6, and 12 years later). The results are discussed and suggestions for future research are provided.     

Dual imputation model for incomplete longitudinal data

Missing values are a practical issue in the analysis of longitudinal data. Multiple imputation (MI) is a well‐known likelihood‐based method that has optimal properties in terms of efficiency and consistency if the imputation model is correctly specified. Doubly robust (DR) weighing‐based methods protect against misspecification bias if one of the models, but not necessarily both, for the data or the mechanism leading to missing data is correct. We propose a new imputation method that captures the simplicity of MI and protection from the DR method. This method integrates MI and DR to protect against misspecification of the imputation model under a missing at random assumption. Our method avoids analytical complications of missing data particularly in multivariate settings, and is easy to implement in standard statistical packages. Moreover, the proposed method works very well with an intermittent pattern of missingness when other DR methods can not be used. Simulation experiments show that the proposed approach achieves improved performance when one of the models is correct. The method is applied to data from the fireworks disaster study, a randomized clinical trial comparing therapies in disaster‐exposed children. We conclude that the new method increases the robustness of imputations.     

Modeling competence development in the presence of selection bias

A major challenge for representative longitudinal studies is panel attrition, because some respondents refuse to continue participating across all measurement waves. Depending on the nature of this selection process, statistical inferences based on the observed sample can be biased. Therefore, statistical analyses need to consider a missing-data mechanism. Because each missing-data model hinges on frequently untestable assumptions, sensitivity analyses are indispensable to gauging the robustness of statistical inferences. This article highlights contemporary approaches for applied researchers to acknowledge missing data in longitudinal, multilevel modeling and shows how sensitivity analyses can guide their interpretation. Using a representative sample of N = 13,417 German students, the development of mathematical competence across three years was examined by contrasting seven missing-data models, including listwise deletion, full-information maximum likelihood estimation, inverse probability weighting, multiple imputation, selection models, and pattern mixture models. These analyses identified strong selection effects related to various individual and context factors. Comparative analyses revealed that inverse probability weighting performed rather poorly in growth curve modeling. Moreover, school-specific effects should be acknowledged in missing-data models for educational data. Finally, we demonstrated how sensitivity analyses can be used to gauge the robustness of the identified effects.     

Multiple imputation for item scores when test data are factorially complex

Multiple imputation under a two‐way model with error is a simple and effective method that has been used to handle missing item scores in unidimensional test and questionnaire data. Extensions of this method to multidimensional data are proposed. A simulation study is used to investigate whether these extensions produce biased estimates of important statistics in multidimensional data, and to compare them with lower benchmark listwise deletion, two‐way with error and multivariate normal imputation. The new methods produce smaller bias in several psychometrically interesting statistics than the existing methods of two‐way with error and multivariate normal imputation. One of these new methods clearly is preferable for handling missing item scores in multidimensional test data.     

A Comparison of Multilevel Imputation Schemes for Random Coefficient Models: Fully Conditional Specification and Joint Model Imputation with Random Covariance Matrices

Abstract

Literature addressing missing data handling for random coefficient models is particularly scant, and the few studies to date have focused on the fully conditional specification framework and “reverse random coefficient” imputation. Although it has not received much attention in the literature, a joint modeling strategy that uses random within-cluster covariance matrices to preserve cluster-specific associations is a promising alternative for random coefficient analyses. This study is apparently the first to directly compare these procedures. Analytic results suggest that both imputation procedures can introduce bias-inducing incompatibilities with a random coefficient analysis model. Problems with fully conditional specification result from an incorrect distributional assumption, whereas joint imputation uses an underparameterized model that assumes uncorrelated intercepts and slopes. Monte Carlo simulations suggest that biases from these issues are tolerable if the missing data rate is 10% or lower and the sample is composed of at least 30 clusters with 15 observations per group. Furthermore, fully conditional specification tends to be superior with intraclass correlations that are typical of crosssectional data (e.g., ICC?=?.10), whereas the joint model is preferable with high values typical of longitudinal designs (e.g., ICC?=?.50).     

Imputation methods for missing data in educational diagnostic evaluation

In the diagnostic evaluation of educational systems, self-reports are commonly used to collect data, both cognitive and orectic. For various reasons, in these self-reports, some of the students' data are frequently missing. The main goal of this research is to compare the performance of different imputation methods for missing data in the context of the evaluation of educational systems. On an empirical database of 5,000 subjects, 72 conditions were simulated: three levels of missing data, three types of loss mechanisms, and eight methods of imputation. The levels of missing data were 5%, 10%, and 20%. The loss mechanisms were set at: Missing completely at random, moderately conditioned, and strongly conditioned. The eight imputation methods used were: listwise deletion, replacement by the mean of the scale, by the item mean, the subject mean, the corrected subject mean, multiple regression, and Expectation-Maximization (EM) algorithm, with and without auxiliary variables. The results indicate that the recovery of the data is more accurate when using an appropriate combination of different methods of recovering lost data. When a case is incomplete, the mean of the subject works very well, whereas for completely lost data, multiple imputation with the EM algorithm is recommended. The use of this combination is especially recommended when data loss is greater and its loss mechanism is more conditioned. Lastly, the results are discussed, and some future lines of research are analyzed.     

Best practices for missing data management in counseling psychology

This article urges counseling psychology researchers to recognize and report how missing data are handled, because consumers of research cannot accurately interpret findings without knowing the amount and pattern of missing data or the strategies that were used to handle those data. Patterns of missing data are reviewed, and some of the common strategies for dealing with them are described. The authors provide an illustration in which data were simulated and evaluate 3 methods of handling missing data: mean substitution, multiple imputation, and full information maximum likelihood. Results suggest that mean substitution is a poor method for handling missing data, whereas both multiple imputation and full information maximum likelihood are recommended alternatives to this approach. The authors suggest that researchers fully consider and report the amount and pattern of missing data and the strategy for handling those data in counseling psychology research and that editors advise researchers of this expectation.     

A Cautious Note on Auxiliary Variables That Can Increase Bias in Missing Data Problems

The treatment of missing data in the social sciences has changed tremendously during the last decade. Modern missing data techniques such as multiple imputation and full-information maximum likelihood are used much more frequently. These methods assume that data are missing at random. One very common approach to increase the likelihood that missing at random is achieved consists of including many covariates as so-called auxiliary variables. These variables are either included based on data considerations or in an inclusive fashion; that is, taking all available auxiliary variables. In this article, we point out that there are some instances in which auxiliary variables exhibit the surprising property of increasing bias in missing data problems. In a series of focused simulation studies, we highlight some situations in which this type of biasing behavior can occur. We briefly discuss possible ways how one can avoid selecting bias-inducing covariates as auxiliary variables.     

Biases in multilevel analyses caused by cluster-specific fixed-effects imputation

When datasets are affected by nonresponse, imputation of the missing values is a viable solution. However, most imputation routines implemented in commonly used statistical software packages do not accommodate multilevel models that are popular in education research and other settings involving clustering of units. A common strategy to take the hierarchical structure of the data into account is to include cluster-specific fixed effects in the imputation model. Still, this ad hoc approach has never been compared analytically to the congenial multilevel imputation in a random slopes setting. In this paper, we evaluate the impact of the cluster-specific fixed-effects imputation model on multilevel inference. We show analytically that the cluster-specific fixed-effects imputation strategy will generally bias inferences obtained from random coefficient models. The bias of random-effects variances and global fixed-effects confidence intervals depends on the cluster size, the relation of within- and between-cluster variance, and the missing data mechanism. We illustrate the negative implications of cluster-specific fixed-effects imputation using simulation studies and an application based on data from the National Educational Panel Study (NEPS) in Germany.     

Methods for Mediation Analysis with Missing Data

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.     

Simple imputation methods versus direct likelihood analysis for missing item scores in multilevel educational data

Missing data, such as item responses in multilevel data, are ubiquitous in educational research settings. Researchers in the item response theory (IRT) context have shown that ignoring such missing data can create problems in the estimation of the IRT model parameters. Consequently, several imputation methods for dealing with missing item data have been proposed and shown to be effective when applied with traditional IRT models. Additionally, a nonimputation direct likelihood analysis has been shown to be an effective tool for handling missing observations in clustered data settings. This study investigates the performance of six simple imputation methods, which have been found to be useful in other IRT contexts, versus a direct likelihood analysis, in multilevel data from educational settings. Multilevel item response data were simulated on the basis of two empirical data sets, and some of the item scores were deleted, such that they were missing either completely at random or simply at random. An explanatory IRT model was used for modeling the complete, incomplete, and imputed data sets. We showed that direct likelihood analysis of the incomplete data sets produced unbiased parameter estimates that were comparable to those from a complete data analysis. Multiple-imputation approaches of the two-way mean and corrected item mean substitution methods displayed varying degrees of effectiveness in imputing data that in turn could produce unbiased parameter estimates. The simple random imputation, adjusted random imputation, item means substitution, and regression imputation methods seemed to be less effective in imputing missing item scores in multilevel data settings.     

Investigation of Multiple Imputation in Low-Quality Questionnaire Data

The performance of multiple imputation in questionnaire data has been studied in various simulation studies. However, in practice, questionnaire data are usually more complex than simulated data. For example, items may be counterindicative or may have unacceptably low factor loadings on every subscale, or completely missing subscales may complicate computations. In this article, it was studied how well multiple imputation recovered the results of several psychometrically important statistics in a data set with such properties. Analysis of this data set revealed that multiple imputation was able to recover the results of these analyses well. Also, a simulation study showed that multiple imputation produced small bias in these statistics for simulated data sets with the same properties.