A Class of Distribution-Free Models for Longitudinal Mediation Analysis

Mediation analysis constitutes an important part of treatment study to identify the mechanisms by which an intervention achieves its effect. Structural equation model (SEM) is a popular framework for modeling such causal relationship. However, current methods impose various restrictions on the study designs and data distributions, limiting the utility of the information they provide in real study applications. In particular, in longitudinal studies missing data is commonly addressed under the assumption of missing at random (MAR), where current methods are unable to handle such missing data if parametric assumptions are violated. In this paper, we propose a new, robust approach to address the limitations of current SEM within the context of longitudinal mediation analysis by utilizing a class of functional response models (FRM). Being distribution-free, the FRM-based approach does not impose any parametric assumption on data distributions. In addition, by extending the inverse probability weighted (IPW) estimates to the current context, the FRM-based SEM provides valid inference for longitudinal mediation analysis under the two most popular missing data mechanisms; missing completely at random (MCAR) and missing at random (MAR). We illustrate the approach with both real and simulated data.     

认知诊断缺失数据处理方法的比较：零替换、多重插补与极大似然估计法

数据缺失在测验中经常发生, 认知诊断评估也不例外, 数据缺失会导致诊断结果的偏差。首先, 通过模拟研究在多种实验条件下比较了常用的缺失数据处理方法。结果表明：(1)缺失数据导致估计精确性下降, 随着人数与题目数量减少、缺失率增大、题目质量降低, 所有方法的PCCR均下降, Bias绝对值和RMSE均上升。(2)估计题目参数时, EM法表现最好, 其次是MI, FIML和ZR法表现不稳定。(3)估计被试知识状态时, EM和FIML表现最好, MI和ZR表现不稳定。其次, 在PISA2015实证数据中进一步探索了不同方法的表现。综合模拟和实证研究结果, 推荐选用EM或FIML法进行缺失数据处理。     

Postmodeling Sensitivity Analysis to Detect the Effect of Missing Data Mechanisms

Incomplete or missing data is a common problem in almost all areas of empirical research. It is well known that simple and ad hoc methods such as complete case analysis or mean imputation can lead to biased and/or inefficient estimates. The method of maximum likelihood works well; however, when the missing data mechanism is not one of missing completely at random (MCAR) or missing at random (MAR), it too can result in incorrect inference. Statistical tests for MCAR have been proposed, but these are restricted to a certain class of problems. The idea of sensitivity analysis as a means to detect the missing data mechanism has been proposed in the statistics literature in conjunction with selection models where conjointly the data and missing data mechanism are modeled. Our approach is different here in that we do not model the missing data mechanism but use the data at hand to examine the sensitivity of a given model to the missing data mechanism. Our methodology is meant to raise a flag for researchers when the assumptions of MCAR (or MAR) do not hold. To our knowledge, no specific proposal for sensitivity analysis has been set forth in the area of structural equation models (SEM). This article gives a specific method for performing postmodeling sensitivity analysis using a statistical test and graphs. A simulation study is performed to assess the methodology in the context of structural equation models. This study shows success of the method, especially when the sample size is 300 or more and the percentage of missing data is 20% or more. The method is also used to study a set of real data measuring physical and social self-concepts in 463 Nigerian adolescents using a factor analysis model.     

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。     

SEM with Missing Data and Unknown Population Distributions Using Two-Stage ML: Theory and Its Application

This article provides the theory and application of the 2-stage maximum likelihood (ML) procedure for structural equation modeling (SEM) with missing data. The validity of this procedure does not require the assumption of a normally distributed population. When the population is normally distributed and all missing data are missing at random (MAR), the direct ML procedure is nearly optimal for SEM with missing data. When missing data mechanisms are unknown, including auxiliary variables in the analysis will make the missing data mechanism more likely to be MAR. It is much easier to include auxiliary variables in the 2-stage ML than in the direct ML. Based on most recent developments for missing data with an unknown population distribution, the article first provides the least technical material on why the normal distribution-based ML generates consistent parameter estimates when the missing data mechanism is MAR. The article also provides sufficient conditions for the 2-stage ML to be a valid statistical procedure in the general case. For the application of the 2-stage ML, an SAS IML program is given to perform the first-stage analysis and EQS codes are provided to perform the second-stage analysis. An example with open- and closed-book examination data is used to illustrate the application of the provided programs. One aim is for quantitative graduate students/applied psychometricians to understand the technical details for missing data analysis. Another aim is for applied researchers to use the method properly.     

Missing data techniques for structural equation modeling

As with other statistical methods, missing data often create major problems for the estimation of structural equation models (SEMs). Conventional methods such as listwise or pairwise deletion generally do a poor job of using all the available information. However, structural equation modelers are fortunate that many programs for estimating SEMs now have maximum likelihood methods for handling missing data in an optimal fashion. In addition to maximum likelihood, this article also discusses multiple imputation. This method has statistical properties that are almost as good as those for maximum likelihood and can be applied to a much wider array of models and estimation methods.     

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.     

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.     

A Bayesian Approach for Estimating Mediation Effects With Missing Data

Methodologists have developed mediation analysis techniques for a broad range of substantive applications, yet methods for estimating mediating mechanisms with missing data have been understudied. This study outlined a general Bayesian missing data handling approach that can accommodate mediation analyses with any number of manifest variables. Computer simulation studies showed that the Bayesian approach produced frequentist coverage rates and power estimates that were comparable to those of maximum likelihood with the bias-corrected bootstrap. We share an SAS macro that implements Bayesian estimation and use 2 data analysis examples to demonstrate its use.     

MISSING DATA: A CONCEPTUAL REVIEW FOR APPLIED PSYCHOLOGISTS

There has been conspicuously little research concerning missing data problems in the applied psychology literature. Fortunately, other fields have begun to investigate this issue. These include survey research, marketing, statistics, economics, and biometrics. A review of this literature suggests several trends for applied psychologists. For example, listwise deletion of data is often the least accurate technique to deal with missing data. Other methods for estimating missing data scores may be more accurate and preserve more data for investigators to analyze. Further, the literature reveals that the amount of missing data and the reasons for deletion of data impact how investigators should handle the problem. Finally, there is a great need for more investigation of strategies for dealing with missing data, especially when data are missing in nonrandom or systematic patterns.     

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

Researchers have developed missing data handling techniques for estimating interaction effects in multiple regression. Extending to latent variable interactions, we investigated full information maximum likelihood (FIML) estimation to handle incompletely observed indicators for product indicator (PI) and latent moderated structural equations (LMS) methods. Drawing on the analytic work on missing data handling techniques in multiple regression with interaction effects, we compared the performance of FIML for PI and LMS analytically. We performed a simulation study to compare FIML for PI and LMS. We recommend using FIML for LMS when the indicators are missing completely at random (MCAR) or missing at random (MAR) and when they are normally distributed. FIML for LMS produces unbiased parameter estimates with small variances, correct Type I error rates, and high statistical power of interaction effects. We illustrated the use of these methods by analyzing the interaction effect between advanced cancer patients’ depression and change of inner peace well-being on future hopelessness levels.     

The impact of four missing data techniques on validity estimates in human resource management

While missing data are a commo problem in field settings, there is relatively little information in human Resource Management to guide researchers when they conduct analyses with incomplete data. This article discusses four techniques to deal wih missing data. The implications of using listwise deletion, pariwise deletion, mean substitution, and regression estimation are demonstrated in an applied selection situation. The importance of the manner in which data were missing is analyzed and discussed.The authors would like to thank Patricia G. Roth and Tim Summers (bot of Clemson University) as well as Joe Ward (University of Texas-Sa Antonio) for substantive comments on drafts of this article. The authors also appreciate the comments of Rich Arvey on the field of missing data. Diane Segal deserves thanks for her help conducting analyses. Their efforts have greatly enhanced the quality of this article.     

Estimating Incremental Validity Under Missing Data

A common form of missing data is caused by selection on an observed variable (e.g., Z). If the selection variable was measured and is available, the data are regarded as missing at random (MAR). Selection biases correlation, reliability, and effect size estimates when these estimates are computed on listwise deleted (LD) data sets. On the other hand, maximum likelihood (ML) estimates are generally unbiased and outperform LD in most situations, at least when the data are MAR. The exception is when we estimate the partial correlation. In this situation, LD estimates are unbiased when the cause of missingness is partialled out. In other words, there is no advantage of ML estimates over LD estimates in this situation. We demonstrate that under a MAR condition, even ML estimates may become biased, depending on how partial correlations are computed. Finally, we conclude with recommendations about how future researchers might estimate partial correlations even when the cause of missingness is unknown and, perhaps, unknowable.     

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.     

Meta-analytic methods of pooling correlation matrices for structural equation modeling under different patterns of missing data

Three methods of synthesizing correlations for meta-analytic structural equation modeling (SEM) under different degrees and mechanisms of missingness were compared for the estimation of correlation and SEM parameters and goodness-of-fit indices by using Monte Carlo simulation techniques. A revised generalized least squares (GLS) method for synthesizing correlations, weighted-covariance GLS (W-COV GLS), was compared with univariate weighting with untransformed correlations (univariate r) and univariate weighting with Fisher's z-transformed correlations (univariate z). These 3 methods were crossed with listwise and pairwise deletion. Univariate z and W-COV GLS performed similarly, with W-COV GLS providing slightly better estimation of parameters and more correct model rejection rates. Missing not at random data produced high levels of relative bias in correlation and model parameter estimates and higher incorrect SEM model rejection rates. Pairwise deletion resulted in inflated standard errors for all synthesis methods and higher incorrect rejection rates for the SEM model with univariate weighting procedures.     

Pairwise likelihood estimation for confirmatory factor analysis models with categorical variables and data that are missing at random

Methods for the treatment of item non-response in attitudinal scales and in large-scale assessments under the pairwise likelihood (PL) estimation framework and under a missing at random (MAR) mechanism are proposed. Under a full information likelihood estimation framework and MAR, ignorability of the missing data mechanism does not lead to biased estimates. However, this is not the case for pseudo-likelihood approaches such as the PL. We develop and study the performance of three strategies for incorporating missing values into confirmatory factor analysis under the PL framework, the complete-pairs (CP), the available-cases (AC) and the doubly robust (DR) approaches. The CP and AC require only a model for the observed data and standard errors are easy to compute. Doubly-robust versions of the PL estimation require a predictive model for the missing responses given the observed ones and are computationally more demanding than the AC and CP. A simulation study is used to compare the proposed methods. The proposed methods are employed to analyze the UK data on numeracy and literacy collected as part of the OECD Survey of Adult Skills.     

Handling of missing values in path models for opinions or attitudes

In this article we show, by means of a practical example of a path model to explain opinions or attitudes and using a dataset well-known in The Netherlands, that the intercorrelations of the variables may be highly dependent on the number of variables and the corresponding number of missing data involved. As a consequence, differences could arise in the results of multiple regressions and path analyses. (The role of a suppressant variable in a path model will be touched on in passing.) Subsequently, the way that the character of the sample can change when a more rigid listwise selection of cases is applied is demonstrated. Since a practical example is involved, substantive arguments may be used for choosing a strategy of handling of the missing values. In our view, with reference to path models of opinions or attitudes, these arguments lead not to the use of one of the current imputation techniques or sophisticated methods to estimate the population values of the model parameters, but to what may be called a differentiated listwise selection.     

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.     

Using Principal Components as Auxiliary Variables in Missing Data Estimation

To deal with missing data that arise due to participant nonresponse or attrition, methodologists have recommended an “inclusive” strategy where a large set of auxiliary variables are used to inform the missing data process. In practice, the set of possible auxiliary variables is often too large. We propose using principal components analysis (PCA) to reduce the number of possible auxiliary variables to a manageable number. A series of Monte Carlo simulations compared the performance of the inclusive strategy with eight auxiliary variables (inclusive approach) to the PCA strategy using just one principal component derived from the eight original variables (PCA approach). We examined the influence of four independent variables: magnitude of correlations, rate of missing data, missing data mechanism, and sample size on parameter bias, root mean squared error, and confidence interval coverage. Results indicate that the PCA approach results in unbiased parameter estimates and potentially more accuracy than the inclusive approach. We conclude that using the PCA strategy to reduce the number of auxiliary variables is an effective and practical way to reap the benefits of the inclusive strategy in the presence of many possible auxiliary variables.     

Fisher transformations for correlations corrected for selection and missing data

The validity of a test is often estimated in a nonrandom sample of selected individuals. To accurately estimate the relation between the predictor and the criterion we correct this correlation for range restriction. Unfortunately, this corrected correlation cannot be transformed using Fisher'sZ transformation, and asymptotic tests of hypotheses based on small or moderate samples are not accurate. We developed a Fisherr toZ transformation for the corrected correlation for each of two conditions: (a) the criterion data were missing due to selection on the predictor (the missing data were MAR); and (b) the criterion was missing at random, not due to selection (the missing data were MCAR). The twoZ transformations were evaluated in a computer simulation. The transformations were accurate, and tests of hypotheses and confidence intervals based on the transformations were superior to those that were not based on the transformations.