A Bayesian Approach for Analyzing Hierarchical Data With Missing Outcomes Through Structural Equation Models

Structural equation models are widely appreciated in behavioral, social, and psychological research to model relations between latent constructs and manifest variables, and to control for measurement errors. Most applications of structural equation models are based on fully observed data that are independently distributed. However, hierarchical data with a correlated structure are common in behavioral research, and very often, missing data are encountered. In this article, we propose a 2-level structural equation model for analyzing hierarchical data with missing entries, and describe a Bayesian approach for estimation and model comparison. We show how to use WinBUGS software to get the solution conveniently. The proposed methodologies are illustrated through a simulation study, and a real application in relation to organizational and management research concerning the study of the interrelationships of the latent constructs about job satisfaction, job responsibility, and life satisfaction for citizens in 43 countries.     

Brief Research Report: Growth Models With Small Samples and Missing Data

Small samples are common in growth models due to financial and logistical difficulties of following people longitudinally. For similar reasons, longitudinal studies often contain missing data. Though full information maximum likelihood (FIML) is popular to accommodate missing data, the limited number of studies in this area have found that FIML tends to perform poorly with small-sample growth models. This report demonstrates that the fault lies not with how FIML accommodates missingness but rather with maximum likelihood estimation itself. We discuss how the less popular restricted likelihood form of FIML, along with small-sample-appropriate methods, yields trustworthy estimates for growth models with small samples and missing data. That is, previously reported small sample issues with FIML are attributable to finite sample bias of maximum likelihood estimation not direct likelihood. Estimation issues pertinent to joint multiple imputation and predictive mean matching are also included and discussed.     

带有缺失数据线性回归模型的变量选择

通过惩罚估计方程,对响应变量随机缺失下的线性回归模型,给出了一个变量选择方法,并结合局部二次逼近,得到了一个迭代算法,证明了此变量选择方法是相合的并且所得估计达到最优的参数收敛速度,最后通过数据模拟研究了此方法的有限样本性质.     

A Primer on Maximum Likelihood Algorithms Available for Use With Missing Data

Maximum likelihood algorithms for use with missing data are becoming commonplace in microcomputer packages. Specifically, 3 maximum likelihood algorithms are currently available in existing software packages: the multiple-group approach, full information maximum likelihood estimation, and the EM algorithm. Although they belong to the same family of estimator, confusion appears to exist over the differences among the 3 algorithms. This article provides a comprehensive, nontechnical overview of the 3 maximum likelihood algorithms. Multiple imputation, which is frequently used in conjunction with the EM algorithm, is also discussed.     

指数分布无失效数据的Bayes估计及多层Bayes估计

本文讨论了指数分布无失效数据的失效率的先验分布为贝塔指数分布时,失效率的Bayes估计及多层Bayes估计,并给出了可靠度的估计,最后结合实际问题进行了计算,说明了估计的有效性.     

A Review of Methods for Missing Data

This paper reviews methods for handling missing data in a research study. Many researchers use ad hoc methods such as complete case analysis, available case analysis (pairwise deletion), or single-value imputation. Though these methods are easily implemented, they require assumptions about the data that rarely hold in practice. Model-based methods such as maximum likelihood using the EM algorithm and multiple imputation hold more promise for dealing with difficulties caused by missing data. While model-based methods require specialized computer programs and assumptions about the nature of the missing data, these methods are appropriate for a wider range of situations than the more commonly used ad hoc methods. The paper provides an illustration of the methods using data from an intervention study designed to increase students’ ability to control their asthma symptoms.     

Model Selection in Structural Equation Models With Continuous and Polytomous Data

Recently, analysis of structural equation models with polytomous and continuous variables has received a lot of attention. However, contributions to the selection of good models are limited. The main objective of this article is to investigate the maximum likelihood estimation of unknown parameters in a general LISREL-type model with mixed polytomous and continuous data and propose a model selection procedure for obtaining good models for the underlying substantive theory. The maximum likelihood estimate is obtained by a Monte Carlo Expectation Maximization algorithm, in which the E step is evaluated via the Gibbs sampler and the M step is completed via the method of conditional maximization. The convergence of the Monte Carlo Expectation Maximization algorithm is monitored by the bridge sampling. A model selection procedure based on Bayes factor and Occam's window search strategy is proposed. The effectiveness of the procedure in accounting for the model uncertainty and in picking good models is discussed. The proposed methodology is illustrated with a real example.     

Factor Retention in Exploratory Factor Analysis With Missing Data

Determining the number of factors in exploratory factor analysis is arguably the most crucial decision a researcher faces when conducting the analysis. While several simulation studies exist that compare various so-called factor retention criteria under different data conditions, little is known about the impact of missing data on this process. Hence, in this study, we evaluated the performance of different factor retention criteria—the Factor Forest, parallel analysis based on a principal component analysis as well as parallel analysis based on the common factor model and the comparison data approach—in combination with different missing data methods, namely an expectation-maximization algorithm called Amelia, predictive mean matching, and random forest imputation within the multiple imputations by chained equations (MICE) framework as well as pairwise deletion with regard to their accuracy in determining the number of factors when data are missing. Data were simulated for different sample sizes, numbers of factors, numbers of manifest variables (indicators), between-factor correlations, missing data mechanisms and proportions of missing values. In the majority of conditions and for all factor retention criteria except the comparison data approach, the missing data mechanism had little impact on the accuracy and pairwise deletion performed comparably well as the more sophisticated imputation methods. In some conditions, especially small-sample cases and when comparison data were used to determine the number of factors, random forest imputation was preferable to other missing data methods, though. Accordingly, depending on data characteristics and the selected factor retention criterion, choosing an appropriate missing data method is crucial to obtain a valid estimate of the number of factors to extract.     

一种基于RBF的时序缺失数据修复方法

提出了一种基于RBF的时序缺失数据修复方法,利用RBF构建模板数据和当前存在缺失的数据之间的训练关系,并通过该训练关系修复缺失数据．实验表明,该方法能够应用于刚性体以及非刚形体运动或形变追踪,是一种有效的时序缺失数据修复方法．     

Evaluation of Test Statistics for Robust Structural Equation Modeling With Nonnormal Missing Data

A 2-stage robust procedure as well as an R package, rsem, were recently developed for structural equation modeling with nonnormal missing data by Yuan and Zhang (2012). Several test statistics that have been used for complete data analysis are employed to evaluate model fit in the 2-stage robust method. However, properties of these statistics under robust procedures for incomplete nonnormal data analysis have never been studied. This study aims to systematically evaluate and compare 5 test statistics, including a test statistic derived from normal-distribution-based maximum likelihood, a rescaled chi-square statistic, an adjusted chi-square statistic, a corrected residual-based asymptotical distribution-free chi-square statistic, and a residual-based F statistic. These statistics are evaluated under a linear growth curve model by varying 8 factors: population distribution, missing data mechanism, missing data rate, sample size, number of measurement occasions, covariance between the latent intercept and slope, variance of measurement errors, and downweighting rate of the 2-stage robust method. The performance of the test statistics varies and the one derived from the 2-stage normal-distribution-based maximum likelihood performs much worse than the other four. Application of the 2-stage robust method and of the test statistics is illustrated through growth curve analysis of mathematical ability development, using data on the Peabody Individual Achievement Test mathematics assessment from the National Longitudinal Survey of Youth 1997 Cohort.     

Effects of Missing Data Methods in Structural Equation Modeling With Nonnormal Longitudinal Data

The purpose of this study is to investigate the effects of missing data techniques in longitudinal studies under diverse conditions. A Monte Carlo simulation examined the performance of 3 missing data methods in latent growth modeling: listwise deletion (LD), maximum likelihood estimation using the expectation and maximization algorithm with a nonnormality correction (robust ML), and the pairwise asymptotically distribution-free method (pairwise ADF). The effects of 3 independent variables (sample size, missing data mechanism, and distribution shape) were investigated on convergence rate, parameter and standard error estimation, and model fit. The results favored robust ML over LD and pairwise ADF in almost all respects. The exceptions included convergence rates under the most severe nonnormality in the missing not at random (MNAR) condition and recovery of standard error estimates across sample sizes. The results also indicate that nonnormality, small sample size, MNAR, and multicollinearity might adversely affect convergence rate and the validity of statistical inferences concerning parameter estimates and model fit statistics.     

Analysis of Longitudinal Studies With Missing Data Using Covariance Structure Modeling With Full-Information Maximum Likelihood

A didactic discussion of covariance structure modeling in longitudinal studies with missing data is presented. Use of the full-information maximum likelihood method is considered for model fitting, parameter estimation, and hypothesis testing purposes, particularly when interested in patterns of temporal change as well as its covariates and predictors. The approach is illustrated with an application of the popular level-and-shape model to data from a cognitive intervention study of elderly adults.     

A SAS Interface for Bayesian Analysis With WinBUGS

Bayesian methods are becoming very popular despite some practical difficulties in implementation. To assist in the practical application of Bayesian methods, we show how to implement Bayesian analysis with WinBUGS as part of a standard set of SAS routines. This implementation procedure is first illustrated by fitting a multiple regression model and then a linear growth curve model. A third example is also provided to demonstrate how to iteratively run WinBUGS inside SAS for Monte Carlo simulation studies. The SAS codes used in this study are easily extended to accommodate many other models with only slight modification. This interface can be of practical benefit in many aspects of Bayesian methods because it allows the SAS users to benefit from the implementation of Bayesian estimation and it also allows the WinBUGS user to benefit from the data processing routines available in SAS.     

缺失数据的处理和挑战

在数据挖掘研究中,缺失数据是一个非常普遍的问题,如何处理缺失数据也是一个热门的研究领域．介绍了缺失数据产生的原因,分类总结了缺失数据的处理方法,最后,提出了处理缺失数据的一些挑战性课题。     

Bayesian Analysis of Structural Equation Model With Fixed Covariates

This article extends the LISREL model to incorporate fixed covariates at both the measurement and the structural equations of the model. A Bayesian procedure with conjugate type prior distributions is established. The joint Bayesian estimates of the latent variables and the structural parameters that involve the regression coefficients of the covariates, the variances, covariances and causations among the manifest and latent variables are obtained via the Gibbs sampler algorithm. It is shown that the conditional distributions required in the Gibbs sampler are familiar distributions, hence the algorithm is very efficient. A goodness of fit statistic for assessing the proposed model is presented. An illustrative example with some real data is presented.     

残缺数据的指数平滑模型

在实际的预测工作中,搜集到的数据,通常不是等时间间距的而是残缺的,怎样利用这种不完整的数据,是预测工作中的难题。本文分析了残缺数据对预测精度的影响并就如何有效地利用MAGNE模型进行了论述。     

基于改进评价准则的贝叶斯网络模型选择

现有的贝叶斯网络模型选择的各种评价准则中,评价准则最为常用.然而在大样本前提下近似推导得到的准则,在基于小样本条件下进行贝叶斯网络模型选择时,得到的网络过于稀疏,网络连通性不足.而准则在小样本条件下学习贝叶斯网络,选择出的网络往往比较复杂.针对这一情况,本文通过引入调整因子,将与准则的惩罚项进行加权,提出了评价准则,从而提高了小样本情况下贝叶斯网络模型选择的效果.理论分析和实验结果反映了改进的准则的合理性和有效性.     

Bayesian Model Selection Methods for Multilevel IRT Models: A Comparison of Five DIC‐Based Indices

Model selection is important in any statistical analysis, and the primary goal is to find the preferred (or most parsimonious) model, based on certain criteria, from a set of candidate models given data. Several recent publications have employed the deviance information criterion (DIC) to do model selection among different forms of multilevel item response theory models (MLIRT). The majority of the practitioners use WinBUGS for implementing MCMC algorithms for MLIRT models, and the default version of DIC provided by WinBUGS focused on the measurement‐level parameters only. The results herein show that this version of DIC is inappropriate. This study introduces five variants of DIC as a model selection index for MLIRT models with dichotomous outcomes. Considering a multilevel IRT model with three levels, five forms of DIC are formed: first‐level conditional DIC computed from the measurement model only, which is the index given by many software packages such as WinBUGS; second‐level marginalized DIC and second‐level joint DIC computed from the second‐level model; and top‐level marginalized DIC and top‐level joint DIC computed from the entire model. We evaluate the performance of the five model selection indices via simulation studies. The manipulated factors include the number of groups, the number of second‐level covariates, the number of top‐level covariates, and the types of measurement models (one‐parameter vs. two‐parameter). Considering the computational viability and interpretability, the second‐level joint DIC is recommended for MLIRT models under our simulated conditions.