矩法估计与极大似然估计的比较

以二项分布为例,讨论了其分布参数的矩法估计与极大似然估计,并比较了两种估计的优良性质,主要讨论它们的一致性、无偏性和有效性．     

A Bayesian Approach to Multilevel Structural Equation Modeling With Continuous and Dichotomous Outcomes

Multilevel Structural equation models are most often estimated from a frequentist framework via maximum likelihood. However, as shown in this article, frequentist results are not always accurate. Alternatively, one can apply a Bayesian approach using Markov chain Monte Carlo estimation methods. This simulation study compared estimation quality using Bayesian and frequentist approaches in the context of a multilevel latent covariate model. Continuous and dichotomous variables were examined because it is not yet known how different types of outcomes—most notably categorical—affect parameter recovery in this modeling context. Within the Bayesian estimation framework, the impact of diffuse, weakly informative, and informative prior distributions were compared. Findings indicated that Bayesian estimation may be used to overcome convergence problems and improve parameter estimate bias. Results highlight the differences in estimation quality between dichotomous and continuous variable models and the importance of prior distribution choice for cluster-level random effects.     

Instrumental Variables Two-Stage Least Squares (2SLS) vs. Maximum Likelihood Structural Equation Modeling of Causal Effects in Linear Regression Models

In the presence of omitted variables or similar validity threats, regression estimates are biased. Unbiased estimates (the causal effects) can be obtained in large samples by fitting instead the Instrumental Variables Regression (IVR) model. The IVR model can be estimated using structural equation modeling (SEM) software or using Econometric estimators such as two-stage least squares (2SLS). We describe 2SLS using SEM terminology, and report a simulation study in which we generated data according to a regression model in the presence of omitted variables and fitted (a) a regression model using ordinary least squares, (b) an IVR model using maximum likelihood (ML) as implemented in SEM software, and (c) an IVR model using 2SLS. Coverage rates of the causal effect using regression methods are always unacceptably low (often 0). When using the IVR model, accurate coverage is obtained across all conditions when N = 500. Even when the IVR model is misspecified, better coverage than regression is generally obtained. Differences between 2SLS and ML are small and favor 2SLS in small samples (N ≤ 100).     

Multilevel Factorial Invariance in n-Level Structural Equation Modeling (nSEM)

We present a multigroup multilevel confirmatory factor analysis (CFA) model and a procedure for testing multilevel factorial invariance in n-level structural equation modeling (nSEM). Multigroup multilevel CFA introduces a complexity when the group membership at the lower level intersects the clustered structure, because the observations in different groups but in the same cluster are not independent of one another. nSEM provides a framework in which the multigroup multilevel data structure is represented with the dependency between groups at the lower level properly taken into account. The procedure for testing multilevel factorial invariance is illustrated with an empirical example using an R package xxm2.     

广义线性模型极大似然估计弱相合性的数值模拟

广义线性模型是经典线性模型的自然推广,它是一类应用广泛的统计模型.选择Probit模型,利用matlab编程,通过数值模拟的方法验证了广义线性模型的极大似然估计的弱相合性.     

Multivariate Meta-Analysis as Structural Equation Models

Multivariate meta-analysis has become increasingly popular in the educational, social, and medical sciences. It is because the outcome measures in a meta-analysis can involve more than one effect size. This article proposes 2 mathematically equivalent models to implement multivariate meta-analysis in structural equation modeling (SEM). Specifically, this article shows how multivariate fixed-, random- and mixed-effects meta-analyses can be formulated as structural equation models. metaSEM (a free R package based on OpenMx) and Mplus are used to implement the proposed procedures. A real data set is used to illustrate the procedures. Formulating multivariate meta-analysis as structural equation models provides many new research opportunities for methodological development in both meta-analysis and SEM. Issues related to and extensions on the SEM-based meta-analysis are discussed.     

极大似然估计及其应用

讨论了极大似然估计原理及求法,说明了极大似然估计在不同分布和参数空间的应用,阐述了生命表中在单风险非完整样本数据环境中表格生存模型的极大似然估计方法．     

Brief Research Report: Bayesian Versus REML Estimations With Noninformative Priors in Multilevel Single-Case Data

Abstract

Recently, researchers have used multilevel models for estimating intervention effects in single-case experiments that include replications across participants (e.g., multiple baseline designs) or for combining results across multiple single-case studies. Researchers estimating these multilevel models have primarily relied on restricted maximum likelihood (REML) techniques, but Bayesian approaches have also been suggested. The purpose of this Monte Carlo simulation study was to examine the impact of estimation method (REML versus Bayesian with noninformative priors) on the estimation of treatment effects (relative bias, root mean square error) and on the inferences about those effects (interval coverage) for autocorrelated multiple-baseline data. Simulated conditions varied with regard to the number of participants, series length, and distribution of the variance within and across participants. REML and Bayesian estimation led to estimates of the fixed effects that showed little to no bias but that differentially impacted the inferences about the fixed effects and the estimates of the variances. Implications for applied researchers and methodologists are discussed.     

A Bayesian Approach for Estimating Multilevel Latent Contextual Models

In many applications of multilevel modeling, group-level (L2) variables for assessing group-level effects are generated by aggregating variables from a lower level (L1). However, the observed group mean might not be a reliable measure of the unobserved true group mean. In this article, we propose a Bayesian approach for estimating a multilevel latent contextual model that corrects for measurement error and sampling error (i.e., sampling only a small number of L1 units from a L2 unit) when estimating group-level effects of aggregated L1 variables. Two simulation studies were conducted to compare the Bayesian approach with the maximum likelihood approach implemented in Mplus. The Bayesian approach showed fewer estimation problems (e.g., inadmissible solutions) and more accurate estimates of the group-level effect than the maximum likelihood approach under problematic conditions (i.e., small number of groups, predictor variable with a small intraclass correlation). An application from educational psychology is used to illustrate the different estimation approaches.     

Automated Selection of Robust Individual-Level Structural Equation Models for Time Series Data

In order to analyze intensive longitudinal data collected across multiple individuals, researchers frequently have to decide between aggregating all individuals or analyzing each individual separately. This paper presents an R package, gimme, which allows for the automatic specification of individual-level structural equation models that combine group-, subgroup-, and individual-level information. This R package is a complement of the GIMME program currently available via a combination of MATLAB and LISREL. By capitalizing on the flexibility of R and the capabilities of the existing structural equation modeling package lavaan, gimme allows for the automated specification and estimation of group-, subgroup-, and individual-level relations in time series data from within a structural equation modeling framework. Applications include daily diary data as well as functional magnetic resonance imaging data.     

Identifying Aberrant Data in Structural Equation Models With IRLS-ADF

In structural equation models, outliers could result in inaccurate parameter estimates and misleading fit statistics when using traditional methods. To robustly estimate structural equation models, iteratively reweighted least squares (IRLS; Yuan & Bentler, 2000) has been proposed, but not thoroughly examined. We explore the large-sample properties of IRLS and its effect on parameter recovery, model fit, and aberrant data identification. A parametric bootstrap technique is proposed to determine the tuning parameters of IRLS, which results in improved Type I error rates in aberrant data identification, for data sets generated from homogenous populations. Scenarios concerning (a) simulated data, (b) contaminated data, and (c) a real data set are studied. Results indicate good parameter recovery, model fit, and aberrant data identification when noisy observations are drawn from a real data set, but lackluster parameter recovery and identification of aberrant data when the noise is parametrically structured. Practical implications and further research are discussed.     

Bayesian Meta-Analytic SEM: A One-Stage Approach to Modeling Between-Studies Heterogeneity in Structural Parameters

Meta-analytic structural equation modeling (MASEM) refers to a set of meta-analysis techniques for combining and comparing structural equation modeling (SEM) results from multiple studies. Existing approaches to MASEM cannot appropriately model between-studies heterogeneity in structural parameters because of missing correlations, lack model fit assessment, and suffer from several theoretical limitations. In this study, we address the major shortcomings of existing approaches by proposing a novel Bayesian multilevel SEM approach. Simulation results showed that the proposed approach performed satisfactorily in terms of parameter estimation and model fit evaluation when the number of studies and the within-study sample size were sufficiently large and when correlations were missing completely at random. An empirical example about the structure of personality based on a subset of data was provided. Results favored the third factor structure over the hierarchical structure. We end the article with discussions and future directions.     

Maximum Likelihood Estimation of Structural Equation Models for Continuous Data: Standard Errors and Goodness of Fit

Classical accounts of maximum likelihood (ML) estimation of structural equation models for continuous outcomes involve normality assumptions: standard errors (SEs) are obtained using the expected information matrix and the goodness of fit of the model is tested using the likelihood ratio (LR) statistic. Satorra and Bentler (1994) introduced SEs and mean adjustments or mean and variance adjustments to the LR statistic (involving also the expected information matrix) that are robust to nonnormality. However, in recent years, SEs obtained using the observed information matrix and alternative test statistics have become available. We investigate what choice of SE and test statistic yields better results using an extensive simulation study. We found that robust SEs computed using the expected information matrix coupled with a mean- and variance-adjusted LR test statistic (i.e., MLMV) is the optimal choice, even with normally distributed data, as it yielded the best combination of accurate SEs and Type I errors.     

The Role of Centering for Interaction of Level 1 Variables in Multilevel Structural Equation Models

This article examined the role of centering in estimating interaction effects in multilevel structural equation models. Interactions are typically represented by product term of 2 variables that are hypothesized to interact. In multilevel structural equation modeling (MSEM), the product term involving Level 1 variables is decomposed into within-cluster and between-cluster random components. The choice of centering affects the decomposition of the product term, and therefore affects the sample variance and covariance associated with the product term used in the maximum likelihood fitting function. The simulation study showed that for an interaction between a Level 1 variable and a Level 2 variable, the product term of uncentered variables or the product term of grand mean centered variables produced unbiased estimates in both Level 1 and Level 2 models. The product term of cluster mean centered variables produced biased estimates in the Level 1 model. For an interaction between 2 Level 1 variables, the product term of cluster mean centered variables produced unbiased estimates in the Level 1 model, whereas the product term of grand mean centered variables produced unbiased estimates for the Level 1 model. Recommendations for researchers who wish to estimate interactions in MSEM are provided.     

Revisiting the Model Size Effect in Structural Equation Modeling

Fitting a large structural equation modeling (SEM) model with moderate to small sample sizes results in an inflated Type I error rate for the likelihood ratio test statistic under the chi-square reference distribution, known as the model size effect. In this article, we show that the number of observed variables (p) and the number of free parameters (q) have unique effects on the Type I error rate of the likelihood ratio test statistic. In addition, the effects of p and q cannot be fully explained using degrees of freedom (df). We also evaluated the performance of 4 correctional methods for the model size effect, including Bartlett’s (1950), Swain’s (1975), and Yuan’s (2005) corrected statistics, and Yuan, Tian, and Yanagihara’s (2015) empirically corrected statistic. We found that Yuan et al.’s (2015) empirically corrected statistic generally yields the best performance in controlling the Type I error rate when fitting large SEM models.     

Using the Standardized Root Mean Squared Residual (SRMR) to Assess Exact Fit in Structural Equation Models

We examine the accuracy of p values obtained using the asymptotic mean and variance (MV) correction to the distribution of the sample standardized root mean squared residual (SRMR) proposed by Maydeu-Olivares to assess the exact fit of SEM models. In a simulation study, we found that under normality, the MV-corrected SRMR statistic provides reasonably accurate Type I errors even in small samples and for large models, clearly outperforming the current standard, that is, the likelihood ratio (LR) test. When data shows excess kurtosis, MV-corrected SRMR p values are only accurate in small models (p = 10), or in medium-sized models (p = 30) if no skewness is present and sample sizes are at least 500. Overall, when data are not normal, the MV-corrected LR test seems to outperform the MV-corrected SRMR. We elaborate on these findings by showing that the asymptotic approximation to the mean of the SRMR sampling distribution is quite accurate, while the asymptotic approximation to the standard deviation is not.     

Maximum Likelihood Dynamic Factor Modeling for Arbitrary N and T Using SEM

This article has 3 objectives that build on each other. First, we demonstrate how to obtain maximum likelihood estimates for dynamic factor models (the direct autoregressive factor score model) with arbitrary T and N by means of structural equation modeling (SEM) and compare the approach to existing methods. Second, we go beyond standard time series analysis (T large and N = 1) and conventional SEM (N large and T = 1 or small) by integrating both approaches. The resulting combined model offers a variety of new modeling options including a direct test of the ergodicity hypothesis, according to which the factorial structure of an individual observed at many time points is identical to the factorial structure of a group of individuals observed at a single point in time. Third, we illustrate the flexibility of SEM time series modeling by extending the approach to account for complex error structures. We end with a discussion of current limitations and future applications of SEM-based time series modeling for arbitrary T and N.     

semPlot: Unified Visualizations of Structural Equation Models

Structural equation modeling (SEM) has a long history of representing models graphically as path diagrams. This article presents the freely available semPlot package for R, which fills the gap between advanced, but time-consuming, graphical software and the limited graphics produced automatically by SEM software. In addition, semPlot offers more functionality than drawing path diagrams: It can act as a common ground for importing SEM results into R. Any result usable as input to semPlot can also be represented in any of the 3 popular SEM frameworks, as well as translated to input syntax for the R packages sem (Fox, Nie, & Byrnes, 2013) and lavaan (Rosseel, 2012). Special considerations are made in the package for the automatic placement of variables, using 3 novel algorithms that extend the earlier work of Boker, McArdle, and Neale (2002). The article concludes with detailed instructions on these node-placement algorithms.     

Testing Time-Invariance of Variable Specificity in Repeated Measure Designs Using Structural Equation Modeling

A structural equation modeling method for examining time-invariance of variable specificity in longitudinal studies with multiple measures is outlined, which is developed within a confirmatory factor-analytic framework. The approach represents a likelihood ratio test for the hypothesis of stability in the specificity part of the residual term associated with repeated administration of each measure. The procedure can be used in the search for parsimonious versions of multiwave multiple-indicator models, to test for variable specificity in them, and to examine assumptions underlying particular parameter estimation procedures in repeated measure designs. The outlined method is illustrated with empirical data.