On Latent Change Model Choice in Longitudinal Studies

An interval estimation procedure for proportion of explained observed variance in latent curve analysis is discussed, which can be used as an aid in the process of choosing between linear and nonlinear models. The method allows obtaining confidence intervals for the R ² indexes associated with repeatedly followed measures in longitudinal studies. In addition to facilitating evaluation of local model fit, the approach is helpful for purposes of differentiating between plausible models stipulating different patterns of change over time, and in particular in empirical situations characterized by large samples and high statistical power. The procedure is also applicable in cross-sectional studies, as well as with general structural equation models. The method is illustrated using data from a nationally representative study of older adults.     

Book Review of Using Multivariate Statistics by Barbara G. Tabachnick and Linda S. Fidell

In applications of structural equation modeling, it is often desirable to obtain measures of uncertainty for special functions of model parameters. This article provides a didactic discussion of how a method widely used in applied statistics can be employed for approximate standard error and confidence interval evaluation of such functions. The described approach is illustrated with data from a cognitive intervention study, in which it is used to estimate time-invariant reliability in multiwave, multiple indicator models.     

Inference and Interval Estimation Methods for Indirect Effects With Latent Variable Models

Although much is known about the performance of recent methods for inference and interval estimation for indirect or mediated effects with observed variables, little is known about their performance in latent variable models. This article presents an extensive Monte Carlo study of 11 different leading or popular methods adapted to structural equation models with latent variables. Manipulated variables included sample size, number of indicators per latent variable, internal consistency per set of indicators, and 16 different path combinations between latent variables. Results indicate that some popular or previously recommended methods, such as the bias-corrected bootstrap and asymptotic standard errors had poorly calibrated Type I error and coverage rates in some conditions. Likelihood-based confidence intervals, the distribution of the product method, and the percentile bootstrap emerged as leading methods for both interval estimation and inference, whereas joint significance tests and the partial posterior method performed well for inference.     

Alternative Methods for Assessing Mediation in Multilevel Data: The Advantages of Multilevel SEM

Multilevel modeling (MLM) is a popular way of assessing mediation effects with clustered data. Two important limitations of this approach have been identified in prior research and a theoretical rationale has been provided for why multilevel structural equation modeling (MSEM) should be preferred. However, to date, no empirical evidence of MSEM's advantages relative to MLM approaches for multilevel mediation analysis has been provided. Nor has it been demonstrated that MSEM performs adequately for mediation analysis in an absolute sense. This study addresses these gaps and finds that the MSEM method outperforms 2 MLM-based techniques in 2-level models in terms of bias and confidence interval coverage while displaying adequate efficiency, convergence rates, and power under a variety of conditions. Simulation results support prior theoretical work regarding the advantages of MSEM over MLM for mediation in clustered data.     

Sensitivity Analysis of the No-Omitted Confounder Assumption in Latent Growth Curve Mediation Models

Latent growth curve mediation models are increasingly used to assess mechanisms of behavior change. For latent growth mediation model, like any another mediation model, even with random treatment assignment, a critical but untestable assumption for valid and unbiased estimates of the indirect effects is that there should be no omitted variable that confounds indirect effects. One way to address this untestable assumption is to conduct sensitivity analysis to assess whether the inference about an indirect effect would change under varying degrees of confounding bias. We developed a sensitivity analysis technique for a latent growth curve mediation model. We compute the biasing effect of confounding on point and confidence interval estimates of the indirect effects in a structural equation modeling framework. We illustrate sensitivity plots to visualize the effects of confounding on each indirect effect and present an empirical example to illustrate the application of the sensitivity analysis.     

Are Robust Standard Errors the Best Approach for Interval Estimation With Nonnormal Data in Structural Equation Modeling?

When the multivariate normality assumption is violated in structural equation modeling, a leading remedy involves estimation via normal theory maximum likelihood with robust corrections to standard errors. We propose that this approach might not be best for forming confidence intervals for quantities with sampling distributions that are slow to approach normality, or for functions of model parameters. We implement and study a robust analog to likelihood-based confidence intervals based on inverting the robust chi-square difference test of Satorra (2000). We compare robust standard errors and the robust likelihood-based approach versus resampling methods in confirmatory factor analysis (Studies 1 & 2) and mediation analysis models (Study 3) for both single parameters and functions of model parameters, and under a variety of nonnormal data generation conditions. The percentile bootstrap emerged as the method with the best calibrated coverage rates and should be preferred if resampling is possible, followed by the robust likelihood-based approach.     

Graphical Extension of Sample Size Planning With AIPE on RMSEA Using R

Kelley and Lai (2011) recently proposed the use of accuracy in parameter estimation (AIPE) for sample size planning in structural equation modeling. The sample size that reaches the desired width for the confidence interval of root mean square error of approximation (RMSEA) is suggested. This study proposes a graphical extension with the AIPE approach, abbreviated as GAIPE, on RMSEA to facilitate sample size planning in structural equation modeling. GAIPE simultaneously displays the expected width of a confidence interval of RMSEA, the necessary sample size to reach the desired width, and the RMSEA values covered in the confidence interval. Power analysis for hypothesis tests related to RMSEA can also be integrated into the GAIPE framework to allow for a concurrent consideration of accuracy in estimation and statistical power to plan sample sizes. A package written in R has been developed to implement GAIPE. Examples and instructions for using the GAIPE package are presented to help readers make use of this flexible framework. With the capacity of incorporating information on accuracy in RMSEA estimation, values of RMSEA, and power for hypothesis testing on RMSEA in a single graphical representation, the GAIPE extension offers an informative and practical approach for sample size planning in structural equation modeling.     

Nested structural equation models: Noncentrality and power of restriction test

This article is concerned with the difference in noncentrality parameters of nested structural equation models and their utility in evaluating statistical power associated with the pertinent restriction test. Based on the seminal work by Browne and Du Toit (1992), Steiger (1989, 1990), Steiger and Lind (1980), and Steiger, Shapiro, and Browne (1985), asymptotic confidence intervals for that difference are discussed. The intervals represent a useful adjunct to widely employed goodness‐of‐fit indexes and test statistics when assessing plausibility of constraints in nested models. The approach also permits estimating power of the test of validity of the nesting restrictions (cf. MacCallum, Browne, & Sugawara, 1996). It is illustrated on data from a 2‐group cognitive training study.     

Confidence Intervals of Fit Indexes by Inverting a Bootstrap Test

Fit indexes are an important tool in the evaluation of model fit in structural equation modeling (SEM). Currently, the newest confidence interval (CI) for fit indexes proposed by Zhang and Savalei (2016) is based on the quantiles of a bootstrap sampling distribution at a single level of misspecification. This method, despite a great improvement over naive and model-based bootstrap methods, still suffers from unsatisfactory coverage. In this work, we propose a new method of constructing bootstrap CIs for various fit indexes. This method directly inverts a bootstrap test and produces a CI that involves levels of misspecification that would not be rejected in a bootstrap test. Similar in rationale to a parametric CI of root mean square error of approximation (RMSEA) based on a noncentral χ2 distribution and a profile-likelihood CI of model parameters, this approach is shown to have better performance than the approach of Zhang and Savalei (2016), with more accurate coverage and more efficient widths.     

基于ARIMA-GARCH模型的城市主干道行程时间时变置信区间预测(英文)

为了提高行程时间预测的可靠性,构建了自回归综合移动平均与广义自回归条件异方差性(ARIMAGARCH)模型进行城市主干道行程时间动态置信区间预测,其中ARIMA模型作为GARCH模型的均值方程用于捕获行程时间均值,GARCH模型用于捕获行程时间条件方差.运用昆山市交通监测系统中采集的实际交通流数据进行验证和评估.结果表明,相较于传统的ARIMA模型,提出的方法虽然不能显著提升行程时间均值的预测性能,但是在行程时间波动性预测方面具有较大的优势.该方法可捕获行程时间异方差,从而能够预测出比ARIMA模型预测的固定置信区间更能反映行程时间观测值波动性的动态置信区间.     

Bayesian Analysis of Mixture Structural Equation Models With an Unknown Number of Components

Multivariate heterogenous data with latent variables are common in many fields such as biological, medical, behavioral, and social-psychological sciences. Mixture structural equation models are multivariate techniques used to examine heterogeneous interrelationships among latent variables. In the analysis of mixture models, determination of the number of mixture components is always an important and challenging issue. This article aims to develop a full Bayesian approach with the use of reversible jump Markov chain Monte Carlo method to analyze mixture structural equation models with an unknown number of components. The proposed procedure can simultaneously and efficiently select the number of mixture components and conduct parameter estimation. Simulation studies show the satisfactory empirical performance of the method. The proposed method is applied to study risk factors of osteoporotic fractures in older people.     

Using the friedman method of ranks for model comparison in structural equation modeling

Empirical researchers maximize their contribution to theory development when they compare alternative theory‐inspired models under the same conditions. Yet model comparison tools in structural equation modeling—χ² difference tests, information criterion measures, and screening heuristics—have significant limitations. This article explores the use of the Friedman method of ranks as an inferential procedure for evaluating competing models. This approach has attractive properties, including limited reliance on sample size, limited distributional assumptions, an explicit multiple comparison procedure, and applicability to the comparison of nonnested models. However, this use of the Friedman method raises important issues regarding the lack of independence of observations and the power of the test.     

A Comparison of Latent Growth Models for Constructs Measured by Multiple Items

A multiple testing approach is outlined that can be used to examine the assumption of underlying normal variables in latent variable models with categorical indicators. The method is based on an application of the increasingly popular Benjamini–Hochberg multiple testing procedure, and is readily applicable with widely circulated software. The discussed method is especially useful for ascertaining this assumption that is very often made in research based on structural equation modeling using models containing discrete outcomes. The described approach is illustrated with numerical data.     

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.     

Bayesian Quantile Structural Equation Models

Structural equation modeling is a common multivariate technique for the assessment of the interrelationships among latent variables. Structural equation models have been extensively applied to behavioral, medical, and social sciences. Basic structural equation models consist of a measurement equation for characterizing latent variables through multiple observed variables and a mean regression-type structural equation for investigating how explanatory latent variables influence outcomes of interest. However, the conventional structural equation does not provide a comprehensive analysis of the relationship between latent variables. In this article, we introduce the quantile regression method into structural equation models to assess the conditional quantile of the outcome latent variable given the explanatory latent variables and covariates. The estimation is conducted in a Bayesian framework with Markov Chain Monte Carlo algorithm. The posterior inference is performed with the help of asymmetric Laplace distribution. A simulation shows that the proposed method performs satisfactorily. An application to a study of chronic kidney disease is presented.     

Regularized Structural Equation Modeling

A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM’s utility.     

Monte Carlo Confidence Intervals for Complex Functions of Indirect Effects

One challenge in mediation analysis is to generate a confidence interval (CI) with high coverage and power that maintains a nominal significance level for any well-defined function of indirect and direct effects in the general context of structural equation modeling (SEM). This study discusses a proposed Monte Carlo extension that finds the CIs for any well-defined function of the coefficients of SEM such as the product of k coefficients and the ratio of the contrasts of indirect effects, using the Monte Carlo method. Finally, we conduct a small-scale simulation study to compare CIs produced by the Monte Carlo, nonparametric bootstrap, and asymptotic-delta methods. Based on our simulation study, we recommend researchers use the Monte Carlo method to test a complex function of indirect effects.     

误差和数据处理

“误差和数据处理”是分析化学数材中的重要部分,近年来教材中内容变化较大。本文就这方面的内容、重点和关键问题作了讨论。误差是测定值与真实值之差,表示测定的准确度;偏差是测定值与平均值之差,表示测定的精密度,二者既有联系又有差别。按高斯误差分布方程作图可得偶然误差的正态分布曲线,根据方程和曲线可得偶然误差的分布规律,可求出偶然误差出现的概率,从概率可得到平均值的置信区间。     

Implementing Restricted Maximum Likelihood Estimation in Structural Equation Models

Structural equation modeling (SEM) is now a generic modeling framework for many multivariate techniques applied in the social and behavioral sciences. Many statistical models can be considered either as special cases of SEM or as part of the latent variable modeling framework. One popular extension is the use of SEM to conduct linear mixed-effects modeling (LMM) such as cross-sectional multilevel modeling and latent growth modeling. It is well known that LMM can be formulated as structural equation models. However, one main difference between the implementations in SEM and LMM is that maximum likelihood (ML) estimation is usually used in SEM, whereas restricted (or residual) maximum likelihood (REML) estimation is the default method in most LMM packages. This article shows how REML estimation can be implemented in SEM. Two empirical examples on latent growth model and meta-analysis are used to illustrate the procedures implemented in OpenMx. Issues related to implementing REML in SEM are discussed.