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Structural equation modeling (SEM) is widely used in behavioral, social, and education research. Drawing publication-ready path diagrams for SEM is not a pleasant task with the existing software. The article introduces an open-source web-based graphical application, "semdiag," for drawing WYSIWYG SEM path diagrams interactively. The application is an on-going project developed by the authors using JavaScript, and it can be used in major web browsers, both online and offline. Several examples are provided to demonstrate how to use the application.

The normal-distribution-based likelihood ratio statistic T[subscript ml] = nF[subscript ml] is widely used for power analysis in structural Equation modeling (SEM). In such an analysis, power and sample size are computed by assuming that T[subscript ml] follows a central chi-square distribution under H[subscript 0] and a noncentral chi-square distribution under H[subscript a]. However, with either violation of normality or not a large enough sample size, both empirical and analytical results indicate that the chi-square distribution assumptions are not realistic and consequently methods of power analysis based on such assumptions are not valid. This article describes a Monte Carlo (MC) method for power analysis. A measure of effect size for characterizing the power property of different rescaled statistics is also provided. Robust methods are proposed to increase the power of T[subscript ml] and other statistics. Simulation results show that the MC method reliably controls Type I errors and robust estimation methods effectively increase the power, and their combination is thus recommended for conducting power analysis in SEM.

Data in social and behavioral sciences typically contain measurement errors and do not have predefined metrics. Structural equation modeling (SEM) is widely used for the analysis of such data, where the scales of the manifest and latent variables are often subjective. This article studies how the model, parameter estimates, their standard errors (SEs), and the corresponding z-statistics are affected by the scales of the manifest and latent variables. Analytical and empirical results show that (1) the normal-distribution-based likelihood ratio statistic is scale-invariant with respect to scale changes of manifest and latent variables as well as to anchor change of latent variables; (2) the normal-distribution-based maximum likelihood (NML) parameter estimates are scale-equivariant with respect to scale-change of manifest and latent variables as well as to anchor change of latent variables; (3) standard errors (SEs) following the NML method are parallel-scale-equivariant with respect to scale changes of the manifest and latent variables; and (4) the z-statistics are scale-invariant with respect to scale changes of the manifest and latent variables. However, only (1) and (2) hold if latent variables are rescaled by changing anchors. Nevertheless, parameters that are not directly related to latent variables with changing anchors are still scale-equivariant and their z-statistics are still scale-invariant. The results are expected to advance understanding of SEM analysis, and also facilitate result interpretation and comparison across studies as in meta analysis. [ABSTRACT FROM AUTHOR]

A necessary step in applying bi-factor models is to evaluate the need for domain factors with a general factor in place. The conventional null hypothesis testing (NHT) was commonly used for such a purpose. However, the conventional NHT meets challenges when the domain loadings are weak or the sample size is insufficient. This article proposes using minimal-effect testing (MET) and equivalence testing (ET) to analyze bi-factor models. A key element in conducting MET and ET is the minimal size of factor loadings that can be regarded as noteworthy in practice, termed as minimal noteworthy size. This article presents two approaches to formulating the minimal noteworthy size and compares the pros and cons of MET, ET, and the conventional NHT. Analysis shows that MET, ET, and the conventional NHT are complementary. Combining them to test the noteworthiness of domain loadings can help researchers make a comprehensive judgment. Real and simulated datasets illustrate the applications of the three methods. Monte Carlo results show that MET and ET can control type I errors reasonably well while maintaining good statistical power. [ABSTRACT FROM AUTHOR]

Cronbach's coefficient alpha is a widely used reliability measure in social, behavioral, and education sciences. It is reported in nearly every study that involves measuring a construct through multiple items. With non-tau-equivalent items, McDonald's omega has been used as a popular alternative to alpha in the literature. Traditional estimation methods for alpha and omega often implicitly assume that data are complete and normally distributed. This study proposes robust procedures to estimate both alpha and omega as well as corresponding standard errors and confidence intervals from samples that may contain potential outlying observations and missing values. The influence of outlying observations and missing data on the estimates of alpha and omega is investigated through two simulation studies. Results show that the newly developed robust method yields substantially improved alpha and omega estimates as well as better coverage rates of confidence intervals than the conventional nonrobust method. An R package coefficientalpha is developed and demonstrated to obtain robust estimates of alpha and omega. [This paper was published in "Educational and Psychological Measurement" (EJ1099054).]

Multilevel structural equation models (MSEM) are typically evaluated on the basis of goodness of fit indices. A problem with these indices is that they pertain to the entire model, reflecting simultaneously the degree of fit for all levels in the model. Consequently, in cases that lack model fit, it is unclear which level model is misspecified. Although some researchers have suggested that model fit be assessed separately at all levels, this is not always ideal as it depends on the choice of fit indices, cutoff points, and magnitude of the intraclass correlation. Equivalence testing has recently been recommended as a superior way to evaluate the goodness of fit of models. The approach has also been heralded as a way to advance the inferential nature of structural equation modeling as a confirmatory tool. The purpose of this paper is to illustrate the use of equivalence testing for assessing multilevel model fit in educational research. Although the described procedures can be easily generalized to models with more than two levels, for simplicity we explicitly consider only two-level models.

Meta-analysis plays a key role in combining studies to obtain more reliable results. In social, behavioral, and health sciences, measurement units are typically not well defined. More meaningful results can be obtained by standardizing the variables and via the analysis of the correlation matrix. Structural equation modeling (SEM) with the combined correlations, called meta-analytical SEM (MASEM), is a powerful tool for examining the relationship among latent constructs as well as those between the latent constructs and the manifest variables. Three classes of methods have been proposed for MASEM: (1) generalized least squares (GLS) in combining correlations and in estimating the structural model, (2) normal-distribution-based maximum likelihood (ML) in combining the correlations and then GLS in estimating the structural model (ML-GLS), and (3) ML in combining correlations and in estimating the structural model (ML). The current article shows that these three methods are equivalent. In particular, (a) the GLS method for combining correlation matrices in meta-analysis is asymptotically equivalent to ML, (b) the three methods (GLS, ML-GLS, ML) for MASEM with correlation matrices are asymptotically equivalent, (c) they also perform equally well empirically, and (d) the GLS method for SEM with the sample correlation matrix in a single study is asymptotically equivalent to ML, which has being discussed extensively in the SEM literature regarding whether the analysis of a correlation matrix yields consistent standard errors and asymptotically valid test statistics. The results and analysis suggest that a sample-size weighted GLS method is preferred for combining correlations and for MASEM.

Latent class analysis (LCA) is a widely used technique for detecting unobserved population heterogeneity in cross-sectional data. Despite its popularity, the performance of LCA is not well understood. In this study, we evaluate the performance of LCA with binary data by examining classification accuracy, parameter estimation accuracy, and coverage rates of confidence intervals (CIs) through Monte Carlo simulation studies. We address the issue of label switching with a distance-based relabeling approach and introduce an index to measure separation among latent classes. Our results show that classification accuracy, parameter estimation accuracy, and CI coverage rates are primarily influenced by class separation and the number of indicators used for LCA. We recommend using a large sample size to mitigate the effects of tiny class sizes. Additionally, the study finds that the parametric bootstrap CIs perform comparably well or better when compared with the CIs based on the standard maximum likelihood method. [ABSTRACT FROM AUTHOR]

Indefinite symmetric matrices that are estimates of positive definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based theory for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.

The vast majority of structural equation models contain no mean structure, that is, the population means are estimated at the sample means and are then eliminated from modeling consideration. Generalized least squares methods are proposed to estimate potential mean structure parameters and to evaluate whether the given model can be successfully augmented with a mean structure. A simulation evaluates the performance of some alternative tests. A method that takes variability due to estimation of covariance structure parameters into account in the mean structure estimator, as well as in the weight matrix of the generalized least squares function, performs best. In small samples, the F-test and Yuan-Bentler adjusted X2 test perform best.