This article examines the adequacy of the “rules of thumb” conventional cutoff criteria and several new alternatives for various fit indexes used to evaluate model fit in practice. Using a 2‐index presentation strategy, which includes using the maximum likelihood (ML)‐based standardized root mean squared residual (SRMR) and supplementing it with either Tucker‐Lewis Index (TLI), Bollen's (1989) Fit Index (BL89), Relative Noncentrality Index (RNI), Comparative Fit Index (CFI), Gamma Hat, McDonald's Centrality Index (Mc), or root mean squared error of approximation (RMSEA), various combinations of cutoff values from selected ranges of cutoff criteria for the ML‐based SRMR and a given supplemental fit index were used to calculate rejection rates for various types of true‐population and misspecified models; that is, models with misspecified factor covariance(s) and models with misspecified factor loading(s). The results suggest that, for the ML method, a cutoff value close to .95 for TLI, BL89, CFI, RNI, and G...

Cutoff criteria for fit indexes in covariance structure analysis : Conventional criteria versus new alternatives

In this article, we provide guidance for substantive researchers on the use of structural equation modeling in practice for theory testing and development. We present a comprehensive, two-step modeling approach that employs a series of nested models and sequential chi-square difference tests. We discuss the comparative advantages of this approach over a one-step approach. Considerations in specification, assessment of fit, and respecification of measurement models using confirmatory factor analysis are reviewed. As background to the two-step approach, the distinction between exploratory and confirmatory analysis, the distinction between complementary approaches for theory testing versus predictive application, and some developments in estimation methods also are discussed.

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Structural equation modeling in practice: a review and recommended two-step approach

This study evaluated the sensitivity of maximum likelihood (ML)-, generalized least squares (GLS)-, and asymptotic distribution-free (ADF)-based fit indices to model misspecification, under conditions that varied sample size and distribution. The effect of violating assumptions of asymptotic robustness theory also was examined. Standardized root-mean-square residual (SRMR) was the most sensitive index to models with misspecified factor covariance(s), and Tucker-Lewis Index (1973; TLI), Bollen's fit index (1989; BL89), relative noncentrality index (RNI), comparative fit index (CFI), and the MLand GLS-based gamma hat, McDonald's centrality index (1989; Me), and root-mean-square error of approximation (RMSEA) were the most sensitive indices to models with misspecified factor loadings. With ML and GLS methods, we recommend the use of SRMR, supplemented by TLI, BL89, RNI, CFI, gamma hat, Me, or RMSEA (TLI, Me, and RMSEA are less preferable at small sample sizes). With the ADF method, we recommend the use of SRMR, supplemented by TLI, BL89, RNI, or CFI. Finally, most of the ML-based fit indices outperformed those obtained from GLS and ADF and are preferable for evaluating model fit.

Fit indices in covariance structure modeling : Sensitivity to underparameterized model misspecification

Practical problems that are frequently encountered in applications of covariance structure analysis are discussed and solutions are suggested. Conceptual, statistical, and practical requirements fo...

Practical Issues in Structural Modeling

We provide a comprehensive and user-friendly compendium of standards for the use and interpretation of structural equation models (SEMs). To both read about and do research that employs SEMs, it is necessary to master the art and science of the statistical procedures underpinning SEMs in an integrative way with the substantive concepts, theories, and hypotheses that researchers desire to examine. Our aim is to remove some of the mystery and uncertainty of the use of SEMs, while conveying the spirit of their possibilities.

Specification, evaluation, and interpretation of structural equation models

http://www.fss.uu.nl/pubs/pgmvanderheijden/48.aleast-squaresalgorithmforamixturemodelforcompositionaldata..pdf

A least squares algorithm for a mixture model for compositional data

Starting with Kenny and Judd (Psychol. Bull. 96:201–210, 1984) several methods have been introduced for analyzing models with interaction terms. In all these methods more information from the data than just means and covariances is required. In this paper we also use more than just first- and second-order moments; however, we are aiming to adding just a selection of the third-order moments. The key issue in this paper is to develop theoretical results that will allow practitioners to evaluate the strength of different third-order moments in assessing interaction terms of the model. To select the third-order moments, we propose to be guided by the power of the goodness-of-fit test of a model with no interactions, which varies with each selection of third-order moments. A theorem is presented that relates the power of the usual goodness-of-fit test of the model with the power of a moment test for the significance of third-order moments; the latter has the advantage that it can be computed without fitting a model. The main conclusion is that the selection of third-order moments can be based on the power of a moment test, thus assessing the relevance in the analysis of different sets of third-order moments can be computationally simple. The paper gives an illustration of the method and argues for the need of refraining from adding into the analysis an excess of higher-order moments.

/pdf/moment-testing-for-interaction-terms-in-structural-equation-3z6p6743ko.pdf

Moment testing for interaction terms in structural equation modeling

The psychometric properties of the Dutch Teacher’s Report Form (TRF) for teachers of Unaccompanied Refugee Minors (URM) were evaluated in this study. The teachers (n = 486) that participated received a Dutch TRF to report on the mental health of the unaccompanied minor. Hierarchical confirmative factor analysis and individual confirmatory factor analyses support the a priori structure of the Dutch TRF. However, the Thought Problems subscale could not be verified in this study, suggesting that some of the problem behavior reported by teachers of URM differs from that of parent reports or that the item constellation of the Dutch TRF is different for teachers of URM. The total Internalizing and Externalizing scales show good internal consistency. The construct and concurrent validity of the Dutch TRF were found to be acceptable. The results suggest that the Dutch TRF is a reliable and valid instrument to assess emotional and behavior problems of URM.

Validation of the Teacher’s Report Form for Teachers of Unaccompanied Refugee Minors

The authors propose some new log-bilinear models for the analysis of asymmetry in square contingency tables. In these models the logarithm of the expected frequency is split up into two parts: (a) a symmetric or a quasi-symmetric part, and (b) a skew-symmetric part. The skew-symmetric part is decomposed using the so-called Gower decomposition of skew-symmetric matrices. It is shown how graphical representations of this decomposition can simplify the interpretation.

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Some new log bilinear models for the analysis of asymmetry in a square contingency table

We discuss structural equation models for non-normal variables. In this situation the maximum likelihood and the generalized least-squares estimates of the model parameters can give incorrect estimates of the standard errors and the associated goodness-of-fit chi-squared statistics. If the sample size is not large, for instance smaller than about 1000, asymptotic distribution-free estimation methods are also not applicable. This paper assumes that the observed variables are transformed to normally distributed variables. The non-normally distributed variables are transformed with a Box–Cox function. Estimation of the model parameters and the transformation parameters is done by the maximum likelihood method. Furthermore, the test statistics (i.e. standard deviations) of these parameters are derived. This makes it possible to show the importance of the transformations. Finally, an empirical example is presented.

Ab Mooijaart

Papers

A least squares algorithm for a mixture model for compositional data

Moment testing for interaction terms in structural equation modeling

Validation of the Teacher’s Report Form for Teachers of Unaccompanied Refugee Minors

Some new log bilinear models for the analysis of asymmetry in a square contingency table

Estimating structural equation models with non-normal variables by using transformations