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

Structural equation modeling (SEM) is a vast field and widely used by many applied researchers in the social and behavioral sciences. Over the years, many software packages for structural equation modeling have been developed, both free and commercial. However, perhaps the best state-of-the-art software packages in this field are still closed-source and/or commercial. The R package lavaan has been developed to provide applied researchers, teachers, and statisticians, a free, fully open-source, but commercial-quality package for latent variable modeling. This paper explains the aims behind the development of the package, gives an overview of its most important features, and provides some examples to illustrate how lavaan works in practice.

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lavaan: An R Package for Structural Equation Modeling

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

Despite the widespread use of exploratory factor analysis in psychological research, researchers often make questionable decisions when conducting these analyses. This article reviews the major design and analytical decisions that must be made when conducting a factor analysis and notes that each of these decisions has important consequences for the obtained results. Recommendations that have been made in the methodological literature are discussed. Analyses of 3 existing empirical data sets are used to illustrate how questionable decisions in conducting factor analyses can yield problematic results. The article presents a survey of 2 prominent journals that suggests that researchers routinely conduct analyses using such questionable methods. The implications of these practices for psychological research are discussed, and the reasons for current practices are reviewed.

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Evaluating the use of exploratory factor analysis in psychological research.

For structural equation models, a huge variety of fit indices has been developed. These indices, however, can point to conflicting conclusions about the extent to which a model actually matches the observed data. The present article provides some guidelines that should help applied researchers to evaluate the adequacy of a given structural equation model. First, as goodness-of-fit measures depend on the method used for parameter estimation, maximum likelihood (ML) and weighted least squares (WLS) methods are introduced in the context of structural equation modeling. Then, the most common goodness-of-fit indices are discussed and some recommendations for practitioners given. Finally, we generated an artificial data set according to a "true" model and analyzed two misspecified and two correctly specified models as examples of poor model fit, adequate fit, and good fit.

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Evaluating the Fit of Structural Equation Models: Tests of Significance and Descriptive Goodness-of-Fit Measures.

Monte Carlo computer simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood )~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the Satorra-Bentler rescaled X 2 (SB) were examined under varying conditions of sample size, model specification, and multivariate distribution. For properly specified models, ML and SB showed no evidence of bias under normal distributions across all sample sizes, whereas ADF was biased at all but the largest sample sizes. ML was increasingly overestimated with increasing nonnormality, but both SB (at all sample sizes) and ADF (only at large sample sizes) showed no evidence of bias. For misspecified models, ML was again inflated with increasing nonnormality, but both SB and ADF were underestimated with increasing nonnormality. It appears that the power of the SB and ADF test statistics to detect a model misspecification is attenuated given nonnormally distributed data.

The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis.

Structural equation models with non-normal variables: Problems and remedies

Structural equation models with nonnormal variables: Problems and remedies.

The investigation of personality structure: Statistical models

Following a quantitative review examining the relative influence of positive and negative social exchanges on emotional health, a second-order factor model of negative social exchange was tested and supported in a sample of 906 college students. Structural equation analysis exploring the unique effects of negative social exchange and perceived support satisfaction on depression revealed both variables to predict this outcome, over and above the contribution of personality and coping variables. In addition to the direct effects of the Big Five personality dimensions on depression, indirect effects of the Big Five via negative social exchange, support satisfaction, and avoidant coping also were observed. Although the prevailing belief among support researchers is that the negative effects of conflictual social interactions outweigh the positive effects of social support, the results of the quantitative review suggest boundary conditions that qualify this conclusion. In addition to corroborating a hierarchical measurement model of negative social exchange, the results of the present investigation suggest that how social support and social negativity are measured substantially influences conclusions concerning their relative impact on psychological distress.

John F. Finch

Papers

The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis.

Structural equation models with non-normal variables: Problems and remedies

Structural equation models with nonnormal variables: Problems and remedies.

The investigation of personality structure: Statistical models

A Comparison of the Influence of Conflictual and Supportive Social Interactions on Psychological Distress