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

This article is concerned with measures of fit of a model. Two types of error involved in fitting a model are considered. The first is error of approximation which involves the fit of the model, wi...

Alternative Ways of Assessing Model Fit

Normed and nonnormed fit indexes are frequently used as adjuncts to chi-square statistics for evaluating the fit of a structural model A drawback of existing indexes is that they estimate no known population parameters A new coefficient is proposed to summarize the relative reduction in the noncentrality parameters of two nested models Two estimators of the coefficient yield new normed (CFI) and nonnormed (FI) fit indexes CFI avoids the underestimation of fit often noted in small samples for Bentler and Bonett's (1980) normed fit index (NFI) FI is a linear function of Bentler and Bonett's non-normed fit index (NNFI) that avoids the extreme underestimation and overestimation often found in NNFI Asymptotically, CFI, FI, NFI, and a new index developed by Bollen are equivalent measures of comparative fit, whereas NNFI measures relative fit by comparing noncentrality per degree of freedom All of the indexes are generalized to permit use of Wald and Lagrange multiplier statistics An example illustrates the behavior of these indexes under conditions of correct specification and misspecification The new fit indexes perform very well at all sample sizes

Comparative fit indexes in structural models

Relationship marketing—establishing, developing, and maintaining successful relational exchanges—constitutes a major shift in marketing theory and practice. After conceptualizing relationship marke...

The Commitment-Trust Theory of Relationship Marketing

Measurement invariance is usually tested using Multigroup Confirmatory Factor Analysis, which examines the change in the goodness-of-fit index (GFI) when cross-group constraints are imposed on a measurement model. Although many studies have examined the properties of GFI as indicators of overall model fit for single-group data, there have been none to date that examine how GFIs change when between-group constraints are added to a measurement model. The lack of a consensus about what constitutes significant GFI differences places limits on measurement invariance testing. We examine 20 GFIs based on the minimum fit function. A simulation under the two-group situation was used to examine changes in the GFIs (ΔGFIs) when invariance constraints were added. Based on the results, we recommend using Δcomparative fit index, ΔGamma hat, and ΔMcDonald's Noncentrality Index to evaluate measurement invariance. These three ΔGFIs are independent of both model complexity and sample size, and are not correlated with the o...

Evaluating Goodness-of-Fit Indexes for Testing Measurement Invariance

Discusses how current goodness-of-fit indices fail to assess parsimony and hence disconfirmability of a model and are insensitive to misspecifications of causal relations (a) among latent variables when measurement model with many indicators is correct and (b) when causal relations corresponding to free parameters expected to be nonzero turn out to be zero or near zero. A discussion of philosophy of parsimony elucidates relations of parsimony to parameter estimation, disconfirmability, and goodness of fit. AGFI in LISREL is rejected. A method of adjusting goodness-of-fit indices by a parsimony ratio is described. Also discusses less biased estimates of goodness of fit and a relative normedfit index for testing fit of structural model exclusive of the measurement model. By a goodness-of-fit index, in structural equations modeling, we mean an index for assessing the fit of a model to data that ranges in possible value between zero and unity, with zero indicating a complete lack of fit and unity indicating perfect fit. Although chi-square statistics are often used as goodness-of-fit indices, they range between zero and infinity, with zero indicating perfect fit and a large number indicating extreme lack of fit. We prefer to call chi-square and other indices with this property lack-of-fit indices. For a recent discussion of both lack-of-fit and goodness-of-fit indices, see Wheaton (I 988). In this article we evaluate the use of goodness-of-fit indices for the assessment of the fit of structural equation models to data. Our aim is to review their rationales and to assess their strengths and weaknesses. We also consider other aspects of the problem of evaluating a structural equation model with goodness-of-fit indices. For example, are certain goodness-of-fit indices to be used only in certain stages of research (a contention of Sobel & Bohrnstedt, 1985)? Or, how biased are estimates of goodness of fit in small samples? What bearing does parsimony have on assessing the goodness of fit of the model? Can goodness-of-fit indices focus on the fit of certain aspects of a model as opposed to the fit of the overall model? For example, to what extent do current goodness-of-fit indices fail to reveal poor fits in the structural submodel among the latent variables because of good fits in the measurement model relating latent variables to manifest indicators? We describe a goodness-of-fit index now

https://psycnet.apa.org/journals/bul/105/3/430.pdf

C. Dean Stilwell

Papers

Evaluation of goodness-of-fit indices for structural equation models