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Journal ArticleDOI

Comparative fit indexes in structural models

01 Mar 1990-Psychological Bulletin (American Psychological Association)-Vol. 107, Iss: 2, pp 238-246
TL;DR: A new coefficient is proposed to summarize the relative reduction in the noncentrality parameters of two nested models and two estimators of the coefficient yield new normed (CFI) and nonnormed (FI) fit indexes.
Abstract: 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
Citations
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Journal ArticleDOI
TL;DR: In this article, the adequacy of the conventional cutoff criteria and several new alternatives for various fit indexes used to evaluate model fit in practice were examined, and the results suggest that, for the ML method, a cutoff value close to.95 for TLI, BL89, CFI, RNI, and G...
Abstract: 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...

76,383 citations


Cites background or methods from "Comparative fit indexes in structur..."

  • ...an index developed by Bollen (BL89; 1989), Bender's (1989, 1990) and McDonald and Marsh's (1990) Relative Noncentrality Index (RNI), and Bender's Comparative Fit Index (CFI)....

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  • ...As noted by Bentler and Bonett (1980), fit indexes were designed to avoid some of the problems of sample size and distributional misspecification associated with the conventional overall test of fit (the % statistic) in the evaluation of a model....

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  • ...Hu and Bentler (1997) found that a designated cutoff value may not work equally well with various types of fit indexes, sample sizes, estimators, or distributions....

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  • ...Bentler (1999) Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives, Structural Equation Modeling: A Multidisciplinary Journal, 6:1, 1-55, DOI: 10....

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Journal ArticleDOI
TL;DR: Relationship marketing, established, developing, and maintaining successful relational exchanges, constitutes a major shift in marketing theory and practice as mentioned in this paper, after conceptualizing relationship relationships as a set of relationships.
Abstract: Relationship marketing—establishing, developing, and maintaining successful relational exchanges—constitutes a major shift in marketing theory and practice. After conceptualizing relationship marke...

19,920 citations


Cites methods from "Comparative fit indexes in structur..."

  • ...The proposed structural model's comparative fit index, CFI (Bentler 1990), of ....

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Journal ArticleDOI
TL;DR: In this paper, the authors examined the change in the goodness-of-fit index (GFI) when cross-group constraints are imposed on a measurement model and found that the change was independent of both model complexity and sample size.
Abstract: 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...

10,597 citations


Cites methods from "Comparative fit indexes in structur..."

  • ...Incremental GFIs, including the Normed Fit Index (Bentler & Bonett, 1980), Relative Fit Index (RFI; Bollen, 1986), Incremental Fit Index (IFI; Bollen 1989a), TLI (Tucker & Lewis, 1973), CFI (Bentler, 1990), and Relative Noncentrality Index (RNI; McDonald & Marsh, 1990)....

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  • ...Some in common use include the comparative fit index (CFI; Bentler, 1990), Tucker–Lewis Index (TLI; Tucker & Lewis, 1973), Normed Fit Index (NNFI; Bentler & Bonett, 1980), and root mean squared 234 CHEUNG AND RENSVOLD 2 min ˆ( 1) (1)N Fχ error of approximation (RMSEA; Steiger, 1989)....

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  • ...Some in common use include the comparative fit index (CFI; Bentler, 1990), Tucker–Lewis Index (TLI; Tucker & Lewis, 1973), Normed Fit Index (NNFI; Bentler & Bonett, 1980), and root mean squared 234 CHEUNG AND RENSVOLD...

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Journal ArticleDOI
TL;DR: In this article, 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.
Abstract: 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.

9,249 citations


Cites methods or result from "Comparative fit indexes in structur..."

  • ...A noncentrality fit index usually involves first defining a population-fit-index parameter and then using estimators of this parameter to define the sample-fit index (Bentler, 1990; McDonald, 1989; McDonald & Marsh, 1990; Steiger, 1989)....

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  • ...As a consequence, alternative measures of fit, namely, so-called fit indices, were developed and recommended as plausible additional measures of model fit (e.g., Akaike, 1987; Bentler, 1990; Bentler & Bonett, 1980; Bollen, 1986, 1989; lames, Mulaik, & Brett, 1982; JOreskog & Sorbom, 1981; Marsh, Balla, & McDonald, 1988; McDonald, 1989; McDonald & Marsh, 1990; Steiger & Lind, 1980; Tanaka, 1987; Tanaka & Huba, 1985; Tucker & Lewis, 1973)....

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  • ...This is consistent with the theoretically predicted asymptotic properties and has been noted previously in several other studies (e.g., Bearden et al., 1982; Bentler, 1990; La Du & Tanaka, 1989)....

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Journal ArticleDOI
TL;DR: In this article, a selection of fit indices that are widely regarded as the most informative indices available to researchers is presented, along with guidelines on their use and strategies for their use.
Abstract: The following paper presents current thinking and research on fit indices for structural equation modelling. The paper presents a selection of fit indices that are widely regarded as the most informative indices available to researchers. As well as outlining each of these indices, guidelines are presented on their use. The paper also provides reporting strategies of these indices and concludes with a discussion on the future of fit indices.

7,904 citations


Cites background or methods or result from "Comparative fit indexes in structur..."

  • ...This index was first introduced by Bentler (1990) and subsequently included as part of the fit indices in his EQS program (Kline, 2005)....

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  • ...The Comparative Fit Index (CFI: Bentler, 1990) is a revised form of the NFI which takes into account sample size (Byrne, 1998) that performs well even when sample size is small (Tabachnick and Fidell, 2007)....

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  • ...However in situations were small samples are used, the value of the NNFI can indicate poor fit despite other statistics pointing towards good fit (Bentler, 1990; Kline, 2005; Tabachnick and Fidell, 2007)....

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  • ...A major drawback to this index is that it is sensitive to sample size, underestimating fit for samples less than 200 (Mulaik et al, 1989; Bentler, 1990), and is thus not recommended to be solely relied on (Kline, 2005)....

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References
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Journal ArticleDOI
TL;DR: In this article, a general null model based on modified independence among variables is proposed to provide an additional reference point for the statistical and scientific evaluation of covariance structure models, and the importance of supplementing statistical evaluation with incremental fit indices associated with the comparison of hierarchical models.
Abstract: Factor analysis, path analysis, structural equation modeling, and related multivariate statistical methods are based on maximum likelihood or generalized least squares estimation developed for covariance structure models. Large-sample theory provides a chi-square goodness-of-fit test for comparing a model against a general alternative model based on correlated variables. This model comparison is insufficient for model evaluation: In large samples virtually any model tends to be rejected as inadequate, and in small samples various competing models, if evaluated, might be equally acceptable. A general null model based on modified independence among variables is proposed to provide an additional reference point for the statistical and scientific evaluation of covariance structure models. Use of the null model in the context of a procedure that sequentially evaluates the statistical necessity of various sets of parameters places statistical methods in covariance structure analysis into a more complete framework. The concepts of ideal models and pseudo chi-square tests are introduced, and their roles in hypothesis testing are developed. The importance of supplementing statistical evaluation with incremental fit indices associated with the comparison of hierarchical models is also emphasized. Normed and nonnormed fit indices are developed and illustrated.

16,420 citations

Book
01 Jan 1989

9,143 citations


"Comparative fit indexes in structur..." refers methods in this paper

  • ...They are routinely available in a public computer program (Bentler, 1986, 1989) and are typically applied to compare nested submodels....

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Journal ArticleDOI
TL;DR: In this paper, a reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis, which can indicate that an otherwise acceptable factor model does not exactly represent the interrelations between the attributes for a population.
Abstract: Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution.

6,359 citations

Journal ArticleDOI
TL;DR: In this article, the entropy-based information criterion (AIC) has been extended in two ways without violating Akaike's main principles: CAIC and CAICF, which make AIC asymptotically consistent and penalize overparameterization more stringently.
Abstract: During the last fifteen years, Akaike's entropy-based Information Criterion (AIC) has had a fundamental impact in statistical model evaluation problems. This paper studies the general theory of the AIC procedure and provides its analytical extensions in two ways without violating Akaike's main principles. These extensions make AIC asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the “true” models. These selection criteria are called CAIC and CAICF. Asymptotic properties of AIC and its extensions are investigated, and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different Monte Carlo experiments under different conditions.

3,850 citations

Journal ArticleDOI
TL;DR: In this paper, the influence of sample size on different goodness-of-fit indices used in confirmatory factor analysis (CFA) was examined and the results are consistent with the observation that the amount of random, unexplained variance varies inversely with sample size.
Abstract: This investigation examined the influence of sample size on different goodness-of-fit indices used in confirmatory factor analysis (CFA). The first two data sets were derived from large normative samples of responses to a multidimensional self-concept instrument and to a multidimensional instrument used to assess students' evaluations of teaching effectiveness. In the third set, data were simulated and generated according to the model to be tested. In the fourth, data were simulated and generated according to a three-factor model that did not have a simple structure. Twelve fit indicators were used to assess goodness-offit in all CFAs. All analyses were conducted with the LISREL V package. One-way ANOVAs and a visual inspection of graphs were used to assess the sample size effect on each index for the four data sets. Despite the inconsistency of the findings with previous claims, the results are consistent with the observation that the amount of random, unexplained variance varies inversely with sample size. Appendices include a set of computed statements, an explanation and listing of the 12 goodness-of-fit indicators, a bibliography, a table of results, and figures showing sample size effect. (Author/LMO) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ***********************************************************************

3,746 citations