Comparative Fit Indices in Structural Models

otJ.x?tNrtr;A;rtt7e- eta, Comparative Fit lndexes in Structural Models P. M. Bentler University of California, Los Angeles Norincd and nonnormed fit indexes are frequently used as adjuncts to chi-square statistics for evalu- ating the fit of it structural model A drawback of existing indexes is that they estimate no known poptilatioit pzirtimetcrs A new coethcieni is proposed to summarize the relative reduction in the noncentrality parameters oftwo nested models. Two estimators olthe coefficient yield new normed (CH) and nonnormed (Fl) fit indexes. CFI avoids the underestimation of fit often noted in small samples for Bentler and Bonctts (l9es'0) normed fit index (l\lFl). F1 is a linear function of Berttlcr and Bonetts non—normed fit index (NNFI) that avoids the extreme underestimation and overestima- tion often found in NNFI. Asymptotically, CFl, Fl, NFL and a new index developed by Bollen are equivalent measures of comparative fit. whereas NNFl measures relative fit by comparing noncon- trality per degree offreedom. All ofthc indexes are generalized to permit use of Wald and Lagrange multiplier statistics. An example illustrates the behavior ofthese indexes under conditions ofcorrect specification and misspecification. The new fit indexes perform very well at all sample sizes. As is well known, the goodness-of—fit test statistic T used in -C‘-ql the ad; , ‘_ ., ferred to the chi-square distribution to determine acceptance or rejection ofa specific null hypothesis, 2 = 2(6). In the context of covariance structure analysis, 2 is the population covariance matrix and 6 is a vector of more basic parameters, for example, the factor loadings and intercorrelations and unique variances in a confirmatory factor analysis. The statistic T reflects the closeness of E ‘='‘ 2((}), based on the estimator 5, to the sample matrix S, the sample covariance matrix in covariance structure analysis, in the chi-square metric. Acceptance or rejection of the null hypothesis via a test based on T may be inappropriate or incomplete in model evaluation for several reasons: cl‘ 3 sirtzcttirnl n 1. Some basic assumptions underlying T may be false and the distribution of the statistic may not be robust to violation ofthese assumptions. 2. No specific model 2(6) may be assumed to exist in the population, and Tis intended to provide a summary regarding closeness ofZ to S, but not necessarily a test of E = 2(6). 3. in small samples, T may not be chi-square distributed; hence, the probability values used to evaluate the null hypothe- sis may not be correct. This research was supported in part by United States Public Health Service Grants DA0107O and DAOOOI7 and is based on a February 1988 technical report and a paper presented at the Psychoineiric Society meetings, June I988, Los Angeles. 1 Helpful dlSCUSSl0uS with J. de Leeuw, R. I. Jennrich. T. A. B. Snijders. and J. A. Woodward; the computer assistance ofShinn-Tzong Wu: and the production assistance of Julie Specl-tart are gratefully acknowl- edged. Correspondence concerning this article should be addressed to P. M. Bentler, Department Cf Psychology, University ofCalifornia, Los Ange- les. California 90024-1563. 4. in large samples, any a priori hypothesis 2 = 2(6), al- , may be rejected. As a consequence, the statistic T may not be clearly interpret- able, and transformations of Tdesigned to map it into a more interpretable 0-1, or approximate 0-1, range have been devel- oped. Those indexes are usually called goodness-of-fit indexes (e.g., Bentler, 1983, p. 507; loreskog & Sijrborn, 1984, p. 1.40). A related class of indexes, here called comparative goodness-of fit indexes, assess T in relation to the fit of a more restrictive model. These comparative fit indexes, formalized by Bentler and Bonett {I980}, are very widely used (Bentler dc Bonett, 1987) and are the sole object of this article. Alternative ap- proaches to evaluating model adequacy are reviewed elsewhere (e.g.., Bollen & Liang, l985; Bozdogan, i987; i,aDu & Tanaka, in press; Wheaten, l987). Although covariance structure analy- sis is emphasized, the methods developed here hold for any type of structural model, including, for example, mean-covariance structures and log—linear models. Although more than 30 fit indexes have been reported and their empirical behavior studied (Marsh, Balla, it McDonald, i988), and although new ones continue to be developed {Bollen, 1989). it is surprising to note that they have been developed as purely descripti‘ 3 statistics. Apparently, no population parame- ter has been defined that is being estimated by any ofthe exist- ing indexes. ln this article, I define an explicit population com- parative ftt coefficient, provide two alternative estimators of the coefficient, and investigate the asymptotic relations between the new and previously defined comparative fit indexes. Further- more, new indexes based on Wald and Lagrange multiplier sta- tistics are developed. Nested Models and Comparative Fit In evaluating comparative model fit, it is helpful to focus on more than one pair of models. Consider a series of nested models, Psychological Bulletin. 1990, Vol. l07, No. 2, 238 -246 Copynght I990 by the American Psychological Association. 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