8/12/2019 Bentler & Bonner- Significante Resrs and Goodness of Fit in the Analysis of Covariance

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Statistical methods for covariance structureanal-ysis were reviewedand found to be inadequatein large samplesdu e to almostcertain rejectionofany a prioristructural hypothesis. Fit indicesless influenced by sample size, ranging fromzeroto one, wereproposed asad ditional guidesfor model evaluation. Pseudo chi-square testsfor evaluating structuralm isspecificationwereintroduced. [The S C I and the S S C ! indicatethat this paper ha s been cited in over 260publications.1

Peter M. BentlerDepartment of Psychology

University of California

Los Angeles, CA 90024-1563and

Douglas G . BonettDepartment of Statistics

Universityof WyomingLaramie, WY 82071

july24, 1987

Th e problem ofmodel evaluationaddressedin this paper was widely known informallyand, in the caseofexploratory factoranalysis,had a long history.

Whentheir assumptionsare met, statisticalmethods for evaluating models provide cleardecision rules regardingmodel adequacy. Yet,in large samples virtuallyany model may berejected, even ifth e degree of misspecificationisvery minor. In factor analysis widespreadexperience had suggested that reliance on astatisticaldecision rule might not beoptimalfor determining thenumber of factors, i.e., inmodelchoice. Similar susp icionswere begin-fling toarise in the newly developing area ofcovariance structure analysis, as epitomizedbyLISREL,a computer program. These meth-od swere becomingpopularnot onlybecausethey could answer new types of questionsabout data, especially nonexperimental data,butalso becausethey provided an aura ofob-

jectivityd ue to their use of maximum likeli-

hood methods.However, many researcherswere becoming frustrated with using thesemethods because they had toreject modelsthat seemedtohave only minordiscrepanciesbetween the estimated model and data.Clearly, whatw as needed was a new way toevaluate model adequacy.

InAugust1979we recognized that generalhierarchical modelcomparisons, when stan-dardized to a baseline or null model, couldplay acriticalrole in defining normed andnon-

normed fit indices thatare less influenced bysamplesize than a statistical goodness-of-fittest. We immediately understood the poten-tiaJimpact ofour approach,and we were abletotestthe ideas and write them upwithin afew weeksfasterthan any other paper eitherof u s has everwritten. Infact, the basic ideaswere drafted overa weekend. Developmentof the pseudo chi-square test took much ofourtime.Our fit indices have become very popular,

and the citationstoourarticle largelyreflectthispo pularity.Variants that penalize numberof parameters differently and that utilize al-ternative nullmodels or a nonnormed indexthat may be less influenced bysample size

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have also been proposed. Different types ofindices that do no t depend on a null model,a s developedmore recentlyby LISRELand gen-eralized byBentler,

3havebeen proposed, but

the evidence appears tobe that our originalindices perform aswella s or better thanthe

neweralternatives.45Whileweappreciatethe impact that ourar-

ticle ha s had, we are alsofrustrated by thefactthat its ke yoriginal statistical contribution, thepseudo chi-square test for evaluating modelmisspeciuication, h asbeen largelyoverlooked.However,recent workhasacknowledged theimportanceofthistest,espe cially in locatinga fundamental misspecification of the mea-surement model.

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1987bylSl C U R R E N T CONTENTS

This Weeks Citation C las s ic SE 1987[ Bentler P M & Bonett D G. Significance tests a n d goodness of lit in the analysis ofcovariance structures. Psycho!. Buff. 88:588-606. 1980.

ltiniversity of Cal)forrtia. Los Ange l e s . CA l

I. Sobel 51 K & Bohrnstedt C W. Use of null modets in evaluating th e fit ol cooariance structure models. (Tuma N B. ed)Sociological methodology 1985. San Francisco: Jossey-Bass. 985. p. 52-78.

2. Bollen K A. Sample sizeand Bernice an d Boneits nonnormed (it index. Psvchomerrika 51:375-7, 1986.

3. Broiler P M. Somecontributions to efliciern Statistics ~flstructural models: specif ication an d estimation of momentstructures. Psychontcrrika 48:493-517. 1983. (Cited 2 5 times.)

4. Graham . 1 W& Collins 1 . sI. Samplesize and fit indices inanalssisof covariance structureanalysis. Unpublished paperpresented at the meetings of th e Psychometric Society. J u n e 987, Montreal. Canada.

5. Wheaton B. Assessnsenl of fit in overidentilied models with latent variabt~s.(Long I S. cd) Common problents inquantiratisc social research. Beverly Hills: Sage. (In press.)

6. Anderson J C & G~rbingD W. Structural equat ion modelingin practice: a review an d recommended two-stepapproach.Psychol. Bull. In press.)