A scaled difference chisquare test statistic for moment structure analysis
 Albert Satorra,
 Peter M. Bentler
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A family of scaling corrections aimed to improve the chisquare approximation of goodnessoffit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, SatorraBentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model sayM _{0} implies on a less restricted oneM _{1}. IfT _{0} andT _{1} denote the goodnessoffit test statistics associated toM _{0} andM _{1}, respectively, then typically the differenceT _{ d }=T _{0}−T _{1} is used as a chisquare test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the modelsM _{0} andM _{1}. As in the case of the goodnessoffit test, it is of interest to scale the statisticT _{ d } in order to improve its chisquare approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chisquare statistic from the scaled goodnessoffit test statistics of modelsM _{0} andM _{1}. A Monte Carlo study is provided to illustrate the performance of the competing statistics.
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 Title
 A scaled difference chisquare test statistic for moment structure analysis
 Journal

Psychometrika
Volume 66, Issue 4 , pp 507514
 Cover Date
 20011201
 DOI
 10.1007/BF02296192
 Print ISSN
 00333123
 Online ISSN
 18600980
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 momentstructures
 goodnessoffit test
 chisquare difference test statistic
 chisquare distribution
 nonnormality
 Industry Sectors
 Authors

 Albert Satorra ^{(1)}
 Peter M. Bentler ^{(2)}
 Author Affiliations

 1. Universitat Pompeu Fabra, Ramon Trias Fargas, 25–27, 08005, Barcelona, Spain
 2. University of California, Los Angeles