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A scaled difference chi-square test statistic for moment structure analysis

Listed author(s):
  • Albert Satorra


  • Peter Bentler

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler'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 say ${\cal M}_0$ implies on a less restricted one ${\cal M}_1$. If $T_0$ and $T_1$ denote the goodness-of-fit test statistics associated to ${\cal M}_0$ and ${\cal M}_1$, respectively, then typically the difference $T_d = T_0 - T_1$ is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models ${\cal M}_0$ and ${\cal M}_1$. As in the case of the goodness-of-fit test, it is of interest to scale the statistic $T_d$ in order to improve its chi-square approximation in realistic, i.e., nonasymptotic and nonnormal, applications. In a recent paper, Satorra (1999) shows that the difference between two Satorra- Bentler 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 chi-square statistic from the scaled goodness-of-fit test statistics of models ${\cal M}_0$ and ${\cal M}_1$. A Monte Carlo study is provided to illustrate the performance of the competing statistics.
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Article provided by Springer & The Psychometric Society in its journal Psychometrika.

Volume (Year): 66 (2001)
Issue (Month): 4 (December)
Pages: 507-514

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Handle: RePEc:spr:psycho:v:66:y:2001:i:4:p:507-514
DOI: 10.1007/BF02296192
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  1. Albert Satorra, 1991. "Asymptotic robust inferences in the analysis of mean and covariance structures," Economics Working Papers 3, Department of Economics and Business, Universitat Pompeu Fabra.
  2. Albert Satorra, 1999. "Scaled and adjusted restricted tests in multi-sample analysis of moment structures," Economics Working Papers 395, Department of Economics and Business, Universitat Pompeu Fabra.
  3. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer;The Psychometric Society, vol. 54(1), pages 131-151, March.
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