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Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic


A scaled difference test statistic \(\tilde{T}{}_{d}\) that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic \(\tilde{T}_{d}\) is asymptotically equivalent to the scaled difference test statistic \(\bar{T}_{d}\) introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic \(\tilde{T}_{d}\) has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic \(\bar{T}_{d}\) that avoids negative chi-square values.

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Author information

Correspondence to Albert Satorra.

Additional information

Research supported by grants SEJ2006-13537 and PR2007-0221 from the Spanish Ministry of Science and Technology and by USPHS grants DA00017 and DA01070.

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Satorra, A., Bentler, P.M. Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic. Psychometrika 75, 243–248 (2010).

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  • moment-structures
  • goodness-of-fit test
  • chi-square difference test statistic
  • chi-square distribution
  • nonnormality