, Volume 75, Issue 2, pp 243–248

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

Theory and Methods

DOI: 10.1007/s11336-009-9135-y

Cite this article as:
Satorra, A. & Bentler, P.M. Psychometrika (2010) 75: 243. doi:10.1007/s11336-009-9135-y


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.


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

Copyright information

© The Psychometric Society 2009

Authors and Affiliations

  1. 1.Department of Economics and BusinessUniversitat Pompeu FabraBarcelonaSpain
  2. 2.Department of PsychologyUniversity of California, Los AngelesLos AngelesUSA

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