Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic |
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Authors: | Albert Satorra Peter M Bentler |
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Institution: | (1) Department of Child and Adolescent Psychiatry, Erasmus MC, Sophia Children’s Hospital, Rotterdam, The Netherlands;(2) Department of Basic Psychology and Methodology, Faculty of Psychology, University of Murcia, Espinardo Campus, 30100 Murcia, Spain |
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Abstract: | A scaled difference test statistic
(T)\tilde]d\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
(T)\tilde]d\tilde{T}_{d}
is asymptotically equivalent to the scaled difference test statistic
`(T)]d\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
(T)\tilde]d\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
`(T)]d\bar{T}_{d}
that avoids negative chi-square values. |
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Keywords: | |
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