ENSURING POSITIVENESS OF THE SCALED DIFFERENCE CHI-SQUARE TEST STATISTIC

A scaled difference test statistic (T) over tilde (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) over tilde (d) is asymptotically equivalent to the scaled difference test statistic (T) over bar (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) over tilde (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) over bar (d) that avoids negative chi-square values.