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 ...

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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