Noncentrality-Based Indices of Fit - Extensions to Multiple Group Analysis
When more than one group is analyzed, the Chi-square statistic is a weighted sum of the discrepancy functions obtained from the individual groups. If the sample sizes are equal, the noncentrality-based indices discussed above generalize in a way that is completely straightforward. When sample sizes are unequal, this is not so, although SEPATH will still compute modified versions of the indices as described below, and these will still be of considerable value in assessing model fit.
With K independent samples, the overall Chi-square statistic is of the form
(111)
where
(112)
and
(113)
The Chi-square statistic is then computed as
(114)
This statistic has, under the assumptions of Steiger, Shapiro, and Browne (1985) a large sample distribution that is approximated by a noncentral Chi-square distribution, with n degrees of freedom, and a noncentrality parameter equal to
(115)
where 
is the population discrepancy function for the kth group.
One can estimate this noncentrality parameter and set confidence intervals on it. However, inference to relevant population quantities is less straightforward. Consider, for example, the point estimate analogous to the single sample case. The statistic
(116)
has an expected value of approximately
(117)
or
(118)
where ck is as defined above. This demonstrates that we can estimate a weighted average of the discrepancies for each sample, where the weights sum to 1, and are a function of sample size. If the sample sizes are equal, the weighted average becomes the simple arithmetic average, or mean, and so we can also estimate the unweighted sum of discrepancies.
How one should this information to produce multiple group versions of the RMSEA, and population gamma indices is open to some question when sample sizes are not equal. Perhaps the most natural candidates for the population RMSEA would be an "unweighted" index,
(119)
and a "weighted" index
(120)
When sample sizes are equal, both are the same.
Unfortunately, since we can only estimate the weighted average of population discrepancies, we must choose the second option when sample sizes are unequal. SEPATH currently reports point and interval estimates for the weighted coefficient, which represents the square root of the ratio of a weighted average of discrepancies to an average number of degrees of freedom.
In calculating analogs of the population gamma indices, SEPATH substitutes K times the estimate of the weighted average of discrepancies in place of F* in equations 107 and 110.