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 *k*th 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.