The *Chi-square* statistics based on ML
and GLS estimation procedures assume multivariate normality of the data.
When this assumption is violated, the resulting test statistic will, in
general, no longer have a c2 distribution, and can therefore
mislead during the fit evaluation process. Browne (1982, 1984) proposed
procedures which will lead to a correct *Chi-square* statistic under
much more general conditions.

Suppose the covariance structure model is expressed in the form

**S** = **S(q) + E**

For any *p ´ p* symmetric matrix **A**, let ** vecs**(

**s** = ** vecs
**(

** s(q)
= vecs (S(q))
** (63)

**e**
= ** vecs **(

one can express the model of Equation 61 alternatively as

**s** = **s(q) **+ **e** (65)

Browne (1984) showed that

**e***
= (*N* - 1)1/2**e** (66)

has, under a true null hypothesis, an asymptotic
distribution which is multivariate normal with null mean vector and variance-covariance
matrix **Y***. *Moreover,
the generalized least squares discrepancy function of the form

(67)

has the property that if **U** is selected
to be a consistent estimator of **Y**,
then

c2
= (*N* - 1)F(**s**,
**s(q)**) (68)

will have an asymptotic *Chi-square* distribution
under very general distributional assumptions. Both the IRGLS and GLS
discrepancy functions discussed above can be expressed in the form of
Equation 67. However, they incorporate a **U **matrix which is generally
a consistent estimator of **Y**
only under the assumption of multivariate normality. Modifying **U **so
that it is a consistent estimator of **Y**
under more general assumptions will allow the assumption of multivariate
normality to be dispensed with, thus leading to "asymptotically distribution
free" (ADF) procedures for the analysis of covariance structures.

ADF procedures have seldom been used in practice, although it seems that the assumption of multivariate normality is frequently contestable with data in the behavioral sciences. One reason for the lack of popularity of ADF procedures is that they were not implemented in widely available computer software like LISREL VI, EzPATH 1.0, or COSAN.

There are other serious practical problems
with ADF estimation procedures. First, **U** can, in practice, be a
very large matrix, thus imposing practical limits on the size of the problem
which can be processed. Second, the elements of **U** require estimates
of second and fourth-order moments of the manifest variables. Such estimates
have large sampling variability at small to moderate sample sizes, so,
in general, one might expect the *Chi-square* test statistic (and
associated estimates) based on ADF estimation to converge somewhat more
slowly to the asymptotic behavior than comparable normal theory estimation
procedures.

However, there seems little justification for
performing normal theory estimation and testing procedures on data which
are clearly non-normal, especially when the sample size is large and the
numbers of variables and unknowns moderate. The current version of *SEPATH*
therefore supports two versions of ADF estimation.

Browne (1984) showed that, with the typical definitions for the first, second, and fourth-order sample moments about the mean, i.e.,

(69)

(70)

(71)

(72)

a typical element of **Ug**,
a consistent and Gramian (but not unbiased) estimator of **Y**,
is given by

(73)

The ADFG option in *SEPATH* minimizes
the discrepancy function in Equation 67 with **U = Ug**.

Browne (1984) also showed how to obtain unbiased
estimates of the elements of **Y**.
A matrix **Uun** containing
these unbiased estimates has typical element

(74)

The "ADF Unbiased" option in *SEPATH*
minimizes the discrepancy function in Equation 67 with**
U = Uun**.