Let **S** be the sample covariance matrix
based on *N* observations, and for notational convenience, define
*n *= *N* - 1.
**S(q)** is the attempt
to reproduce **S** with a particular model and a particular parameter
vector **q. S(q***ML***)** is the corresponding matrix
constructed from the vector of maximum likelihood estimates *ML* obtained by minimizing the
discrepancy function of Equation 56. Equations 56 and 58 give two alternative
discrepancy functions which lead to *Chi-square* statistics for testing
structural models. Suppose one has obtained maximum likelihood estimates.
Then under conditions (i.e., the "population drift" conditions
in Steiger, Shapiro, and Browne, 1985) designed to simulate the situation
where the model fits well but not perfectly, * n*F

Interestingly, if one divides by the noncentrality
parameter by *n*, one obtains a measure of population badness-of-fit
which depends only on the model, **S**,
and the method of estimation.

If one has a single observation from a noncentral
*Chi-square* distribution, it is very easy to obtain an unbiased
estimate of the noncentrality parameter. By well known theory, if noncentral
*Chi-square* variate *X* has noncentrality parameter *l* and degrees of freedom *n*, the expected value of *X*
is given by

* E*(

whence it immediately follows that an unbiased
estimate of *l* is simply
*X* - *n*. Consequently
a large sample "biased corrected" estimate of F*** is

(*X - n*)/*n. *Since
F* *can never be negative, the simple unbiased estimator is generally
modified in practice by converting negative values to zero. The estimate

is the result.

It is also possible, by a variety of methods,
to obtain, from a single observation from a non-central *Chi-square*
distribution with *n*
degrees of freedom, a maximum likelihood estimate of the noncentrality
parameter *l,* and confidence
intervals for *l* as
well. (See, e.g.,

Before continuing, recall some very basic statistical principles.

1. Under very general conditions, if
is a maximum likelihood estimator for a parameter *q*,
then for any monotonic strictly increasing function f (

2. Moreover, if *q*,

These principles immediately imply that, since
one can obtain a maximum likelihood estimate and confidence interval for
*n*.

*SEPATH* obtains
a point estimate and confidence interval for *Chi-square* statistic at the 100(a/2) and
100(1-a/2) percentile
points of a c 2n ,l distribution. The "point estimate" of the
"population noncentrality index" printed by *SEPATH* is
the simple bias-corrected estimate F+ (see Equation 89) recommended by
McDonald (1988).

When the IRGLS or GLS estimation methods are
employed, the population noncentrality index is a quadratic form, and
as such is a weighted sum of squares of the residuals. Suppose you were
to place the non-redundant elements of **S**
in a vector **s = ****S**). Recalling the
result of Equation 67, the discrepancy function is of the form