Analysis of covariance structures can be extended
to handle modeling of the means of variables as well as their variances
and covariances. However, some important restrictions apply to these model
extensions. In particular, in order to obtain a *Chi-square* statistic,
the assumption of multivariate normality of the observed variables is
required. The estimates obtained are "maximum normal likelihood"
estimates, rather than "maximum Wishart likelihood" estimates,
as in standard covariance structure models.

The extended *SEPATH* model has the same
form as the model in Equation 36, except the variables mx,

(158)

(159)

(160)

(161)

where

Under these conditions, the observed manifest
variables and will no longer have zero means, but rather will have expected
values that are a straightforward function of the model parameters in
**B**, **G**,
tx,
tn,
kx,
and kn.
Define

(162)

(163)

and

(164)

If the population distribution is multivariate normal, then sample means and covariances are statistically independent, and two following discrepancy functions are of particular interest.

The *Maximum Normal Likelihood (ML) *discrepancy
function is

(165)

where is a vector of sample means, and

(166)

is the (biased) maximum normal likelihood estimate
of the covariance matrix. The *Normal Theory Generalized Least Squares
(GLS)* discrepancy function is

(167)

Some straightforward algebra shows how the
mathematics developed for covariance structure models can be employed
to fit models for mean and covariance structures. Specifically, if one
uses the standard ML discrepancy function in Equation 56, *but replaces
***S** *with the augmented moment matrix ***MA**, and **S **with

(168)

one obtains a result identical to the discrepancy function in Equation 165. A similar result holds for the GLS discrepancy functions

For a set of *p* variables, the augmented
moment matrix **MA **is a (p+1)x(p+1)
square matrix. The first *p* rows and columns contain the matrix
of moments about zero, while the last row and column contain the sample
means for the *p* variables. The matrix is therefore of the form:

(169)

where **M **is a matrix with element

(170)

and is a vector with the means of the variables.