For practical purposes it is usually not enough
to have a particular model which, when expressed in the framework of Equation
41, reproduces **S**.
For a model to be of much conceptual or practical value, its parameters
must be identified. That is, there must exist *only one* parameter
vector **q** for which**
S** =

Perhaps the simplest example of a covariance structure model which is not identified is a common factor analysis model with two manifest variables and one common factor. In this case (assuming the common factor has a variance of 1) the covariance structure model becomes

**S
**= **ff¢
**+ **U**2 (75)

In this case the parameter vector **q** has 4 elements, the two elements
in **f **and the two diagonal elements of **U**2.

Suppose

(76)

If

(77)

**S
**= **ff¢
**+ **U**2, and the model
fits perfectly. In this case

(78)

But there are other values of **q
**which will reproduce **S
**equally well. In fact there are infinitely many such values.

For example, let

(79)

If **U**2
is restricted to be positive definite, clearly any two values for the
first two elements of **q**
which have a product of .5, and are both less than one in absolute value
will produce a discrepancy function value of zero*. *The diagonal
elements of **U**2 are
then obtained by subtracting the square of the corresponding element of
**f** from 1.0.

Note that this is *not* a problem of the
well known "rotational indeterminacy" in factor analysis. (With
only one factor, there is no rotation.) Rather it is an example of a lesser
known phenomenon, namely, that the elements of **U**2
may not be identified in the common factor model. If **U**2
is not identified, then there may exist common factor patterns which reproduce
**S** equally well, but
which are not obtainable from each other by rotation.

Even in the relatively comfortable confines of the common factor model, the phenomena of model identification are not well understood. Some of the most significant textbooks on factor analysis have failed to ever mention the problem. Moreover, several authoritative figures in the history of psychometrics have produced "results" on model identification in factor analysis which they have later had to retract or correct.

In general, necessary and sufficient conditions
for identification are not available. However, it is often possible to
determine that a model is *not* identified by showing that a necessary
condition is violated.

There are some results available on when **U**2 in the factor model is definitely
*not* identified. One of the best-known was given by Anderson and
Rubin (1956). They showed that if, in unrestricted factor analysis,** ***under any orthogonal or
oblique rotation*, there existed a factor pattern with only 2 non-zero
elements in any column, then **U**2
is not identified. Clearly then, if such a situation exists (see Everitt,
1984, pages 45-49 for an example), additional constraints will have to
be imposed to yield an identified solution.

The Anderson-Rubin result has an important
implication which is often overlooked in discussions of the identification
issue. Namely, *it may not be possible to prove identification in the
population *without knowing S!
In other words the same model may be identified for one **S**,
but not for another. One cannot prove identification merely by counting
equations and unknowns.

For some (relatively simple) models, it may
be possible to prove identification by deriving unique equations, showing
each parameter as a function of the elements of **S**.
Unfortunately this approach is often impractical, and so checking for
identification usually involves two stages.

First, very obvious sources of lack of identification
should be removed. The most obvious source of

**S
= FWF¢
+ U**2 (80)

The variances of the common factors are found
on the diagonal of **W**. The factor loading coefficient for manifest
variable *i* on factor *j* is found in element **F***ij. *It is easy to show that unless
restrictions are imposed on this model, the variance for factor *j*
and the loadings on this factor are jointly indeterminate. To see why,
suppose you were to multiply all the factor variances by 2. If you were
to multiply all the columns of **F** by .7071, you would have exactly
the same **S**. More generally,
if we were to scale the diagonal of **W** with a diagonal scaling matrix
**D**, we could compensate by scaling the columns of **F** with
**D**-1.
In other words, for positive definite **D**,

(81)

so that for any **F** and **W** there
are infinitely many **F **and **W **which reproduce **S**
equally well.

There are several ways of eliminating the lack of identification problem in practice. One way is to fix the variances of the exogenous latent variables at 1. (This fix may not be sufficient in all cases.) Another approach is to apply some constraint to the factor loading coefficients themselves. This approach is popular in structural models where the main interest is in the relations between latent variables. In this case, identification is often obtained by fixing one of the coefficients on a particular variable to 1.

When *unstandardized* latent variable
models are fit, usually the variance of endogenous latent variables will
not be identified. In such situations, the traditional "fix"
has been to set one of the coefficients from the latent variable to 1.
However, when the "Standardization New" option is used, this
is not necessary, as *SEPATH* imposes internal constraints on the
estimation process which result in all endogenous latent variables having
unit variance.

Once obvious sources of non-identification have been eliminated, it is productive to examine whether either of the following easily tested conditions is violated.

1. The number of degrees of freedom for the
model must be nonnegative. That is *p*(*p* + 1)/2 ³ *t*, where *p* is the order of **S**, and *t* is the number
of free parameters in the model.

2. The Hessian (the matrix of second derivatives of the discrepancy function with respect to the parameters) must be positive definite.

Violation of either of these conditions usually
indicates an identification problem (for exceptions, see Shapiro &
Browne, 1983), and *SEPATH* warns the user if they are violated.