FACTOR Model Statement
You can use the FACTOR statement to specify an exploratory or
analysis of the given covariance or correlation matrix C,
- FACTOR < options > ;
C = FF' + U, U = diag
where U is a diagonal matrix and P is symmetric.
Within this section, n denotes the number of manifest
variables corresponding to the rows and columns of matrix
C, and m denotes the number of latent variables (factors
or components) corresponding to the columns of the loading
C = FPF' + U, P = P'
You can specify only one FACTOR statement with each PROC CALIS
statement. You can specify higher-order factor analysis problems
using a COSAN model specification.
PROC CALIS requires more computing time and memory than PROC
because it is designed
for more general
structural estimation problems and is unable to exploit
the special properties of the unconstrained factor analysis
For default (exploratory) factor analysis, PROC CALIS computes
initial estimates for factor loadings and unique variances
by an algebraic method of approximate factor analysis.
If you use a MATRIX statement together
with a FACTOR model specification, initial values are
computed by McDonald's (McDonald and Hartmann 1992) method (if
For details, see "Using the FACTOR and MATRIX Statements".
If neither of the two methods are appropriate, the initial
values are set by the START= option.
The unrestricted factor analysis model is not identified
because any orthogonal
rotated factor loading
matrix is equivalent to the result F,
To obtain an identified factor solution, the FACTOR statement imposes
zero constraints on the m(m - 1)/2 elements in the upper
triangle of F by default.
The following options are available in the FACTOR statement.
You can specify the MATRIX statement and
the FACTOR statement to compute a confirmatory first-order factor
or component analysis. You can define the elements of
the matrices F, P, and U of
the oblique model,
- COMPONENT | COMP
computes a component
analysis instead of a factor analysis (the diagonal matrix U
in the model is set to 0). Note that the rank of
FF' is equal to the number m of components in F.
If m is smaller than the number of variables
in the moment matrix C, the matrix of predicted model
values is singular and
maximum likelihood estimates for F cannot be computed.
You should compute ULS estimates in this case.
- HEYWOOD | HEY
constrains the diagonal elements of U to be nonnegative;
in other words, the model is replaced by
C = FF' + U2 , U = diag
- N = m
specifies the number of first-order factors or
components. The number m of factors should not exceed the
number n of variables in the covariance or correlation
matrix analyzed. For the saturated model, m=n,
the COMP option should generally be specified for U = 0;
otherwise, df < 0. For m = 0 no factor loadings are
estimated, and the model is C = U, with U = diag.
By default, m=1.
normalizes the rows of the factor pattern for rotation
using Kaiser's normalization.
- ROTATE | R = name
an orthogonal rotation. By default, ROTATE=NONE.
The possible values for name are as follows:
- PRINCIPAL | PC
- specifies a principal axis rotation.
If ROTATE=PRINCIPAL is used with a factor
rather than a component model, the following
rotation is performed:
where the columns of matrix T contain the eigenvectors
of Fold' Fold.
- QUARTIMAX | Q
- specifies quartimax rotation.
- VARIMAX | V
- specifies varimax rotation.
- EQUAMAX | E
- specifies equamax rotation.
- PARSIMAX | P
- specifies parsimax rotation.
- performs no rotation (default).
C = FPF' + U2 , P = P' , U = diag
To specify the structure for matrix F, P, or U, you have
to refer to the matrix _F_ , _P_ , or _U_
in the MATRIX statement.
Matrix names automatically set by PROC CALIS always start
with an underscore. As you name your own matrices or variables,
you should avoid leading underscores.
The default matrix forms are as follows.
- lower triangular matrix (0 upper triangle for problem
identification, removing rotational invariance)
- identity matrix (constant)
- diagonal matrix
For details about specifying the elements in matrices, see
the section "MATRIX Statement". If you are using at least one MATRIX
statement in connection
with a FACTOR model statement, you can also use the BOUNDS or
PARAMETERS statement and program statements to constrain the
parameters named in the MATRIX statement. Initial estimates are
computed by McDonald's (McDonald and Hartmann 1992) method. McDonald's
method of computing initial values works better if you scale
the factors by setting the factor variances to 1 rather than
by setting the loadings of the reference variables equal to 1.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.