FACTOR Model Statement

The CALIS Procedure

FACTOR Model Statement

FACTOR < options > ;

You can use the FACTOR statement to specify an exploratory or confirmatory first-order factor analysis of the given covariance or correlation matrix C,

C = FF' + U, U = diag

C = FPF' + U, P = P'

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 matrix F.

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 FACTOR because it is designed for more general structural estimation problems and is unable to exploit the special properties of the unconstrained factor analysis model.

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 possible). For details, see the section "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 $\tilde{F} = F{{\Theta}}$ is equivalent to the result F,

$C= \tilde{F}\tilde{F}^' + U, \tilde{F} = F{{\Theta}}, {where} {{\Theta}}^' {{\Theta}}= {{\Theta}}{{\Theta}}^' = I$

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.

COMPONENT | COMPSTART

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.

NORM

normalizes the rows of the factor pattern for rotation using Kaiser's normalization.

ROTATE | R = name

specifies 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:

$F_{new} = F_{old} T, {with} F_{old}^' F_{old} = T{{\Lambda}}T^'$

where the columns of matrix T contain the eigenvectors of F_old' F_old.

QUARTIMAX | Q

specifies quartimax rotation.

VARIMAX | V

specifies varimax rotation.

EQUAMAX | E

specifies equamax rotation.

PARSIMAX | P

specifies parsimax rotation.

NONE

performs no rotation (default).

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