Structural Equation Models
PROC CALIS can analyze matrix models of the form

C = F_{1} P_{1} F_{1}' + ... + F_{m} P_{m} F_{m}'
where C is a symmetric correlation or covariance
matrix, each matrix F_{k}, k = 1, ... ,m, is the product
of n(k) matrices F_{k1}, ... ,F_{kn(k)},
and each matrix P_{k} is symmetric, that is,

F_{k} = F_{k1} ... F_{kn(k)} and P_{k} = P_{k}' , k = 1, ... ,m
The matrices F_{kj} and P_{k} in the model are
parameterized by the matrices G_{kj} and Q_{k}
where you can specify the type of matrix desired.
The matrices G_{kj} and Q_{k} can contain
 constant values
 parameters to be estimated
 values computed from parameters via programming statements
The parameters can be summarized in a
parameter vector X = (x_{1}, ... , x_{t}).
For a given covariance or correlation matrix C,
PROC CALIS computes the unweighted leastsquares (ULS),
generalized leastsquares (GLS), maximum likelihood (ML),
weighted leastsquares (WLS), or diagonally weighted
leastsquares (DWLS) estimates of the vector X.
Some Special Cases of the Generalized COSAN Model
Original COSAN (Covariance Structure Analysis) Model (McDonald 1978, 1980)
Covariance Structure:

C = F_{1} ... F_{n} PF_{n}' ... F_{1}'
RAM (Reticular Action) Model (McArdle 1980; McArdle and McDonald 1984)
Structural Equation Model:
where A is a matrix of coefficients, and
v and are vectors of random variables.
The variables in v and
can be manifest or latent variables.
The endogenous variables corresponding to the components in v
are expressed as a linear combination of the remaining variables
and a residual component in with covariance matrix P.
Covariance Structure:

C = J(IA)^{1} P((IA)^{1})' J'
with selection matrix J and
LINEQS (Linear Equations) Model (Bentler and Weeks 1980)
Structural Equation Model:
where and are coefficient matrices,
and and are vectors of random variables.
The components of correspond to the endogenous
variables; the components of correspond to
the exogenous variables and to error variables.
The variables in and
can be manifest or latent variables.
The endogenous variables in are
expressed as a linear combination of the remaining
endogenous variables, of the exogenous variables
of , and of a residual component in .
The coefficient matrix describes the
relationships among the endogenous variables of
, and should be nonsingular.
The coefficient matrix describes the
relationships between the endogenous variables of
and the exogenous and error variables of .
Covariance Structure:
with selection matrix J, , and
Keesling  Wiley  Jreskog LISREL (Linear Structural Relationship) Model
Structural Equation Model and Measurement Models:
where and are vectors of latent variables
(factors), and x and y are vectors of manifest variables.
The components of correspond to
endogenous latent variables; the components of
correspond to exogenous latent variables.
The endogenous and exogenous latent variables are
connected by a system of linear equations (the
structural model) with coefficient matrices
B and and an error vector .It is assumed that matrix I B is nonsingular.
The random vectors y and x correspond to manifest
variables that are related to the latent variables
and by two systems of linear equations (the measurement
model) with coefficients and
and with measurement errors and .Covariance Structure:
with selection matrix J,
,,,and .HigherOrder Factor Analysis Models
Firstorder model:

C = F_{1} P_{1} F_{1}' + U^{2}_{1}
Secondorder model:

C = F_{1} F_{2} P_{2} F_{2}' F_{1}' + F_{1} U^{2}_{2} F_{1}' + U^{2}_{1}
Example of McDonald (1980): k=3: Occasions of Measurement;
n=3: Variables (Tests); m=2: Common Factors
For more information on this model, see Example 19.6.
A Structural Equation Example
This example from Wheaton et al. (1977) illustrates
the relationships among the RAM, LINEQS, and LISREL
models.
Different structural models for these data are
in Jreskog and
Srbom (1985) and in
Bentler (1985, p. 28).
The data set contains covariances among six (manifest) variables
collected from 932 people in rural regions of Illinois:
 Variable 1:
 V1, y_{1} : Anomia 1967
 Variable 2:
 V2, y_{2} : Powerlessness 1967
 Variable 3:
 V3, y_{3} : Anomia 1971
 Variable 4:
 V4, y_{4} : Powerlessness 1971
 Variable 5:
 V5, x_{1} : Education (years of schooling)
 Variable 6:
 V6, x_{2} : Duncan's Socioeconomic Index (SEI)
It is assumed that anomia and powerlessness are indicators
of an alienation factor and that education and SEI
are indicators for a socioeconomic status (SES) factor.
Hence, the analysis contains three latent variables:
 Variable 7:
 F1, : Alienation 1967
 Variable 8:
 F2, : Alienation 1971
 Variable 9:
 F3, : Socioeconomic Status (SES)
The following path diagram shows the structural model
used in Bentler (1985, p. 29) and slightly modified in
Jreskog and Srbom (1985, p. 56).
In this notation for the path diagram, regression coefficients
between the variables are indicated as oneheaded arrows.
Variances and covariances among the
variables are indicated as twoheaded arrows.
Indicating error variances and covariances as twoheaded
arrows with the same source and destination (McArdle
1988; McDonald 1985) is helpful in transforming the path
diagram to RAM model list input for the CALIS procedure.
Figure 19.1: Path Diagram of Stability and Alienation Example
Variables in Figure 19.1 are as follows:
 Variable 1:
 V1, y_{1} : Anomia 1967
 Variable 2:
 V2, y_{2} : Powerlessness 1967
 Variable 3:
 V3, y_{3} : Anomia 1971
 Variable 4:
 V4, y_{4} : Powerlessness 1971
 Variable 5:
 V5, x_{1} : Education (years of schooling)
 Variable 6:
 V6, x_{2} : Duncan's Socioeconomic Index (SEI)
 Variable 7:
 F1, : Alienation 1967
 Variable 8:
 F2, : Alienation 1971
 Variable 9:
 F3, : Socioeconomic Status (SES)
RAM Model
The vector v contains the six manifest
variables v_{1} = V1, ... ,v_{6} = V6 and the three
latent variables v_{7}=F1, v_{8}=F2, v_{9}=F3.
The vector contains the corresponding error variables
u_{1} = E1, ... , u_{6} = E6 and u_{7}=D1, u_{8}=D2, u_{9}=D3.
The path diagram corresponds to the following
set of structural equations of the RAM model:
This gives the matrices A and P in the RAM model:
The RAM model
input specification of this example for the CALIS procedure is
in the "RAM Model Specification" section.
LINEQS Model
The vector contains the six endogenous manifest variables
V1, ... , V6 and the two endogenous latent variables F1 and F2.
The vector contains the exogenous error variables
E1, ... , E6, D1, and D2 and the exogenous latent variable
F3. The path diagram corresponds to the following set of structural
equations of the LINEQS model:
This gives the matrices , and in
the LINEQS model:
The LINEQS model
input specification of this example for the CALIS procedure is
in the section "LINEQS Model Specification".
LISREL Model
The vector y contains the four endogenous manifest variables
y_{1} = V1, ... , y_{4} = V4, and the vector x contains the exogenous
manifest variables x_{1}=V5 and x_{2}=V6. The vector contains the error variables corresponding to y, and the vector contains the error
variables and corresponding to x.
The vector contains the endogenous latent variables (factors)
and , while the vector contains the
exogenous latent variable (factor) . The vector contains the errors and in the
equations (disturbance terms) corresponding to .The path diagram corresponds to the following set of structural
equations of the LISREL model:
This gives the matrices , ,B, , and in the LISREL model:
The CALIS procedure does not provide a LISREL
model input specification. However, any model that can be specified
by the LISREL model can also be specified by using the COSAN,
LINEQS, or RAM model specifications in PROC CALIS.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.