Structural Equation Models

Chapter Contents

The CALIS Procedure

Structural Equation Models

The Generalized COSAN Model

PROC CALIS can analyze matrix models of the form

C = F₁ P₁ F₁' + ... + 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_{k_n(k)}, and each matrix P_k is symmetric, that is,

F_k = F_k₁ ... F_{k_n(k)} and P_k = P_k' , k = 1, ... ,m

The matrices F_{k_j} and P_k in the model are parameterized by the matrices G_{k_j} and Q_k

$F_{k_{j}} = \{ \matrix{G_{k_{j}} \cr G^{-1}_{k_{j}} \cr (I- G_{k_{j}})^{-1} \cr} j=1, ... ,n(k) { and } . P_k = \{ \matrix{Q_{k} \cr Q^{-1}_k \cr} .$

where you can specify the type of matrix desired.

The matrices G_{k_j} 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 = (x₁, ... , x_t). For a given covariance or correlation matrix C, PROC CALIS computes the unweighted least-squares (ULS), generalized least-squares (GLS), maximum likelihood (ML), weighted least-squares (WLS), or diagonally weighted least-squares (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 = F₁ ... F_n PF_n' ... F₁'

RAM (Reticular Action) Model (McArdle 1980; McArdle and McDonald 1984)

Structural Equation Model:

$v= A{v}+ {\mu}$

where A is a matrix of coefficients, and v and ${\mu}$ are vectors of random variables. The variables in v and ${\mu}$ 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 ${\mu}$ with covariance matrix P.

Covariance Structure:

C = J(I-A)^-1 P((I-A)^-1)' J'

with selection matrix J and

$C= {\cal E} \{ J{v}v^' J^' \} { and } P= {\cal E} \{{\mu}{\mu}^' \}$

LINEQS (Linear Equations) Model (Bentler and Weeks 1980)

Structural Equation Model:

$eta = {\beta}eta + {\gamma}{\xi}$

where ${\beta}$ and ${\gamma}$ are coefficient matrices, and

and ${\xi}$ are vectors of random variables. The components of

correspond to the endogenous variables; the components of ${\xi}$ correspond to the exogenous variables and to error variables. The variables in

and ${\xi}$ 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 ${\xi}$ , and of a residual component in ${\xi}$ . The coefficient matrix ${\beta}$ describes the relationships among the endogenous variables of

, and $I-{\beta}$ should be nonsingular. The coefficient matrix ${\gamma}$ describes the relationships between the endogenous variables of

and the exogenous and error variables of ${\xi}$ .

Covariance Structure:

$C= J(I- B)^{-1} {{\Gamma}}{{\Phi}}{{\Gamma}}^' ((I- B)^{-1})^' J^'$

with selection matrix J, ${{\Phi}}= {\cal E}\{{\xi}{\xi}^'\}$ , and

$B= ( \matrix{ {\beta}& 0 \cr 0 & 0 \cr} ) { and } {{\Gamma}}= ( \matrix{ {\gamma}\cr I\cr} )$

Keesling - Wiley - J $\ddot{o}$ reskog LISREL (Linear Structural Relationship) Model

Structural Equation Model and Measurement Models:

$eta = Beta + {{\Gamma}}{\xi}+ {\zeta} , y= {{{\Lambda}}}_y eta + {\varepsilon} , x= {{{\Lambda}}}_x {\xi}+ {\delta}$

where

and ${\xi}$ 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 ${\xi}$ 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 ${{\Gamma}}$ and an error vector ${\zeta}$ .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 ${\xi}$ by two systems of linear equations (the measurement model) with coefficients ${{{\Lambda}}}_y$ and ${{{\Lambda}}}_x$ and with measurement errors ${\varepsilon}$ and ${\delta}$ .

Covariance Structure:

$C& = & J(I-A)^{-1} P((I-A)^{-1})^' J^' \A & = & \pmatrix{ 0 & 0 & {{\Lambda}}_y... ...}} \cr & {{{\Theta}}}_{{\delta}} \cr & & {{{\Psi}}} \cr & & & {{{\Phi}}} \cr}$

with selection matrix J, ${{\Phi}}= {\cal E}\{{\xi}{\xi}^'\}$ , ${{\Psi}}= {\cal E} \{ {\zeta}{\zeta}^' \}$ , ${{\Theta}}_{\delta} = {\cal E} \{ {\delta}{\delta}^' \}$ ,and ${{\Theta}}_{\varepsilon} = {\cal E} \{ {\varepsilon}{\varepsilon}^' \}$ .

Higher-Order Factor Analysis Models

First-order model:

C = F₁ P₁ F₁' + U²₁

Second-order model:

C = F₁ F₂ P₂ F₂' F₁' + F₁ U²₂ F₁' + U²₁

First-Order Autoregressive Longitudinal Factor Model

Example of McDonald (1980): k=3: Occasions of Measurement; n=3: Variables (Tests); m=2: Common Factors

$C& = & F_1 F_2 F_3 L{F}_3^{-1} F_2^{-1} P(F_2^{-1})^'(F_3^{-1})^' L^' F_3^' ... ...cr U_{31} & U_{32} & U_{33} \cr} \ S_2 & = & I_2 - D_2^2 , S_3 = I_2 - D_3^2$

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 J $\ddot{o}$ reskog and S $\ddot{o}$ rbom (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, y₁ : Anomia 1967
Variable 2:: V2, y₂ : Powerlessness 1967
Variable 3:: V3, y₃ : Anomia 1971
Variable 4:: V4, y₄ : Powerlessness 1971
Variable 5:: V5, x₁ : Education (years of schooling)
Variable 6:: V6, x₂ : 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, $\eta_1$ : Alienation 1967
Variable 8:: F2, $\eta_2$ : Alienation 1971
Variable 9:: F3, $\xi_1$ : Socioeconomic Status (SES)

The following path diagram shows the structural model used in Bentler (1985, p. 29) and slightly modified in J $\ddot{o}$ reskog and S $\ddot{o}$ rbom (1985, p. 56). In this notation for the path diagram, regression coefficients between the variables are indicated as one-headed arrows. Variances and covariances among the variables are indicated as two-headed arrows. Indicating error variances and covariances as two-headed 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.
$\begin{picture} (440.,550.) \put( 40.,380.){ \framebox (60.,40.){1: V1, y_1}} \... ...ector(0,1)5} \put(360.,55.){\vector(0,1)5} \put(340.,35.){\Theta_4}\end{picture}$

Figure 19.1: Path Diagram of Stability and Alienation Example

Variables in Figure 19.1 are as follows:

Variable 1:: V1, y₁ : Anomia 1967
Variable 2:: V2, y₂ : Powerlessness 1967
Variable 3:: V3, y₃ : Anomia 1971
Variable 4:: V4, y₄ : Powerlessness 1971
Variable 5:: V5, x₁ : Education (years of schooling)
Variable 6:: V6, x₂ : Duncan's Socioeconomic Index (SEI)
Variable 7:: F1, $\eta_1$ : Alienation 1967
Variable 8:: F2, $\eta_2$ : Alienation 1971
Variable 9:: F3, $\xi_1$ : Socioeconomic Status (SES)

RAM Model

The vector v contains the six manifest variables v₁ = V1, ... ,v₆ = V6 and the three latent variables v₇=F1, v₈=F2, v₉=F3. The vector ${\mu}$ contains the corresponding error variables u₁ = E1, ... , u₆ = E6 and u₇=D1, u₈=D2, u₉=D3. The path diagram corresponds to the following set of structural equations of the RAM model:

$v_1 & = & 1.000 v_7 + u_1 \v_2 & = & 0.833 v_7 + u_2 \v_3 & = & 1.000 v_8 + u_3 ... ... = & \gamma_1 v_9 + u_7 \v_8 & = & \beta v_7 + \gamma_2 v_9 + u_8 \v_9 & = & u_9$

This gives the matrices A and P in the RAM model:

$A= ( o & o & o & o & o & o & 1.000 & o & o \ o & o & o & o & o & o & 0.833 & o &... ...o & o & o & o & o & \beta & o & \gamma_2 \ o & o & o & o & o & o & o & o & o )$

$P= (\theta_1 & o & \theta_5 & o & o & o & o & o & o \ o & \theta_2 & o & \theta_... ... & o & o & o & o & o & o & \psi_2 & o \ o & o & o & o & o & o & o & o & \phi )$

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 ${\xi}$ 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:

$V1 & = & 1.0 F1+E1 \ V2 & = & .833 F1+E2 \ V3 & = & 1.0 F2+E3 \ V4 & = & .833 F2... ...\lambda}F3+E6 \ F1 & = & {\gamma}_1 F3+D1 \ F2 & = & {\beta} F1+{\gamma}_2 F3+D2$

This gives the matrices ${\beta}$ , ${\gamma}$ and ${{\Phi}}$ in the LINEQS model:

${\beta}= (o & o & o & o & o & o & 1. & o \ o & o & o & o & o & o & .833 & o \ o ... ...& o & o & o & o & 1 & o & \gamma_1 \ o & o & o & o & o & o & o & 1 & \gamma_2 )$

${{\Phi}}= (\theta_1 & o & \theta_5 & o & o & o & o & o & o \ o & \theta_2 & o & ... ...o & o & o & o & o & o & o & \psi_2 & o \ o & o & o & o & o & o & o & o & \phi )$

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 y₁ = V1, ... , y₄ = V4, and the vector x contains the exogenous manifest variables x₁=V5 and x₂=V6. The vector ${\varepsilon}$ contains the error variables $\varepsilon_1=E1, ... , \varepsilon_4=E4$ corresponding to y, and the vector ${\delta}$ contains the error variables $\delta_1=E5$ and $\delta_2=E6$ corresponding to x. The vector

contains the endogenous latent variables (factors) $\eta_1=F1$ and $\eta_2=F2$ , while the vector ${\xi}$ contains the exogenous latent variable (factor) $\xi_1=F3$ . The vector ${\zeta}$ contains the errors $\zeta_1=D1$ and $\zeta_2=D2$ in the equations (disturbance terms) corresponding to

.The path diagram corresponds to the following set of structural equations of the LISREL model:

$y_1 & = & 1.0 \eta_1 + \epsilon_1 \ y_2 & = & .833 \eta_1 + \epsilon_2 \ y_3 & =... ... \gamma_1 \xi_1 + \zeta_1 \ \eta_2 & = & \beta \eta_1 + \gamma_2 \xi_1 + \zeta_2$

This gives the matrices ${{{\Lambda}}}_y$ , ${{{\Lambda}}}_x$ ,B, ${{\Gamma}}$ , and ${{\Phi}}$ in the LISREL model:

${{\Lambda}}_y = (1. & o \ .833 & o \ o & 1. \ o & .833 ) ,{{\Lambda}}_x = (1. \ \lambda ) , B= (o & o \ \beta & o ) , {{\Gamma}}= (\gamma_1 \ \gamma_2 )$

${{{\Theta}}}^2_{\varepsilon} = (\theta_1 & o & \theta_5 & o \ o & \theta_2 & o &... ...ta_4 & o ) , {{{\Psi}}} = (\psi_1 & o \ o & \psi_2 ) , {{{\Phi}}} = ( \phi )$

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.

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Structural Equation Models

The Generalized COSAN Model

Some Special Cases of the Generalized COSAN Model

Original COSAN (Covariance Structure Analysis) Model (McDonald 1978, 1980)

RAM (Reticular Action) Model (McArdle 1980; McArdle and McDonald 1984)

LINEQS (Linear Equations) Model (Bentler and Weeks 1980)

Keesling - Wiley - Jreskog LISREL (Linear Structural Relationship) Model

Higher-Order Factor Analysis Models

First-Order Autoregressive Longitudinal Factor Model

A Structural Equation Example

RAM Model

LINEQS Model

LISREL Model

Keesling - Wiley - J $\ddot{o}$ reskog LISREL (Linear Structural Relationship) Model