lisrel

Latent Structural Equation Models

Purpose: Combine a econometric structural model with a psychometric measurement model.

The structural model is a causal model that relates the exogenous (independent) variables to the endogenous (dependent) variables. The measurement model is a factor model that relates the observed indicators to the latent factors. All relations are assumed to be linear.

References:

Bollen, K. A. (1989) Structural equations with latent variables. NY: Wiley

Long, J.S. (1983) Covariance structure models: An introduction to LISREL. Beverly Hills: Sage Pubications.

MacCallum,R. C.; Wegener, D. T.; Uchino, B. N.; Fabrigar, L.R. The problem of equivalent models in applications of covariance structure analysis. Psychological-Bulletin.1993 Jul; Vol 114(1): 185-199.

Structural Model:

h = Bh + G x + V

h is a (m x 1) vector of latent endogenous variables

x is a (n x 1) vector of latent exogenous variables

V is a (m x 1) vector of residual variables

B is a matrix of structural coefficients that relate endogenous to endogenous

G is a matrix of structural coefficients that relate endogenous to exogenous

Measurement Model:

Y = L_y h + e

X = L_x x + d

Y is a (p x 1) vector of indicators for the endogenous variables

X is a (q x 1) vector of indicators for the exogenous variables

e is a vector of endogenous measurement errors

d is a vector of exogenous measurement errors

L _y and L _x are coefficients relating latent variables to indicators

Definitions:

F = Cov(x ,x )

Y = Cov(V,V)

Q _y = Cov(e ,e )

Q _x = Cov(d ,d )

Assumptions:

Cov(x ,V) = 0 , Cov(x ,e ) = 0 , Cov(x ,d ) = 0

Cov(e ,V) = 0 , Cov(d ,V) = 0

Cov(e ,d ) = 0

Predictions:

S _xx = Cov(X,X) = L_x FL_x ' + Q _x

S _xy = Cov(X,Y) = L_x F G'(I-B')^-1 L _y '

S _yy = Cov(Y,Y) = L _y(I-B)^-1(GF G' + y )(I-B')^-1 L _y ' + Q _y

No. of data points = (p+q)(p+q+1)/2

No. of parameters = total no. free in q = {F , G, B, y , L _x , L_y , Q_y , Q _x }

df = (no. data points - no. parms)

Estimation:

S = [S _xx S _xy

S _yx S _yy ]

S = [ S_xx S_xy

S_yx S_yy ]

S_xx = (X'X)/(N-1)

S_xy = (X'Y)/(N-1)

S_yy = (Y'Y)/(N-1)

Generalized Least Squares: Find q that minimizes F

F = (.5)Trace[ (I - S^-1 S) (I - S^-1 S)] = (.5)Trace[ (S^-1(S -S) )²]

Model Comparison:

c ² (df) = (N-1) F

Suppose Model B is nested within Model A

Model A has df_A and c ²_A

Model B has df_B and c ²_B

df_B - df_A > 0

Test of Comparison between A versus B:

c ² diff = (c ²_B - c ²_A ) has df = ( df_B - df_A )

Cross Validation:

Divide total sample into two parts, calibration versus test sample.

N = N₁ + N₂

S₁ is the sample cov matrix from first sample

S₂ is the sample cov matrix from second sample

1. Estimate q of each model using S₁ .

2. Use these same parameters to evaluate fit to S₂ .

3. Compare each model using c ² from second test sample.

This method can be used to test non-nested models that differ

in terms of number of parameters.

Other Fit Indices