I have been conducting several simulations that use a covariance matrix. I needed to expand the code that I found in the psych package to have more than 2 latent variables (the code probably allows it but I didn’t figure it out). I ran across Joreskog’s 1971 paper and realized that I could use the confirmatory factor analysis model equation to build the population covariance matrix.

The code below demonstrates a 5 factor congeneric data structure

*fx* is the factor loading matrix, *err* has the error variances on the diagonal of an empty matrix, and *phi* is a matrix of the correlations between the latent variables.

#######################################
###---Population Covariance Generation
#######################################
###---Loadings
fx<-t(matrix(c(
.5,0,0,0,0,
.6,0,0,0,0,
.7,0,0,0,0,
.8,0,0,0,0,
0,.5,0,0,0,
0,.6,0,0,0,
0,.7,0,0,0,
0,.8,0,0,0,
0,0,.5,0,0,
0,0,.6,0,0,
0,0,.7,0,0,
0,0,.8,0,0,
0,0,0,.5,0,
0,0,0,.6,0,
0,0,0,.7,0,
0,0,0,.8,0,
0,0,0,0,.5,
0,0,0,0,.6,
0,0,0,0,.7,
0,0,0,0,.8), nrow=5))
###--Error Variances
err<-diag(c(.6^2,.7^2,.8^2,.9^2,
.6^2,.7^2,.8^2,.9^2,
.6^2,.7^2,.8^2,.9^2,
.6^2,.7^2,.8^2,.9^2,
.6^2,.7^2,.8^2,.9^2))
###---5x5 matrix of factor covariances
phi<-matrix(c(rep(.3, 25)), nrow=5)
diag(phi)<-1
sigma<-(fx%*%phi%*%t(fx)+err)
######################################

For sample data I used the mvrnorm() function from the MASS package

library(MASS)
mvrnorm(100, nrow(fx),sigma)

To simulate parallel form data the values in the *fx* matrix need to be the same and the diagonal in the *err* matrix need to be the same. One could also manipulate the *phi* matrix and thus change the correlations between the latent variables.