# Creating a Covariance Matrix from Scratch

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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.

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