analytical solution for latent variable correlation

[Maximilian Ernst]

Hi together,

I have two dichotomous variables A and B.

I know their joint distribution, e.g. P(A), P(B), P(A and B). (I have no data, but I can simulate it with this information)

Under the assumption that those variables reflect latent variables that are normally distributet, I can use the simulated data and estimate the correlation of the latent Variables with the following model:

modAB <- "

factorA =~ 1*A

factorB =~ 1*B

factorA ~~ factorB

A ~~ 0*A

B ~~ 0*B

factorA ~~ 1*factorA

factorB ~~ 1*factorB

"

fitAB <- sem(modAB, datAB, ordered = c("A", "B"))

However, I have many of those pairs, which makes this approach computationally expensive (first drawing data, then fitting the model, then extracting the parameter of interest.)

I assume that because I only have two dichotomous variables and I know their distribution, there is a way to directly calculate die correlation of the latent variables from the known probabilities. Does someone now how to do that?

[Maximilian Ernst]

A direct follow-up: is the latent variable correlation in this case equivalent to the polychoric correlation of A and B?

[Terrence Jorgensen]

A direct follow-up: is the latent variable correlation in this case equivalent to the polychoric correlation of A and B?

Yes, the tetrachoric correlation since both are binary.  If A and B are endogeous variables, you can simply rely on lavaan to do this.  That is how DWLS estimation works:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3162326/

I assume that because I only have two dichotomous variables and I know their distribution, there is a way to directly calculate die correlation of the latent variables from the known probabilities. Does someone now how to do that?

Googling "how to calculate tetrachoric correlation" returned this:

https://www.statisticshowto.datasciencecentral.com/tetrachoric-correlation/

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen

[Maximilian Ernst]

Thanks, also for the nice paper!! I found the r-package polycor, which computes the tetrachoric correlation from the 2x2 contingency table of a pair of variables. So now instead of fitting a latent variable model each time, I just compute the correlation directly, which speeds up my code extremely.

Best,

Maximilian