In Structural Equation Modeling one approach to deal with categorical Mediators is the "underlying latent variable approach". However using this approach e.g. with the MPlus software, only thresholds are calculated, which can be used to calculate the probabilities for the categories of the categorical variable. The intercept or constant of the underlying continious latent variable is not shown.
Given the Model:
Y on M X X1
M on X X1
M=Categorcial, three categories
I want to calculate the "total effect" when X ist zero. So I want to calculate the Y-value when X is zero and M* (the underlying latent variable) takes on the predicted value of X=0. This would be the intercept or constant of the underlying latent variable (M*) in "M on X". What I don't know is how I get this intercept if I have more then two categories. With two categories it seems to be easy:
For a binary dependent variable, the probit regression model expresses the probability of u given x as,
P (u = 1 | x) = F (a + bx) = F(-t + bx),
where F is the standard normal distribution function, a is the probit regression intercept, b is the probit regression slope, t is the probit threshold where
t = -a, and P (u = 0 | x) = 1 – P (u = 1> | x).
(Muthén & Muthén 2017: 552)
Does anybody know how to calculate the intercept of the underlying latent variable if the observed categorical variable has more than two categories?
Muthén, L.K. and Muthén, B.O. (1998-2017). Mplus User’s Guide. Eighth Edition. Los Angeles, CA: Muthén & Muthén