I have a question regarding a two-wave cross-lagged model with two latent variables with categorical indicators and one observed categorical variable.
I have read that categorical dependent variables should not be related to latent variables by WITH, but rather by regressing the latent variable on the observed categorical. However, in my model I cannot state that observed_categorical_dependent -> latent_continuous at time 2. If I specify it with a WITH statement, the estimates are different. Also, I have related factor indicators over time with WITH statements, although they are categorical.
1. Does it make sense to use WITH for residual covariance of a latent with an observed categorical? 2. Does it make sense to correlate observed categorical items over time with WITH? 3. Is it wrong to covary an exogenous observed with an exogenous latent using a WITH statement? It does not produce a warning or error statement.
The idea that that categorical variables should not be related to latent variables by WITH, but rather by regressing the latent variable on the observed categorical is only relevant if the categorical variable is exogenous. Otherwise, WITH is fine. I assume you use WLSMV. Then the answers are:
3. it is not wrong. It is just that with WLSMV is adds an assumption of underlying normality for the categorical exogenous variable.
So, a categorical factor indicator is not treated differently when it is on the lhs of an ON statement than it is on the rhs of a BY statement? What if what I want to do is to regress the residual variance of y on f2?