I ran example 3.13 which uses WLMV with the theta parameterization. The output shows scale factors for the endogenous variables. When I change to the delta I see residual variances. I think I understand the distinction between the two, but I am puzzled by the output which seems to contradict the explanation in the user manual.
If the theta parameterization is used in a two-group one-factor factor model and the scalar model is believed to be an adequate model for the data, is the following correct? The expressions from page 495 in the Mplus manual P (u = 0 | x) = F (t1 - b*x), P (u = 1 | x) = F (t2 - b*x) - F (t1 - b*x), P (u = 2 | x) = F (- t2 + b*x), with b = the factor loading and x = the factor, can be used to calculate category probabilities on item i for the group in which the residual variance equals one, but have to be modified for the other group by dividing each term by the residual standard deviation for item i.