Identification of two-indicator LC model
Message/Author
 Ryan McCammon posted on Monday, May 22, 2006 - 5:59 am
On page 5 of the following article,

http://spitswww.uvt.nl/~vermunt/ermss2004a.pdf

Vermunt & Magidson suggest that "the
restrictions P(Y` = 1|X = 1) = P(Y` = 2|X = 2) can be used to identify a
two-class model with two dichotomous indicators"

My trouble is that I don't see an easy way to implement this constraint in MPlus. One idea I had was something like this, based on the idea that the maximum logit threshold is 15:

Model c1:
%c1#1%
[u1\$1] (lg1);

Model c2:
%c1#1%
[u1\$1] (lg2);

MODEL CONSTRAINT:
lg2 = 15 - lg1;

This seems far from foolproof. Is there a better way?
 Bengt O. Muthen posted on Monday, May 22, 2006 - 7:12 pm
It is a little tricky since your restriction is on the probability scale and your parameters are on the logit scale. But you can do it as follows.

Note that for a given item,

(1) P(y=2 | x=2) = F(-tau2)

where x is the latent class variable, tau2 is the Mplus threshold parameter for class 2, and F is the logistic function,

F(z) = 1/(1+exp(-z)).

Also,

(2) P(y=1 | x=1) = 1 - P(y=2 | x=1)
= 1 - F(-tau1).

You should give a label to tau1 and tau2 in the Model statements, then use those 2 labels in Model constraint to apply the required contraint

(1) = (2)

i.e.

F(-tau2) = 1 - F(-tau1)

So since the logit z is obtained as

z = log(F(z)/(1-F(z)),

you say

labtau2 = - log((1-F(-labtau1))/F(tau1);

- I think.