Do I have to impose equality constraints on measurement error (i.e., rho) parameters across time?
No, you do not have to impose equality constraints. However, it is often a good idea to do so, because this keeps the meaning of the latent statuses the same over time. This corresponds to the idea of factor invariance in factor analysis. Sometimes, however, you may want to explore the latent class structure separately for each time to get a sense of what underlying groups there may be in your population. You also may want to model multiple times together without restricting measurement to be equal over time. In this case, the number of classes generally has to be equal unless you use a flexible structural equation modeling package that allows you to condition class membership at time 2 on class membership at time 1. One thing to keep in mind is that if a latent class does not exist at time 1 but does at time 2, it is okay for the class membership probability for that class to be (essentially) zero at time 1, with people transitioning into it at time 2. This could be substantively inte
