Hello. I am interested in fitting longitudinal count data (and possibly a growth mixture model of that data) over approximately 10 timepoints. The user manual discusses the Poisson and the Zero inflated Poisson for such data structures, but it does so for a straight-line change curve. When I think of a "Poisson model" I think of a nonlinear relation between "X" and the "counts," where the counts are predicted from X. The "Poisson model" Mplus uses for longitudinal data is much different, it seems. What exactly is Poisson here? Presumably the distribution of the counts conditional on each timepoint, but aren't the Poisson parameters also important? These are not returned with Mplus (since a straight-line model is fit and what is returned are the intercepts and slopes, etc.). Would it not make sense to have a Poisson model over time, where time was a predictor of the Poisson parameter at specific values of time? Maybe I'm missing something? Is there a way to get the Poisson parameters?Can you point me to any theoretical work the Mplus programs relies on for the Poisson longitudinal model (or applications)? Thank you very much.
Poisson regression for a count outcome certainly involves a non-linear relationship between the outcome and the covariate. But that does not mean that growth in a count outcome could not be a linear function of time - that is quite a different matter. Note that the linear growth model for a count outcome means that the log rate parameter of the Poisson model is specified to have linear growth over time. If you have a time-invariant covariate, it will still have a non-linear relationship to the outcome.
A good reference on the statistics for Poisson and ZIP growth modeling in the more general setting of mixtures is the 1999 Roeder et al JASA article. Mplus includes this and much more.