I am exploring a semi-continuous latent growth model across 4 time points for two groups. In the semi-continuous growth model example in the Mplus User's Guide, the latent intercept factor mean for the binary portion is fixed at 0 and thresholds are estimated and equated. Because I want to freely estimate the latent intercept factor mean so that I can test for equality of this mean across groups, I reparameterized the model by fixing all thresholds at 0 and estimating the latent intercept mean. However, I am unsure as how to interpret the mean for the latent intercept factor. Because the model is centered at the 1st time point, I thought that exp(M)/(1+exp(M) should equal the estimated probability for endorsing the 2nd category at the 1st time point, where M is the latent intercept mean. However, my calculation does not equal the estimated probability provided in the output. Thank you!
In the binary part of the two-part model, random effects influence the outcome probability. The probability that you have computed is conditional on a person being at the mean of the random intercept. That is not the same as the (marginal) probability. To get the marginal probability you have to numerically integrate over the random intercept and that is what Mplus does to get the estimated probability.
So does this mean that the interpretation of the random intercept for a growth process with binary indicators is analogous to interpretation of a random intercept for a growth process with continuous indicators? If not, what meaning or interpretation does the random intercept mean with binary indicators have? Or is there no substantively useful meaning or interpretation?
I think it has a meaningful substantive interpretation given that it directly influences the probability. It is similar to the interpretation of an intercept for a continuous outcome. The intercept mean isn't the logit that gives the mean probability, but for people at the intercept mean it is the logit behind the probability for those people. This same phenomenon has been written about in the binary growth literature, for instance in the context of population-averaged and subject-specific differences, e.g. in the GEE literature by Zeger and others. See also text books on it, including the standard multilevel books.