I have arrived at a possibly valid generalization and wanted to share it in case it is valid and might help others. In testing an added growth model, such as in Fig 5 of Muthen & Curran (1997), it seemed to me that there were two ways one could go. One could constrain the added growth factor to be equal across the groups and compare that to a model in which the added factor is free to vary across groups (but the common factors are constrained to be equal across groups). Alternatively, one could omit the added growth factor from both groups and compare that to a model in which the added factor is added to the group it is hypothesized to be specific to. I have done some VERY limited power simulations that suggest that the second approach is more powerful. Though my simulations are limited it makes sense as the correct model in the second approach is a less parameterized version of the correct model (correct assuming the added growth model is valid) and yet accounts for the data as well as does the more parameterized version. If anyone thinks this is faulty logic or likely to be invalid, I would love to hear about it.
In Muthen-Curran we compared 2 approaches. One is your second approach. The other is not using an added growth factor in either group, but just test equality across groups of the regular growth factors means (and variances). We found that this other approach seemed less powerful in pinpointing the treatment effects. Your first approach is a little different from that in that you have the added growth factor in both groups equal and not equal. That approach is a little odd to me in that the control group shouldn't have an added growth factor.
Note also that the Muthen-Curran approach with treatment-baseline interaction is limited to monotonic treatment effects (e.g. low baseline gives low effect, medium gives larger effect, high gives largest effect), whereas growth mixture modeling captures general effects (e.g. low baseline gives large effect, medium gives low effect, high gives large effect).
actually my first approach did involve constraining the added growth factor (and all others) to be equal across the two groups but no matter as it didn't seem to be as powerful and doesn't seem like it should be.
Thanks for the info about growth mixture modeling, I wasn't previously thinking it would be relevant for my applications but your post suggests to me it would be worthwhile for me to learn more about it. It looks like the two upcoming Mplus short courses advertised on the website won't cover growth mixture modeling. I see it is in the Web training course, do you ever cover it in any live training sessions these days or should I just go through the Web course?
We used to teach 5 days straight, but now split up the content in sets of 2-3 days, where we can go a little deeper for each topic. The twice annual Hopkins courses cycle through all topics going from Day 1 to Day 5. The March 2008 course essentially covers Day 1 FA and SEM plus categorical versions of it. August 2008 covers mixture (latent class) modeling (cross-sectional and longitudinal) - so GMM would be taught then. March 2009 covers multilevel versions of all previous models. We may add new topics such as missing data and survival, but the idea is to go forward and build it up like this. The courses given in other countries complement the Hopkins courses. For example, multilevel modeling is taught in London this December.