The intercept should be fixed at zero because the means of the categorical outcomes are modeled by the threshold parameters, so we don't need both types of parameters. By intercept variance I assume you mean the intercept factor variance. The variance should be understood in the metric of the probit regression of each outcome on the growth factors. Variances can then be seen as contributing to the y* variance as discussed in Appendix 1 of the User's Guide. Here, the residual variance is fixed at 1. Ultimately, I think one has to judge the intercept variation in terms of how much the outcome probability changes, e.g. comparing 1 sd up and down from the intercept factor mean. I just saw a nice discussion of variances for categorical outcomes in the new Snijders & Bosker book on Multilevel Analysis, chapter 14. They also refer to the R square measure of McKelvey & Zavoina (1975) which is in line with Appendix 1.
There is an example of a multiple indicator growth model for continuous outcomes in the Mplus User's Guide. It is Example 22.4. You can use this as a guide.
When modeling growth on factors, there are two issues to consider. The first is measurement invariance of the factors. To establish complete measurement invariance of the factors, the intercepts of continuous outcomes or the thresholds of categorical outcomes and the factor loadings should be held equal across time. Deviations from these equalities result in partial measurement invariance.
The second issue is the parameterization of the growth model. To parameterize the growth model, the factor means/intercepts are fixed to zero, the mean of the intercept growth factor is fixed to zero, and the mean of the slope growth factor is free.
I have run a parallel process model with covariates over four time points. One of the covariates is depression, which is measured three of the four time points. So, I re-centered the intercept for those three time points. The structural model includes one slope regressed on the other slope and intercept, as well as the second slope regressed on the other intercept. Interestingly, when I move the intercept from time 1-4, the relationship between the two slopes changes, such that it is significant at times 1 and four, but not the between points. How does one interpret the effect of centering the intercept on such changes? I attended the DC training and learned that centering affects interpretation of intercepts regressed on covariates. However, what's the interpretation when change occurs to the slope or structural model?
bmuthen posted on Saturday, October 19, 2002 - 8:31 am
You are right that the centering changes the interpretation of the intercept. Because of this, in the regression
s2 ON i1 s1
the changed meaning of i1 will therefore change the partial regression coefficient for s1.
When you change the centering and therefore the interpretation of the intercept, you have to also consider whether you believe that the intercept is "causing" the slope of the other process, that is you have to consider time ordering of events. For example, if the centering is at time 4 it does not make sense that the time-4 intercept would influence a slope that refers to change after time 1 because that includes development that has happened earlier than time 4.