I'd like to understand the metric of the INTERCEPT and SLOPE factors for a growth model with categorical outcomes and missing data at some time-points. Each outcome variable is coded (0=Never, 1=Monthly, 2=Weekly, & 3=Daily).
As I understand it, ML estimation uses integration in these circumstances. A trichotomous treatment variable (NEW, STANDARD & NO therapies) predicts the two growth factors, implemented as two dummy variables for NEW (D1) and STD (D2), with NO being the reference category.
(1) Do the regression coefficients for D1 and D2 on the two growth factors indicate the difference in logits between NEW and NO therapy and between STD and NO therapy respectively?
(2) If yes, then if the coefficients for INTERCEPT on D1=2.183, and SLOPE on D1=1.298, how would these be interpreted?
(3) Does in involve using the invariant thresholds for the outcome variables (theses are 3.2, 6.6, & 8.8 respectively), and if so, how?
Yes, the logit is the operative term here. With a binary 0/1 outcome for person i,
(1) logit(ti) = -threshold + int_i + t*slope_i,
where t may be 0, 1, 2, ..., the threshold plays the role of the negative intercept, and int and slope are the growth factors. The latter can be replaced by the means if that is what is of interest. So for example, if t=0 for the first time point, the effect of the Ds on the outcome of that time point is channeled through int only, giving rise to logit differences that can be translated into probability differences. A higher logit gives a higher probability for each of the outcome categories.
With ordered polytomous outcomes, the probability derived from the logit in (1) is for the highest category of the outcome, using the highest threshold. The lower category probabilities are obtained in line with our course notes.