Multinomial regression

[Bill Smith]

I have a data set that looks like this (first few rows):

Dr	Time	Patient	Response
1	1	1	1
1	1	2	3
1	1	3	1
1	1	4	1
1	2	5	2
1	2	6	1
1	2	7	2
2	1	8	1
2	1	9	1
2	2	10	2
2	2	11	2

No patient is repeated from time 1 to time 2. The response is multinomial, 1, 2, 3. In terms of the frequencies, the above data look like this:

Dr	Time 1	Time 2
1	4	3
2	2	2

So there's a structure to these data. Time is nested in doctors and patients in time, but it's not truly longitudinal since I'm not repeating assessments on the same patients from time 1 at time 2. I think this is some kind of multilevel model, but I just can't figure out how to write the equations describing it. I have a grouping variable that will be an IV that I've omiited for simplicity for now. Can anyone help me out?

[Bill Smith]

Thought about this some more. Seems like a repeated cross-sectional analysis would work. That makes this a three level model: individual measurements nested in time nested in physicians. If I include the grouping variable which is at level three, then I have an equation that looks like the following for a random intercept model that treats physicians as random:

where

v_ok is the random effect for physician
u_ojk is the random effect for time
e_ijk is the residual for each person

I'd like to free up the covariance structure for u so that it could be something like AR1 to account for the correlations across time. I guess I could also free up the covariance structure for e to account for the correlations within each time point. Does this make sense?