Another subtract.stratum question

[cabutler10]

Hello,
I am working on a multistate model and I am having some difficulty trying to figure out to constrain specific Psi parameters to be equal. My model has four states (A, B, C, and D) and an individual can only move from A-B, A-C, B-D and C-D (so they can move through a system from A to D by transiting between B OR C only and once in D they stay in D). However, the probability of transiting between states A-B and C-D is the same (and A-C and B-D is the same). I have successfully constrained all of the illogical (for my model) transition parameters to be zero by removing them from the design data and am now at a loss as to how to make Psi A-B = Psi C-D (called sticker) and Psi A-C = Psi B-D (called disc). I have tried to make a new row in the design data for each transition that I want with the following code which does run successfully and gives me the set up I want when I look at Amstrata.ddl$Psi data frame.

Code: Select all

Amstrata.ddl$Psi$sticker=0
Amstrata.ddl$Psi$sticker[Amstrata.ddl$Psi$stratum=="A"&Amstrata.ddl$Psi$tostratum=="B" | Amstrata.ddl$Psi$stratum=="C"&Amstrata.ddl$Psi$tostratum=="D"]=1
Amstrata.ddl$Psi$disc=0
Amstrata.ddl$Psi$disc[Amstrata.ddl$Psi$stratum=="A"&Amstrata.ddl$Psi$tostratum=="C" | Amstrata.ddl$Psi$stratum=="B"&Amstrata.ddl$Psi$tostratum=="D"]=1

But when I look at the results of this model, while I have two beta estimates representing the transition I want, I find that MARK has still estimated different real parameter estimates for each transition so that I have four real estimates(A-B, A-C, C-D, and B-D), not two like I need (A-B=C-D and A-C=B-D). I have read that subtract.stratum can force transitions to be equal and I understand that each is estimated by subtraction, but I cannot figure out what I would subtract to get C-D to equal something different than B-D and still work properly?
I know MARK can run this with no problem and so I feel like I am simply overlooking an obvious answer here and any help you can give me would be much appreciated! Thanks!
Alyssa

[cabutler10]

Yes. The problem is with the way the mlogit is specified. Based on the constraints you specified the real parameters with an mlogit are

pr(A to B) = exp(ab)/(1+exp(ab)+exp(ac))
pr(A to C) = exp(ac)/(1+exp(ab)+exp(ac))

pr(B to D) = exp(ac)/(1+exp(ac))
pr(C to D) = exp(ab)/(1+exp(ab))

where ab is the beta for that transition and ac is the other beta. Now if you use logits

pr(A to B) = exp(ab)/(1+exp(ab))
pr(A to C) = exp(ac)/(1+exp(ac))

pr(B to D) = exp(ac)/(1+exp(ac))
pr(C to D) = exp(ab)/(1+exp(ab))

This gets more complicated when you have more non-zero states. I'm not sure what MARK will do in the case where it has to adjust the real values to sum to 1 and the constraints differ across strata. In this simple case it will work.