[Boost-users] multinomial mixed-effects regression model

Subject: [Boost-users] multinomial mixed-effects regression model
From: Thomas Mang (thomasmang.ng_at_[hidden])
Date: 2011-05-10 09:59:14

Hi,

Consider the need for a  regression model which can handle an ordered 
multinomial response variable. There are, for example, proportional odds 
/ cumulative logit models, but actually the regression should include 
random effects (a mixed model), and I would not be aware of multinomial 
regression model as part of lme4 (am I wrong here ?). Further, the 
constraint of proportional odd models that predictors have the same 
relative impact across all levels does not hold for the study in question.

I was wondering if an ordinary binomial mixed model can be turned in an 
multinomial one through preparing the input data.frame in a different way:
Consider three response levels, A, B, C, ordered. I can accurately 
describe the occurrence of each of these three realizations using one to 
two Bernoulli random variables:

Let
P(X == A) = a
P(X in {B, C}) = 1 - a
P(X == B | X in {B, C}) = (1 - a) * b
P(X == C | X in {B, C}) = (1 - a) * (1 - b)

so the first comparison checks if A or either of B/C is the case, and 
the second, conditional on it's either B/C, checks which of these two 
holds. Sort of traversing sequentially the hierarchy of the ordered levels.
In terms of the likelihood of the desired model, the probabilities on 
the right hand side would be exactly achieved if I use one input row in 
case the random variable takes on the value A and assign the response 
variable the value 0, while in the other cases the probabilities are 
achieved by using two input table rows, with the first one having value 
1 for the response variable so the random variate is either B/C) and a 
second row with response equal to 0 if B is the case, and 1 otherwise, 
that is C is the case.

Certainly, degrees of freedom must be manually adjusted in inferences, 
as every measured response should provide only a single degree of freedom.

Question: Do I overlook here something, or is above outlined way a valid 
method to yield an ordered multinomial mixed model by tweaking the input 
table in such manners ?

many thanks and best,
Thomas