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 most likely 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
