[R] multinomial regression model

From: Thomas Mang <thomasmang.ng_at_gmail.com>
Date: Tue, 10 May 2011 22:11:11 +0200

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

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