st: Workaround to implement apc_ie for a multinomial family problem

[Camila Mendes <cacamendes85@gmail.com>]

Hi everyone,

I want a help with my model.

If Y ~ Multinomial(n,p1,p2,p3,p4) and:

n is the total of individuals.
p1 is the probability of falling into the 1st category.
p2 is the probability of falling into the 2nd category.
p3 is the probability of falling into the 3rd category.
y1 is the amount of people in the 1st category.
y2 is the amount of people in the 2nd category.
y3 is the amont of people in the 3rd category.

It can be demonstrated that:

P(Y) = Bin[n,p1] x Bin[n-y1;p2/(1-p1)] x Bin[n-y1-y2;p3/(1-p1-p2)].

Using apc_ie ado algoritm, I want to run separately each of the
conditional models:

A: Bin[n,p1]
B: Bin[n-y1;p2/(1-p1)]
C: Bin[n-y1-y2;p3/(1-p1-p2)]

as a function of the same covariates (age, period and cohort
parameters). I though of this as a workaround to implement apc_ie in
the context of the multinomial family.

Would this work at all? If so, it is clear to me that it would be
necessary to restrict the samples in each step. However, how to
restrict the probabilities as well?

Many thanks,

Camilla.
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