Any multinomial response model can be viewed as the model for an I × J table. Assume product-multinomial sampling from J independent multinomials each with I categories. Define

πij = pij I

h=i phj so that the continuation ratios introduced in Section 4.2 are

πij 1 − πij

= pij

I h=i+1 phj

.

Let rij = n·j − i−1 h=1 nhj ; show that the product-multinomial likelihood J j=1 %
n·j !
I i=1 nij !
I i=1 p nij ij &
can be written as the product of binomial likelihoods, i.e., J j=1 I −1 i=1  rij nij 
πnij ij (1 − πij )
rij−nij .
Using this result and maximum likelihood estimation, show that a set of continuation ratio models can be fitted simultaneously to the entire table by fitting each continuation ratio model separately. Note that the chi-square statistics for fitting each continuation ratio model can be added to get a chi-square statistic for the entire table.