ABSTRACT
The multinomial distribution is an extension of the binomial where the response can take more than two values. Let Yi be a random variable that falls into one of a finite number of categories, labeled 1,2, . . . ,J. Let pi j = P(Yi = j) so ∑Jj=1 pi j = 1. As with binary data (the case where J = 2), we may encounter both grouped and ungrouped data. Let Yi j be the number of observations falling into category j for group or individual i and let ni = ∑ j Yi j. For ungrouped data, ni = 1 and one and only one of Yi1, . . . ,YiJ is equal to one and the rest are zero. The Yi j, conditional on the total ni, follow a multinomial distribution:
P(Yi1 = yi1, . . . ,YiJ = yiJ) = ni
yi1! · · ·yiJ! p yi1 i1 · · · pyiJiJ
We must also distinguish between nominal multinomial data where there is no natural order to the categories and ordinal multinomial data where there is an order. The multinomial logit model is intended for nominal data. It can be used for ordinal data, but the information about order will not be used.
