5.3. A Model for Three Categories

First, some notation. Define

p_i1 = the probability that WALLET=1 for person i,

p_i2 = the probability that WALLET=2 for person i,

p_i3 = the probability that WALLET=3 for person i.

Let x_i be a column vector of explanatory variables for person i:

x_i = [1 x_i1 x_i2 x_i3 x_i4]’

If this is unfamiliar, you can just think of x_i as a single explanatory variable. In order to generalize the logit model to this three-category case, it’s tempting to consider writing three binary logit models, one for each outcome:

where the β’s are row vectors of coefficients. This turns out to be an unworkable approach, however. Because p_i1+p_i2+p_i3=1, these three equations are inconsistent. If the first two equations are true, for example, the third cannot be true. Instead, we formulate the model as follows:

These equations are mutually consistent and one is redundant. For example, the third equation can be obtained from the first two. Using properties of logarithms, we have

which implies that β₃ = β₁ – β₂. Solving for the three probabilities, we get

Because the three numerators sum to the common denominator, we immediately verify that the three probabilities sum to 1.

As with the binary logit model, the most general approach to estimation is maximum likelihood. I won’t go through the derivation, but it’s very similar to the binary case. Again, the Newton-Raphson algorithm is widely used to get the maximum likelihood estimates. The only SAS procedure that will do this is CATMOD.