5.6. General Form of the Model

To this point, we’ve been dealing with just three categories for the dependent variable. Now let’s generalize the model to J categories, with the running index j= 1, ..., J. Let p_ij be the probability that individual i falls into category j. The model is then

Equation 5.1

where x_i is a column vector of variables describing individual i and β_j is a row vector of coefficients for category j. Note that each category is compared with the highest category J. These equations can be solved to yield

Equation 5.2

Because the probabilities for all J categories must sum to 1, we have

After the coefficients are estimated, the logit equation for comparing any two categories j and k of the dependent variable can be obtained from