Multinomial logit models through loglinear

• Fitting a multinomial logistic model on grouped data is very easy: simply specify a multicategory dependent variable in the logit-loglinear version of the GENLOG command.

• But how do we interpret it? Basically for each category of the dependent variable, there are a set of parameters that show the effects of the independent variables for that category compared to the reference category of the dependent variable.

  That is, it is as if for $n-1$ categories of the dependent variable, separate simultaneous binary logit models are fitted, pairing each category with the null category.

• This is therefore sometimes known as the baseline-category logit model:
  
  $$\log \frac{\pi_{ij}}{\pi_{iJ}} = \alpha_j + \beta_{j}x_i$$
  
  

• Output looks like this, with `constants' for the interaction of all independent variables, followed by terms for interactions between the dependent variable and the terms in the model:

  Parameter   Aliased  Term
  
          1            Constant for [FSEX = 1]*[AGEG = 1.00]
          2            Constant for [FSEX = 1]*[AGEG = 2.00]
          . . .
         13            Constant for [FSEX = 2]*[AGEG = 6.00]
         14            Constant for [FSEX = 2]*[AGEG = 7.00]
         15            [VOTE = 1.00]
         16            [VOTE = 2.00]
         17            [VOTE = 3.00]
         18       x    [VOTE = 4.00]
         19            [VOTE = 1.00]*[FSEX = 1]
         20       x    [VOTE = 1.00]*[FSEX = 2]
         21            [VOTE = 2.00]*[FSEX = 1]
         22       x    [VOTE = 2.00]*[FSEX = 2]
         23            [VOTE = 3.00]*[FSEX = 1]
         24       x    [VOTE = 3.00]*[FSEX = 2]
         25       x    [VOTE = 4.00]*[FSEX = 1]
         26       x    [VOTE = 4.00]*[FSEX = 2]
         27            [VOTE = 1.00]*[AGEG = 1.00]
         28            [VOTE = 1.00]*[AGEG = 2.00]
         29            [VOTE = 1.00]*[AGEG = 3.00]
         30            [VOTE = 1.00]*[AGEG = 4.00]
         31            [VOTE = 1.00]*[AGEG = 5.00]
         32            [VOTE = 1.00]*[AGEG = 6.00]
          . . .
  

• From this we can build equations for each dependent category in contrast with the baseline category. Thus for calculating the expected logit for conservative voters who are male, in their thirties we use the terms for [VOTE = 1.00], [VOTE = 1.00]*[FSEX = 1] and [VOTE = 1.00]*[AGEG = 3.00]

• For individual level data, multinomial logistic models exist with the same interpretation, but a different method for fitting.