Multinomial and Ordinal Logistic Regression

• Extension of binary logistic regression to polytomous (multinomial) and ordinal logistic regression is possible (Agresti Ch 8)
• Multinomial logistic proceeds by comparing the effect of covariates on the possibility of being in each of $n-1$ categories compared with a ``baseline category'':
  
  $$log(\frac{\pi_j}{\pi_J}) = \alpha_j + \beta_jx$$
  
  

• This is conceptually equivalent to fitting $n-1$ separate binary logistic models, comparing category 1 with category n, category 2 with n, and so on
• In practice the software usually estimates a simultaneous model which is more statistically sophisticated
• For each independent variable, we have $n-1$ parameter estimates, each estimating the effect of a one-unit change in this variable on the log odds of being in category $j$ rather than $m$