Ordinal Dependent Variables

• When we have an ordinal dependent variable we have a number of options:
  • Treat it as quantitative and linear and use OLS
  • Treat it as nominal and observe the ordinality in the pattern of the parameter estimates
  • Dichotomise it and fit a binary model, or
  • fit an ordinal logistic regression
• All of these may work well but the last is to be preferred
• We have already considered a number of formulations of ordinal logistic regression:
  • adjacent-category
  • continuation-ratio and
  • proportional odds
• Adjacent-category logistic models effectively fit separate simultaneous logits on subsets of the data, predicting the log odds of being in category $j$ versus $j-1$. The default is to constrain the parameters in these $n-1$ models to be the same: the effect of the independent variable on being in the higher rather than the lower of each adjacent pair
• This assumes a certain (log) linearity in the dependent variable
• However, the possibility of relaxing this constraint means you can check for departures from linearity at the cost of a more complex (imparsimonious) model
• Closely related is the continuation-ratio model:
  
  $$log (\frac{\pi_j}{\sum_{k=1}^{j-1}\pi_k}) =\alpha_j + \beta_j$$
  
  

• If we constrain the parameters to be constant across the comparisons, we have an ordinal model which measures the effect of covariates on the log odds of being ``one category higher''
• If we reverse the direction of this model:
  
  $$log (\frac{\pi_j}{\sum_{k=j+1}^{J}\pi_k}) =\alpha_j + \beta_j$$
  
  
  we have a model with a nice interpretation for cases of sequential selection such as the educational system: an ordinal variable such as highest qualification is generated by a sequential process of selections
  • Staying on after compulsory
  • Completing second level
  • Entering university
  • Completing a degree
• At each of these points a different (sequentially smaller) set of individuals is ``at risk of'' passing the test
• If we were to model the effect of covariates on these progressions it is likely that they would have different effects as each, but it is possible that some effects would be the same
• The (reversed) continuation-ratio model allows us to fit these effects, and test whether they differ by transition
• By fitting the separate logits simultaneously we get a statistically more efficient model, and to the extent that we can constrain certain parameters to be the same across transitions we get a more parsimonous ordinal model