Ordinal regression is used when y is ordinal. This means that the outcome consists of three or more categories that are possible to rank (i.e. ordered categories; see Measurement scales).  

Example Educational level (1=Compulsory; 2=Upper secondary; 3=University)    School marks (1=Low; 2=Average; 3=High)    Self-rated health (1=Excellent; 2=Good; 3=Fair; 4=Poor)    Statement: “Eurovision Song Contest is entertaining” (1=Strongly agree; 2=Agree; 3=Neither agree nor disagree; 4=Disagree; 5=Strongly disagree)

Proportional odds

Ordinal regression is a type of logistic regression that can handle the fact that the outcome has multiple (ordered) outcome categories. Instead of modelling the probability of the outcome being a case, we consider the cumulative probability across the outcome categories. This means that we estimate the odds of being at or above a given threshold across all cumulative splits.  

In the model, each outcome category has its own intercept (at each threshold) but the same coefficient for the overall x-variable. Because of this, we have to assume that the effect of x on the odds of the outcome being a case for each subsequent category is the same for every category. This reflects the notion of proportional odds (sometimes referred to as parallel lines), which is a key assumption behind ordinal regression analysis. Put differently, the proportional odds assumption means that the estimate between each pair of outcome categories are assumed to be the same regardless of which pair is considered. 

Other names for ordinal regression 

Sometimes, ordinal regression analysis is referred to as, e.g., ordered logit regression, ordinal logistic regression, or proportional odds regression.