Linear Regression with Ordered Categorical Outputs

Question

Say I want to create a model to predict outcomes of an emergency room, where the outputs are:

1. Stroke
2. Seizure
3. Overdose

Why can't I run a linear regression model that assigns 1, 2, and 3 to Stroke, Seizure, and Overdose instead of creating a classification model? ISL (Friedman) makes the point that:

Unfortunately, this coding implies an ordering on the outcomes, putting drug overdose in between stroke and epileptic seizure, and insisting that the difference between stroke and drug overdose is the same as the difference between drug overdose and epileptic seizure.

However, why does the "difference", or "distance", matter? Can't I just take the output of the regression and round to the nearest integer and call that my guess?

Answer 1

I'm sorry, but I think there are a lot of issues here.

First, I'm not sure these are even ordered. They certainly aren't mutually exclusive. The third one should not be "overdose". And there need to be more categories - certainly there needs to be "death" and "nothing". Also, overdose of what? Some drug given in the hospital or a drug the patient took on his or her own? In the latter case, it's not an outcome.

Second, if you assume they are ordered, then an OLS model will violate the assumptions about the distributions of the residuals. And the output could be anything. Not only could it be in between 1 and 2, but it could be negative or above 3.

Third, if the responses are ordinal then rounding to the nearest integer is impossible (at least technically) because "nearest" has no meaning. Since you don't know the size of the gaps (much less what's in those gaps) you don't know what's closest to what. (This might be arguable if there was some underlying continuum, but here, there is not).