Kruskal-Wallis and Ordinal Logistic Regression

Question

Many relate Ordinal Logistic Regression (OLR) with Kruskal-Wallis (KW) in the sense that OLR may be used when there are two or more IVs and interaction effect between them are of interest, while KW deals with only 1 IV. This is summarized by the table here: http://www.ats.ucla.edu/stat/sas/whatstat/

However, I don't fully understand one thing: how does OLR handle a continuous DV? Sure, such response values usually contain a natural ordering but my impression is that the DV for OLR must be ordered and discrete 'categories'?

So I have two questions:

First, can you use OLR as long as the values of the DV contain an order (e.g. a continuous DV)? If yes, is this conclusion discussed in any text?

Second, if the answer to above is no, then which regression can be used in case that the assumption(s) of linear regression cannot be met, and, test for interaction effect between IVs is desired?

Answer 1

Regarding your first question, no, ordinal logistic is not appropriate when the dependent variable is continuous. Some people dichotomize or otherwise categorize the dependent variable, but this is often a bad idea.

Regarding your second question, there are a variety of things to do when the assumptions of linear regression are not met. You can try transformations of either the dependent or independent variables (if they make sense), there are robust regression methods (if the assumption being violated is about the shape of the residuals), quantile regression makes no assumptions about the residuals, there are also regression trees and their offshoots.