Which glm family to use for ordinal DV?

Question

I'm trying to test whether duration of time spent on the Internet (ratio scale) can predict behavioural problems (ordinal, with scores ranging from 0-10). I just wanted to double check that an ordinal regression (using generalised linear model) is the right way to go? Also, I was wondering which glm family I should use? From my understanding, binomial wouldn't work as it deals with binary outcomes. My guess is gaussian family and identity link? Unfortunately, I'm not that familiar with this so I'd be grateful if someone could help me explain this?

Thanks so much for the help!

Answer 1

Given you have an ordinal response using the function MASS::polr should be more appropriate; it implements a proportional odds logistic regression routine. A very comprehensive tutorial on the analysis of ordinal response variables can be found here. It is also worth checking this thread on: How to understand output from R's polr function (ordered logistic regression)?. In brief, a proportional odds model instead of modelling the probability of response in a particular category, it models the cumulative probability that the response is not greater than a chosen category.

Your understanding that a binomial family would be too restrictive is fine. A Gaussian with an identity link would be quite unnatural though too; you could not easily constrain it to positive and/or integers responses. You might want to consider using a GLM with a Poisson family but that it is a bit hand-wavy as you need to often define arbitrary categories (so something like glm( ..., family = poisson)). I have seen this being used as illustrative example in some case (eg. Faraway's Extending the Linear Model with R, Chapt. 4.5) but I think it is a bit suboptimal when compared to a real proportional odds logistic regression.

A free and accessible paper on the matter is: Regression models for ordinal responses: a review of methods and applications. from Ananth & Kleinbaum; it is a bit dated (1997) but it is nicely presented and will bring you up-to-speed it relevant terminology.

Answer 2

A multinomial distribution with a cumulative logit link function. You could also use a cumulative probit link function, but that's usually less desirable.