Given the below dependent variable description, should I choose either Ordered or Multinomial logistic regressions?

My dependent variable looks like a range of ranks. It actually might be considered that way. But the ranking is based on subjective non-quantifiable cutoffs. We assessed the behaviors of a group of people (confidentiality doesn't allow me to explain further) and we have categorized their behavior from non-cooperative (the score zero) to fully collaborative (the top score).

There are some cutoffs defined within this range of zero and the top score, making it look like an order of ranks. The cutoffs really exist and are important behavioral factors, based on which we have determined if a subject should get a score of 1, 2, etc. But the problem is that the distance between 1 and 2 is not necessarily equal to the distance between 2 and 3. And by distance, I mean the extent of the improvement in the behavior pattern. That distance is not actually quantifiable.

So I don't know should I use an ordered logit or a multinomial one? My preference would be the ordinal one since its interpretation would be easier! But I don't know if its assumptions allow me or not. The Wikipedia page has bad news apparently "The model only applies to data that meet the proportional odds assumption, that the relationship[clarification needed] between any two pairs of outcome groups is statistically the same. This means that the coefficients that describe the relationship between, say, the lowest versus all higher categories of the response variable are the same as those that describe the relationship between the next lowest category and all higher categories, etc. Because the relationship between all pairs of groups is the same, there is only one set of coefficients." But still I think I should ask, maybe there is some way around.

I also much appreciate any other hints you might have for me.

Lets start with the good news: The proportional odds assumptions does not require that the distances between categories are the same.

So what does the proportional odds assumption imply? Say we have three ordered outcomes (1, 2, 3), then we could model the choice of 1 versus 2 or 3 and the choice of 2 versus 3. The proportional odds assumption says that the effects of all explanatory/independent/x-variables are the same across these two choice comparisons.

This is not a trivial assumption, but you obviously can't get the simplification of getting one set of parameters for multiple outcomes without making some strong assumptions. Test exist for this assumption. The one I most often come across is an approximate test by Brant (1990). Alternatives are the closely related Wolfe and Gould (1998) test and the usual Wald, likelihood ratio and score test. Richard Williams and I have been doing simulations comparing these different tests, to investigate under which circumstances these tests perform well and when they break down. We recently presented first results at the 2013 German Stata Users' meeting, and the slides are here: http://www.maartenbuis.nl/presentations/gsug13.pdf

Brant, R. (1990). Assessing proportionality in the proportional odds model for ordinal logistic regression. Biometrics, 46(4):1171–1178. http://www.jstor.org/stable/2532457

Wolfe, R. and Gould, W. W. (1998). An approximate likelihood-ratio test for ordinal response models. Stata Technical Bulletin, 7(42):24–27. http://www.stata.com/products/stb/journals/stb42.pdf

• Many thanks dear Maarten. I think in the case of most of the IVs, the effects of Xs (IVs) are consistent for Y = 1 vs 2, or 1 vs 3, or 2 vs 3 etc. In some cases, the independent variable might have an effect which can be specifically more powerful for some scores of Y. So I would collapse the range of Y scores into a dichotomized Y variable and do a binary logit. Or maybe I can exclude those certain variables from the ordinal logit model and run it, and then play with the excluded variables in multinomial modelds, or in binary ones. Do you have any other ways in mind? Thanks again. :) – Vic Jun 13 '13 at 13:55
• Notice that the proportional odds assumption looks as 1 versus more , 2 versus more , etc. So it does not look at 1 versus 2, but 1 versus (2, 3, 4, ...) – Maarten Buis Jun 13 '13 at 13:58
• You seem to look for a model that can partially relaxes the proportional odds assumption for some variables. This model already exists: a generalized ordered logit. You can find more about that here: nd.edu/~rwilliam/gologit2 – Maarten Buis Jun 13 '13 at 14:00
• I see you have updated your response. Thanks again for the new advice and links and hints. I am glad I can determine using objective tests that if the assumptions are met or not. That was a very good help. Thanks. – Vic Jun 13 '13 at 14:01
• What you propose in terms of separate binary logistic regression is a reasonable approximation, in fact it is what both the Brant test and the Wolfe-Gould test use, but it is only an approximation. Whenever possible you should estimate the model in one go using maximum likelihood, for example using the gologit2 command in Stata, which is described in the link above – Maarten Buis Jun 13 '13 at 14:02