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Suppose that you've got a logistic regression with multiple nominal outcomes that cannot be ordered in a theoretically meaningful way. Assume further, however, that the proportional odds assumption nonetheless holds true for these categories for a given order. In this scenario, would it be possible to fit an ordered logistic regression model?

What I'm really asking is if there's any reason, apart from the proportional odds assumption, for why a ordered logit needs ordered categories? Is it simply a mathematical assumption or is there something else to it that I'm presently missing?

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The short answer is that you need ordered categories for the model to fit accurately. You raise an interesting question of whether you can solve for a reordering of categories to satisfy the proportional odds assumption will you get a good-enough model fit and can you adjust inferential quantities for such data dredging. That's worth a simulation study.

There is a blend between the prop. odds model and the polytomous logistic model called the partial proportional odds model. See http://www.citeulike.org/user/harrelfe/article/13264679. I believe the R vgam package can fit it. But it needs some kind of prespecification.

A useful exercise is checking the ordinality and proportional odds assumptions separately for each predictor as examplified in my course note handouts available as a link from http://biostat.mc.vanderbilt.edu/rms (R function plot.xmean.ordinaly).

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    $\begingroup$ I understand. Is there any benefit in also considering the nonproportional odds model, or is that simply an extension of the PPO model that implies that the proportional odds assumption has been relaxed for every independent variable? $\endgroup$ – Johan Larsson Jul 23 '15 at 7:38
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    $\begingroup$ If by nonproportional odds you mean polytomous (multinomial) logistic, there is a disadvantage because of the huge number of parameters required when you don't have an ordinal outcome. $\endgroup$ – Frank Harrell Jul 23 '15 at 13:08
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    $\begingroup$ It was a term I encountered here, but I believe it simply referred to a partial proportional odds model with the parallel lines assumption nullified for every predictor. I agree about the disadvantage of the multinomial model; in fact it was the reason for which I posted this question in the first place. $\endgroup$ – Johan Larsson Jul 23 '15 at 17:00
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    $\begingroup$ Partial proportional odds with the proportionality assumption made for none of the predictors is equivalent to polytomous logistic regression. $\endgroup$ – Frank Harrell Jul 23 '15 at 18:20

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