Where is there is only set of odds ratio in ordinal logistic regression?

Question

I am running an ordinal logit regression with health status(poor, fair, good, very good, excellent) as the outcome. I understand the difference between ordinal and multinomial outcomes, and read many tutorials about the proportional odds assumption. But none were plain enough for me to understand how that works, and how/why the model produces only 1 set of odds ratios.

Below is the closest explanation I have found, but it still doesn't clarify the process of producing only 1 set of odds ratios:

The probability of a person's health being in the "Poor or below" category, which includes the "Poor", "Fair", and "Good" categories. This is compared to the reference category of "Excellent or above". The probability of a person's health being in the "Fair or below" category, which includes "Poor", "Fair", "Good", and "Very Good". This is compared to the reference category of "Excellent". The probability of a person's health being in the "Good or below" category, which includes "Poor", "Fair", and "Good". This is compared to the reference category of "Excellent or above". The probability of a person's health being in the "Very Good or below" category, which includes "Poor", "Fair", "Good", and "Very Good". This is compared to the reference category of "Excellent". The probability of a person's health being in the "Excellent" category. This is the reference category.

This question was asked and answered in many instances before, but I hope someone can help clarify this in plain English using this example. Thank you in advance!

Answer 1

The reason that an ordered logit only gives you one set of odds ratios is that it assumes (maybe correctly, maybe incorrectly) that one set are all that you need. The model is governed by something called the "proportional odds assumption." Intuitively, this means that the model assumes that the "impact" any independent variable on the log of the odds of "moving up the scale" is the same no matter what point on the scale you start at. So if your model gives you an odds ratio of 1.25 for the coefficient "age" year that means that with each additional year of age your odds of "moving to the next higher category" increase by 25%. The model assumes that this is true no matter whether we are talking about moving from "good" to "very good" or moving from "poor" to "fair" or whatever.

You can actually test that assumption with something called a brant test, but even if it's violated a bit, an ordered logit usually still tells you a pretty good story. However, if assumption is violated a lot - if the dynamics causing someone to move from poor to fair are totally different from the dynamics causing people to move from good to very good - then an ordered logit model is the wrong tool for the job. In that case you probably want a multinomial logit model, which chooses an explicit reference category and does produce separate sets of coefficients for each category. But for precisely that reason, multinomial logit models are much harder to interpret, and require larger sample sizes.

One more thing, since you referred to the odds ratios as telling you about the "probability" of giving a certain response. In any kind of logit model (binary, ordered, whatever) odds ratios never tell you how the probability of the outcome changes, but how the odds change. So in a binary logit model an odds ratio of 1.25 does means that the odds increase by 25%, but doesn't tell you anything about how the probability changes (that's a "risk ratio"). And odds and probabilities are very different things. This is a common misunderstanding about odds ratios. If you want to actually know how the probability changes, you need to do some extra work, such as calculating average marginal effects.

Answer 2

This is explained in detail here (start with the first link to BBR). If you want to allow ratios of odds of $Y \geq y$ to vary with $y$ without restriction, use the polytomous (multinomial) logistic regression model. This requires a much larger sample size, e.g., the size of the smallest cell of $Y$ needs to be large. Proportional odds and proportional hazards and other semiparametric ordinal models requires an overall large-enough sample size and there to be no more than say $\frac{4}{5}$ of the observations at any one level of $Y$ unless the overall $N$ is large. Sample size issues are covered in one of the links in the first link I provided at the top, and here.