I'm writing an interdisciplinary research paper and I'm having some troubles in clearly explaining my findings. In particular, I applied a proportional odds model with one regressor $x$ and three intercepts (three ordered categories) $\alpha_{j}$ with $j = 1|2, \,\,2|3,\,\, 3|4$.

$${logit}(\pi_{j}) = {ln}\left(\frac{\pi_{j}}{1 - \pi_{j}}\right) = \alpha_{j} + \beta^{T}x.$$

For the estimated coefficient $\beta$ I obtain an odds ratio of about 1.1, which should indicate that for an increase in the value of the regressor the odds of moving from a lower or equal to a higher category increases of about 10%.

As you can see, my explanation is not very clear and may cause some doubts in readers who don't know the proportional odds model and/or lack of proper statistical training.

Can you help me in rephrasing a little to make my findings comprehensible to a broader audience?



I think that the first and biggest hurdle is making sure that people indeed understand logistic regression and what an odds ratio actually is. If they get that far, you simply need to explain that proportional odds models take logistic regression one step further to account for ordered categorical responses.

A naive approach could be to run a logistic regression model for a cut point. You could cut the outcome so that a positive response is a 3 or higher versus a negative response is a 2 or lower. This is a valid data analysis approach, except for that the cutpoint is arbitrary. You might get a slightly different outcome running the same model with a cutpoint at 2 instead of 3.

Proportional odds models, in a sense, "average up" over all possible cutpoint models to maximize the amount of information you can get out of the data. This is very good for modeling the association between one or more continuous or categorical predictors and an ordinal outcome, and it can even be used to predict outcomes somewhat.

An example of proportional odds "in the field" comes from the following paper, where authors examined the relationship between ambient air pollution and asthma severity (on a Likert type scale)

Our results indicate that a 10-μg/m3 increase in particulate matter less than or equal to 2.5 μm (PM2.5) lagged 1 day was associated with a 1.20 times increased odds of having a more serious asthma attack [95% confidence interval (CI), 1.05 to 1.37]

"A more serious asthma attack" here is taken to be the lay interpretation of what the proportional odds model is estimating. It conveys, in essence, a very nice counterfactual interpretation of findings which is why I, as a statistician, like these models so much.

  • $\begingroup$ I really appreciated your answer. Thanks a lot, man! :) $\endgroup$ – stochazesthai Mar 24 '15 at 10:26

A key step is to make sure people understand why log-odds-ratios are useful. To help motivate log-odds-ratios, try the tale of two principals:

  • High School A reduced the dropout rate from 10% to 5%, a dramatic 50% decrease!
  • High School B increased the graduation rate from 90% to 95%, a modest 5.5% increase.

The first principal was lauded by the NYTimes for slashing the drop out rate by half. The other principal got a short mention in the local newspaper. Even though they did the same thing.

Log-odds ratios puts these on even terms:

$$\log \left( \frac{0.95/0.05}{0.90/0.10}\right)=0.32$$ and $$\log \left( \frac{0.05/0.95}{0.10/0.90}\right)=-0.32$$

In fewer words, you can say that log-odds-ratios consider a change from 10% to 5% equivalent to a change from 90% to 95%.

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    $\begingroup$ I edited your question to take advantage of the website's mathjax equation presentation, but your most recent edit seems to have overwritten that. If you look at the edit history, you'll be able to see the code that I used to write your math as math. $\endgroup$ – Sycorax Mar 23 '15 at 19:24
  • $\begingroup$ Sorry! I made a small tweak simultaneously. Yours is better! $\endgroup$ – prototype Mar 23 '15 at 19:24
  • $\begingroup$ No worries. The code is still there so that when you get the post to look how you like, you can just drop it in. $\endgroup$ – Sycorax Mar 23 '15 at 19:25
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    $\begingroup$ This is an important point about the odds ratio and one that could easily be explained with level curves in a labbe plot. However it has no bearing on proportional odds models specifically. $\endgroup$ – AdamO Mar 24 '15 at 21:10

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