Why Odds Ratios Are Odd

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Podcast: Why Odds Ratios Are Odd
Length: 5 minutes 59 seconds
Written and read by the author

This brings us to today’s topic: Odds Ratio (OR) vs. Relative Risk (RR): Odds vs. Probability: why we love them and why these two cousin statistics continue to confuse us. Anyone with a math, science, or medical background has been taught this, and if you are like me, keep forgetting it.

Why should you care? If you regularly throw dice at the Bellagio craps tables or read the occasional medical paper, you care. Popping up as the most common measures of association in epidemiologic studies, we in the veterinary health care professionals come across these when looking at infectious and disease risk. In our current COVID-19 pandemic, no one can avoid epidemiologic values. How do your odds and risk fair?

Don’t worry, this article won’t get too “mathy.” For exact calculations, the CDC has a free pdf called the Principles of Epidemiology: Home Self-Study Course if you want a deep dive. Let’s talk about the main differences between odds and risk. We saw the first difference with the dice example. 1 divided by 5 is 0.2 and is larger than 1 out of 6 or 0.167. Not a huge deal but big enough if you are a numbers person. When the numbers start to stack up, this margin may become relevant.

The Rare Disease Assumption allows OR and RR to be practically interchanged whenever the disease is rare. The cutoff for “rare” differs by the specialist. Here’s why this works, roughly one out of 11 million people will die by a plane crash. The odds of that occurring is 1 divided by 11 million minus the one unfortunate soul. The risk is 1 divided by the full 11 million. Either provide you a similar number with which you can formulate risk.

Having exhaled a sigh of relief, you board the plane and are confronted with an even more frightening concern – the middle seat. Not being rare, the risk is 0.33, while the odds are 0.5. Whenever you mix up terms in the absence of the rare disease assumption, odds and OR will overestimate risk.

In theory, within the same paper, you could see OR and RR being compared for one disease but not for another. If you forget, you could grab the OR from one paper and look at the RR in another. Not only should you always take the comparison of values from separate studies with great, great trepidation, dear friend, but in the absence of the rare disease assumption, you will surely be misled.

With rare diseases, researchers more commonly utilize case-control OR-generating studies and for rare exposure, cohort RR-generating studies. As these and other study designs begin to piece our understanding of a rare disease together, statistics spew forth with more abundance than diarrhea in an IBD study.

Odds and OR is not all bad, though. This value’s flexibility allows for an OR to be calculated whenever an RR can be. RRs remain less flexible; remember, they need prevalence or incidence data. For logistical regression, researchers need OR. Logistical regression sounds fancy, but it just allows you to understand the relationship with a binary value (alive or dead, Cushingoid, or not) with another variable. With the comorbidity of obesity, how does that affect the probability of endocrine disease? You see logistical regression all the time and thus love OR.

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