A common woe of modern day epidemiologists is the interpretation of their work. Epidemiologists write papers on exposure-disease relationships by collecting data on relatively large groups of people. Although there are many ways in which our work can be misinterpreted, a common pitfall is the confusion of an odds ratio with relative risk.
After epidemiologists carefully design a study and collect their data, they run statistical analyses to determine if the risk factor we are interested in was indeed related to the outcome of interest.* The output of this analysis is typically some variation of either a relative risk or odds ratio. On the surface, both are relatively easy to interpret:
If a study finds a relative risk of 1.5, this would mean that risk of disease is 50% higher in the exposed group compared to the unexposed group.
If a study finds an odds ratio of 1.5, this would mean that the odds of disease are 50% higher in the exposed group compared to the unexposed group.
However, risk and odds are not equivalent.
If I told you that you could pick between the following two lotteries, which would you choose?
A) One where you have a 1 to 3 odds of winning
B) One where you have a 33% chance of winning
We can work through a simple mathematical example that allows us to interpret these two statements:
We can think of the odds ratio as how many times we’ll win versus how many times we’ll lose. Hypothetically speaking, if we were able to play lottery A four times, we’d win once, but lose three times.
Odds = 1:3 = 0.33
Probability = (1/(1+odds))
Probability = (1/(1+0.33))
Probability = 0.25
Here, the odds of 1:3 translate to a probability of winning of 25%.
So, given the two options above, you are better off picking lottery B, where you have a 33% chance of winning – much better than a 1 to 3 odds of winning!
So, why is this relevant to epidemiology? Often, epidemiologists are constrained by study-design and can only calculate an odds ratio, and not a relative risk. There’s nothing wrong with odds ratios, yet our concern here is that since odds are always greater than risk, individuals reading the study may misinterpret their odds of disease as their risk of disease.
Often, this can lead to increased concern for a risk factor that may not be as strongly related to a disease as it is seems.
There are some instances in epidemiology when an odds ratio can be interpreted as a relative risk, mainly, when the rare-disease assumption holds.
If an outcome is rare, the odds ratio ends up being almost equivalent to the relative risk, and a mix up of the odds and risk does not end up being very concerning. The key to understanding this assumption is in the column totals in red for the following two examples below:
Here is an example of a common disease from a mock dataset. Here, the disease afflicts 33% of the total population (60/200). In this study, two groups of 100 people each were observed. One group had been exposed to a certain risk factor, and the other had not. The number of diseased individuals was higher in the exposed group.
The relative risk is the ratio between your probability of getting the disease if you are exposed and your probability of getting the disease if you were not exposed.
RR = (40/100)/(20/100)
RR = 2
The odds ratio is the ratio between the odds of getting the disease if you are exposed and the odds you will get the disease if you were not exposed.
Odds = (40/60)/(20/80)
Odds = 2.66
Note, the relative risk and odds ratio are not equal.
Things get interesting when we look at a rare disease (prevalence of 3.33% in the population, or 60/2000)
Here the relative risk is:
=(40/1000)/(20/1000)
=2
And the odds ratio is:
=(40/960)/(20/980)
=2.04
Note that the odds ratio is much closer to the relative risk when the disease is rare. This has to do with the fact that when a disease is rare, the number of non-diseased individuals (in blue) approximates the total number of study participants in each exposure group (in red).
From this example, you can see the odds ratio is a good approximation for the relative risk only when the disease is rare. As a disease becomes more common, the odds ratio will overestimate the relative risk.
Of course, as with any scientific explanation, there is more to this assumption. For a more descriptive and detailed explanation, I would refer readers to Modern Epidemiology, Chapter 8: Study Design and Conduct (Rothman, 2008), from which I drew core content for this post.
Takeaway:
1.) Always double check whether the reported effect estimate is an odds ratio or relative risk.
2.) Words like “more likely” are sometimes used to describe odds, but should only be used to describe risk.
3.) Odds will always be higher than risk, although they can very closely approximate risk when the disease is rare.
*Epidemiologists often use the term “risk factor” to mean any exposure of interest, even those that are positive, like physical activity, or eating fruits and vegetables. “Outcome” can refer to anything of interest to epidemiologists: a disease, a period of time, and even a continuous measure such as heart rate or weight.
Reference
Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern epidemiology (Vol. 3). Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins.