Probabilities and odds in medical science

Probabilities and odds are often confused. The Dutch language doesn’t even have a word for “odds” and instead simply uses the English term. The difference between the two is relevant though. When gambling, for example, and also when trying to understand your risk of getting a disease.

This post is also available in Dutch.

Probabilities and odds are often confused. The Dutch language doesn’t even have a word for “odds” and instead simply uses the English term. The difference between the two is relevant though. When gambling, for example, and also when trying to understand your risk of getting a disease.

The difference between “probabilities” and “odds” is subtle but important. This is especially true in medical science, where confusing relative risk, which uses probabilities, and odds ratios can lead to misleading results. 

Both probabilities and odds reflect the chance of some event happening in the future or not. They are both ratios, calculated using the number of times a certain event occurred in the past. The difference, however, lies in what this number is divided by; probabilities are relative to the total number of observations, whereas odds are relative to the number of events that did not occur.

Probabilities are relative to all observations

The probability of an event happening is calculated as the ratio of the number of times the event happened, and the number of observations that were made.

p(event) = \dfrac{\#\text{events}}{\#\text{observations}}

Probabilities are always between 0 and 1. Something cannot happen with greater than a 100% chance because, at that point, it always happens. Nor can it occur with lower than a 0% chance because, at that point, it never happens.

If you were to toss a fair die, then it would land on any of its sides with a probability of 1/6.

Odds are relative to non-occurrences

The odds of an event happening are calculated as the ratio of the number of times the event happened, and the number of observations where the event did not happen.

odds(event) = \dfrac{\#\text{events}}{\#\text{non-events}}

Odds greater than 1 mean that there were more observations with events than without them.

Odds are useful when you’re gambling and want to know if the “odds are stacked against you”. If you bet that you will throw a 6, then your probability of winning is 1/6, but your odds of winning are 1/5.

So the odds against you are 5 to 1, meaning that it’s five times more likely that you’ll lose than win. Tough bet!

Relative Risk of viral exposure during an epidemic

In the die example, the probabilities and odds are known because the die is fair. In medical sciences, however, we often do not know these things and many studies actually try to determine them, particularly in epidemiology.

Imagine an epidemic where many people are exposed to a virus and several of them die over the course of a year. In the meantime, several people who were not exposed died of other causes.

An interesting question would then be: How much does someone’s risk of dying increase due to exposure? This is addressed by the relative risk (RR), which in our case is the ratio of the probabilities of exposed vs. unexposed people dying.

\text{RR} = \dfrac{p(dying|exposed)}{p(dying|unexposed)} = \dfrac{\frac{\#\text{dead \& exposed}}{\#\text{exposed}}}{\frac{\#\text{dead \& unexposed}}{\#\text{unexposed}}}

The RR says something about the danger of being exposed to the virus (i.e., how strong its effect is) and is relatively easy to interpret. If the probability of death in one year is 60% after exposure but is normally 20%, then the RR will be 3 (0.6/0.2). In that case, death after exposure is 3 times more likely than without it.

What about the odds ratio?

You can also determine how much your odds of dying increase by calculating the odds ratio (OR). The OR is sometimes more appropriate to use and also says something about how strong an effect is.

\text{OR} = \dfrac{\text{odds}(dying|exposed)}{\text{odds}(dying|unexposed)} = \dfrac{\frac{\#\text{dead \& exposed}}{\#\text{alive \& exposed}}}{\frac{\#\text{dead \& unexposed}}{\#\text{alive \& unexposed}}}

The odds of dying when exposed are 3/2 (60% die and 40% stay alive) and 1/4 when not exposed (20% die and 80% stay alive). That gives an OR of 1.5/0.25 = 6.

The OR is more difficult to interpret. That’s why people sometimes interpret the OR as RR, which is more intuitive. However, the OR and RR can be very different, especially when an exposure has a large effect on groups that were already at high risk because, for large probabilities, the odds get much larger in comparison. That’s the case in our example, where the odds (3/2) of dying when exposed are much bigger than the probability (0.6), leading to an OR that is twice as big as the RR.

Choose carefully!

So why not just always use the RR? The OR has some benefits over the RR such as certain statistical properties. Importantly, the OR can be used in some cases where the RR cannot. 

For example, in case-control studies, scientists may select a number of people who have developed some disease (“cases”) and then look back at how often they were exposed to a virus. They then take a group of people who have not developed the same disease and also check how often they were exposed. They can then calculate the OR because they know the number of exposed and non-exposed in each of the groups. However, they cannot calculate an RR because they do not know the total percentage of people in the population who were exposed or not exposed to the virus. 

The OR and RR are both valuable but it’s important to keep in mind that they are different measures which, while often similar, can seriously diverge in some cases. Applying them correctly is key in deciding on appropriate treatments or public health strategies. We need an accurate picture of the world if we are to turn the odds in our favor.

Credits
Author: Jeroen
Buddy: Mónica
Editor: Christienne
Translator: Wessel
Editor translation: Jill

Picture obtained from Alex Chambers via Unsplash

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