Odds ratio

Definitions & Key takeaways

The odds ratio (OR) is a statistical measure used to compare the odds of an event occurring in one group to the odds of the same event occurring in another group. It is often used in retrospective studies, such as case-control studies, to compare the likelihood of an outcome (such as a disease or condition) occurring in one group of people compared to another group.

The odds ratio is calculated by dividing the odds of the event occurring in one group by the odds of the event occurring in the other group. For example, if the odds of a disease occurring in group A are 1 in 10, and the odds of the disease occurring in group B are 1 in 20, the odds ratio would be (1/10) / (1/20) = 2. This indicates that the odds of the disease occurring in group A are twice as high as the odds of the disease occurring in group B.

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In statistics, the words probability and odds are often confused with each other, because they help measure the same thing - the chance that an outcome will occur - and in both cases, we need to know the same two things - the number of times an outcome actually happened, or didn’t happen, and the total number of times an outcome could have happened.
The probability is the number of times an outcome happened divided by the number of times the outcome could have happened, and is often represented by a capital P.
For example, let’s say we want to figure out the probability of having a heart attack for people with hypertension - or high blood pressure.
To do this, we could carry out a cohort study which is where we start with two exposure groups and follow them over time to see if they develop a certain outcome.
We could recruit 100 people with hypertension, The exposed group - and 100 people without hypertension - the non-exposed group - and organize our results in a 2 by 2 table, and keep track of how many of them have heart attacks in the next year.
The two outcomes - heart attack or no heart attack - labeled on the top, and the two exposure groups - hypertension or no hypertension - on the side, and each of the cells inside the box labeled as a, b, c, or d.
Now, let’s say there are 9 people that have heart attacks in the group with hypertension - cell a - and 3 people that have heart attacks in the group without hypertension - cell c.
That means that there are 100 minus 9, or 91, people that didn’t have heart attacks in the group with hypertension - cell b - and 100 minus 3, or 97, people that didn’t have heart attacks in the group without hypertension - cell d.
To calculate the relative risk, we need the probability of having a heart attack in the group with hypertension - so cell a, 9, divided by cell a plus cell b or 100, which gives us 0.09.
We also need the probability of having a heart attack in the group that doesn’t have hypertension - so cell c, 3, divided by cell c plus cell d, 100, which gives us 0.03.
The relative risk is 0.09 divided by 0.03, so 3, which means that people who have hypertension have 3 times the risk of having a heart attack in one year compared to people without hypertension.
The probability of having a heart attack in the past year for people with hypertension is 9 divided by 100, and the result can be written as a decimal - 0.09 - or a percentage - 9%.
On the flip side, we could also find the probability of not having a heart attack for people with hypertension. A simple way to do this is to subtract the probability of having a heart attack from 1, or 100%.
So, the probability of not having a heart attack for people with hypertension is 1 minus 0.09, which equals 0.91, or 91%.
Now, the relative risk can only be calculated when we have information about incidence, or the probability of a new outcome occurring during a certain time period.
That’s because the word risk implies that hypertension is responsible for causing, or at least partially causing, heart attacks to happen.
And, if hypertension causes heart attacks to happen, then a person would have to have hypertension before they have a heart attack, which can only be measured by incidence - which measures new events.
But sometimes information on incidence can’t be collected, so relative risk can’t be calculated. For example, case-control studies start with a case group, that already has the outcome, and a control group, that doesn’t have the outcome - and looks at past exposures.
In this situation, odds and odds ratios come in very handy. Odds compare the probability of an outcome occurring with the probability of an outcome not occurring, so it’s P compared with 1 minus P.
So in our hypertension example, we’d use the probability of having a heart attack - 0.09 - compared with the probability of not having a heart attack, 1 minus 0.09, or 0.91.
Dividing both sides by 0.91, we get 0.0989 compared to 1, and rounding off we can say that it’s 0.1 compared to 1, and multiplying by 10, we get an odds of 1 to 10.
In other words, in one year, for every 1 person with hypertension who has a heart attack, there are 10 people with hypertension who do not have a heart attack.
In fact, the odds will always be slightly higher than the probability of an event, but the odds and the probability end up being very similar when an outcome is rare - specifically when it’s found in less than 10% of the population.
In this population, the probability of heart attacks is only 0.09 - so it is fairly similar to the odds - 0.1. Now, odds help us calculate the odds ratio, which is a comparison of the odds of an event occurring in one group divided by the odds of an event occurring in another group.
And, just like odds are similar to probabilities when the outcome is rare, the odds ratio is also similar to the relative risk when the outcome is rare.
So even if the relative risk can’t be calculated - like when we don’t have information on incidence - if the outcome is rare, we can still get an estimate of the relative risk from the odds ratio.
One type of odds ratio is the outcome odds ratio, and it’s a comparison of the odds of an outcome between one exposure group and a second exposure group.
The outcome odd ratio can be used to estimate the relative risk. To calculate the outcome odds ratio, we need the odds of having a heart attack in the group with hypertension - so cell a divided by cell b, or 9 divided by 91, which is 0.1 - divided by the odds of having a heart attack in the group without hypertension - so cell c divided by cell d, or 3 divided by 97, which is 0.03.
So, 0.1 divided by 0.03 is 3.2, which means that people who have hypertension have 3.2 times the odds of having a heart attack in one year compared to people without hypertension.
Since the outcome is rare enough, the outcome odds ratio ends up being similar to the relative risk, 3. Now, there’s also an exposure odds ratio, which is a comparison of the odds of an exposure between one outcome group and a second outcome group.
Funny thing is, the outcome odds ratio and the exposure odds ratio come up with the same number. Now let’s say we have data from a case-control study which recruited 12 people that had heart attacks, the cases, and 188 people never had a heart attack, the controls.
Again, the numbers in the 2 by 2 table are the same, but the meaning of each cell is now a little different. For cell a, instead of saying that 9 people with hypertension got heart attacks - which would be the incidence, we say that 9 people who had heart attacks have a history of hypertension - which is the prevalence.
So, 91 people in cell b didn’t have a heart attack but do have a history of hypertension, 3 people in cell c had a heart attack but don’t have a history of hypertension, and 97 people in cell d didn’t have a heart attack, and don’t have a history of hypertension.
To calculate the exposure odds ratio, we use the odds of having hypertension for someone who already had a heart attack is cell a divided by cell c - so 9 divided by 3, or 3 - over cell b divided by cell d - so 91 divided by 97, 0.94 - which equals 3.2.
Notice that this number is the same as the outcome odds ratio! To see why that makes sense, we can use a 2 by 2 table is to use the cross-product ratio - so (a times d) divided by (b times c).
This works because there’s a rule for dividing fractions that says that one fraction divided by a second fraction is the same as one fraction times the reciprocal of the second fraction.
So, we can rearrange the outcome odds ratio equation - (a divided by b) over (c divided by d) - into (a divided by b) times (d divided by c), which turns into (a times d) divided by (b times c).
We can also rearrange the exposure odds ratio (a divided by c) over (b divided by d) - into (a divided by c) times (d divided by b), which also gives us the same equation.
In theory, we can use the outcome odds ratio to estimate the relative risk, since they both look at the chance of an outcome occurring in two different exposure groups.
And, since the exposure odds ratio comes up with the same number as the outcome odds ratio, we use the exposure odds ratio to estimate the relative risk.
Alright, as a quick recap, Even though the exposure odds ratio does not measure incidence, it can estimate the relative risk, when the outcome is sufficiently rare.
To quickly calculate the odds ratio you can calculate the cross-product ratio- so (a times d) divided by (b times c). This is particularly helpful in case-control studies.