Interaction

Definitions & Key takeaways

Interaction can refer to biological interaction - which is where two exposures like radon gas and toxins in cigarettes work together to influence an outcome - like lung cancer. It can also refer to statistical interaction, also called effect modification, which is the statistical methodology used to find out if there's a biological interaction.

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The word “interaction” can refer to biological interaction - which is where two exposures like radon gas and toxins in cigarettes work together to influence an outcome - like lung cancer.
But the word “interaction” can also refer to statistical interaction, also called effect modification, which is the statistical methodology used to find out if there’s a biological interaction.
Most diseases are caused by multiple exposures that work together, like our example of radon gas and smoking cigarettes leading to lung cancer.
Radon is a radioactive gas that gets released from the decay of elements like uranium and radium in rocks and soil. It can be found in dust particles in the air, so most people breathe in a low level of radon every day.
Unfortunately, radon causes mutations in DNA and people who breathe in high levels of radon have an increased risk of lung cancer.
Similarly, people who smoke cigarettes have a higher risk of lung cancer because of tobacco contains various toxins that also mutate the DNA.
In addition, cigarette smoke harms the cilia in the lungs. Those are the little hairlike structures that normally clear out things like mucus, dust particles, and chemicals.
Damaged cilia is a big problem for people who are exposed to high levels of radon, because the lungs can’t get rid of radon-containing dust.
This is an example of biological interaction, because even though radon and smoking can both separately cause lung cancer, they also work together to amplify the risk.
Statistical interaction can help us figure out how much the risk increases for people who are exposed to both factors compared to people who are exposed to one factor or the other.
Statistical interaction can be assessed by comparing the effect of one exposure on the outcome in each strata or level of the other exposure.
For example, you could figure out how smoking affects the risk of lung cancer among people exposed to high levels of radon, and how it affects the risk of lung cancer among people exposed to low levels of radon.
To do this, you might start by looking at the crude or unstratified effect, so you’d recruit 100 people who smoke and 100 people who don’t smoke, and compare the proportion of people in each group who develop lung cancer in the next ten years.
Let’s say you find an overall relative risk, or crude effect, of 7.5 - so people that smoke have 7.5 times the risk of developing lung cancer, compared to people that don’t smoke.
After you find the crude effect, you want to see if the risk of lung cancer changes for each level of radon exposure. So, let’s say that overall, there are 140 people in the study that were exposed to low levels of radon and 60 people in the study that were exposed to high levels of radon.
Now, of the 140 people in the low radon group, 60 people smoked and 80 people didn’t smoke. Of the 60 smokers, 40 of them - 67% - developed lung cancer, and of the 80 non-smokers, 35 of them - 44% - developed lung cancer.
This gives us a relative risk of 1.5 in the low radon group, which is lower than the crude relative risk of 7.5. On the flip side, of the 60 people in the high radon group, 40 people smoked and 20 didn’t smoke.
Of the 40 smokers, 35 of them - 88% - developed lung cancer, and of the 20 non-smokers, 5 of them - 25% - developed lung cancer.
This gives us a relative risk of 3.5 in high radon group, which is lower than the crude relative risk, but higher than the relative risk in the low radon group.
We’d call this result a heterogeneity of effects - because each strata has a different risk. One common mistake is distinguishing between confounding and interaction.
In confounding, a third variable is associated with both the outcome and the exposure, whereas in interaction a third variable is associated with the outcome but not the exposure.
For example, let’s say that smoking is the exposure, lung cancer is the outcome, and radon exposure is the third variable.
Here, there’s an association between radon and an increased risk of lung cancer, but people exposed to high levels of radon aren’t more likely to be smokers - so it’s not associated with the exposure.
In contrast, exposure to coal dust would be a confounder, because just like radon it’s associated with an increased risk of lung cancer.
Additionally, exposure to coal dust is also associated with smoking, because coal miners tend to smoke more cigarettes than people in other professions.
Statistical interaction can be used to figure out if the joint or combined effect of the two exposures is different from what we expect them to be, based on their independent effects.
In other words, if we add or multiply the effect of one exposure with the effect of the other exposure, will the joint effect match what we think it should?
We can figure out the expected joint effect of two exposures in two ways. First, we can calculate it on either a multiplicative or additive scale.
Using the multiplicative scale, we can get the expected joint effect from multiplying the observed effect of one exposure by the observed effect of the other exposure.
For example, let’s say we want to see if there’s a multiplicative interaction between smoking and radon exposure, and we observe four groups of people with 100 people in each group.
People that have neither exposure, people that do smoke but aren’t exposed to radon, people who don’t smoke but are exposed to radon, and people who have both exposures, which is the observed joint effect.
For each group, we find the absolute risk of lung cancer, which is calculated by dividing the number of people who got lung cancer in that group by the total number of people in that group.
So, let’s say the observed absolute risk of lung cancer is 3 for the first group, 15 for the second group, 9 for the third group, and 60 for the fourth group.
The absolute risk of lung cancer of 3 in the first group is the background risk since there’s still a risk of lung cancer for people who aren’t exposed to either smoking or radon.
Next, let’s calculate the relative risks for each group by dividing each one by the background risk. So the relative risk of people who only smoke is 15 divided by 3 or 5; and the relative risk for people who are only exposed to radon is 9 divided by 3 or 3, and the relative risk for people who smoke and are exposed to radon is 60 divided by 3 or 20.
We now multiply the relative risk of each of the two exposure groups - 5 times 3 - to get the expected joint effect of 15.
In other words, we expect people who smoke and are exposed to radon to have 15 times the risk of lung cancer compared to people who don’t smoke and aren’t exposed to radon.
But the observed relative risk in this group was 20, which is higher than the expected relative risk of 15, so there’s evidence of interaction.
Specifically, since the observed effect is higher than the expected effect, it’s called synergistic interaction, and it implies that the two variables work together to increase the risk.
On the other hand, if the observed effect is lower than the expected effect, it’s called antagonistic, which implies that the two variables work together to decrease the risk; and if the observed effect is equal to the expected effect, then no interaction is present.
We can also calculate the expected joint effect on the additive scale by adding the observed effect of one exposure group to the observed effect of the other exposure group.
This time we use the attributable risk, or the excess amount of risk associated with a certain exposure. The attributable risk is calculated by taking the absolute risk in an exposure group and subtracting by the amount of the background risk.
So the attributable risk of people who only smoke is 15 minus 3 or 12; and the attributable risk for people who are only exposed to radon is 9 minus 3 or 6, and the attributable risk for people who smoke and are exposed to radon is 60 minus 3 or 57.
We now add the attributable risk of each of the two exposure groups - 12 plus 6 - to get the expected joint effect of 18.
In other words, we expect people who smoke and are exposed to radon to have 18 excess cases of lung cancer compared to people who don’t smoke and aren’t exposed to radon.
But the observed attributable risk in this group was 57, which is higher than the expected attributable risk of 18, so once again, that means that it’s a synergistic interaction.
Most statistical models - like linear regression and logistic regression - are designed to only assess multiplicative interaction - so that’s the one that is usually evaluated for.
This is particularly problematic in situations where there’s antagonistic multiplicative interaction and synergistic additive interaction.
That might happen because the threshold for synergistic multiplicative interaction is typically larger than the threshold for synergistic additive interaction, simply because multiplication results in larger numbers compared to addition.
For example, if the observed absolute risk for people exposed to A is 5, and B is 6, then the threshold for synergistic multiplicative interaction is 30, and the threshold for synergistic additive interaction is only 11.
If the observed absolute risk for people exposed to A and B is somewhere between 12 and 29, then there’s an antagonistic multiplicative interaction and a synergistic additive interaction!
Alright, as a quick recap, interaction can refer to either biological interaction, the mechanism by which two exposures influence an outcome, or statistical interaction, the statistical way to test for biological interaction.
Statistical interaction can be tested by looking at the heterogeneity of the effect of the exposure on the outcome in each strata, or by looking at how the observed joint effects of two exposures compare to the expected joint effects of two exposures on either an additive or multiplicative scale.