Not a trick question: what’s the difference between a risk of one in ten million and one of two in ten million? The official answer is “Not much.” Though I would also have accepted “Almost none”, “Close enough to be the same,” and “Who would care?”
The exceptionally nervous care. They would have said “One is twice as large as the other!” That’s mathematically correct. And it sounds juicier than the banal entries. Might get somebody’s blood going if you can claim to have doubled a risk. Thus, those wanting to excite will talk in terms of relative risk.
Relative risk is the simple formula:
(Probability getting it given condition B)/(Probability getting it given condition A).
“Getting it” means meeting a criterion, such as developing or dying of some malady, but it can be anything. The conditions are the different evidences used to calculate the probabilities. Example: “getting it” means seeing a Head on a coin. Let “condition A” be “flip one coin” and “condition B” be “flip two coins.”
We deduce the probability of Head given A of 1/2, which soars to 3/4 given B (trust me). Thus the relative “risk” of seeing a Head is (3/4)/(1/2) = 1.5. Another way to say it is that the risk of a Head is 1.5 times higher for flipping two coins over just one. Easy, right?
Next suppose “getting it” is developing cancer of the albondigas. Condition B might be those exposed to chemicals in the drug coriandrum sativum (actually sold by many corporations: the FDA does not control it). Condition A are those not exposed.
Now “exposed to” and “not exposed to” are tricky. Problem is they don’t mean anything, or anything specifically. Is just a whiff of coriandrum sativum enough to count as “exposed”? Or does it take daily ingestion? If so, for how long a period and in what quantities? And what about the people “not exposed”? Presumably they were “not exposed” because they were different than the people who were “exposed.” How many ways were they different?
Is life easy by defining “exposed to” as “any exposure of any kind in any amount”, which is certainly plain enough? No, because it begs the question why the folks who did not see any exposure of any kind in any amount didn’t. Were they off on holiday? Did they eat different foods? Come from a different culture? Were they older or younger? Have variant genetics? You can go on forever—literally—and never be sure of identifying all things different between the groups.
Another twist. Suppose God Himself told us that the probability of cancer in the exposed group is 2 in 10 million and the probability of cancer in the not-exposed group is 1 in 10 million. The relative risk is 2. We could write a peer-reviewed paper which says, and says truthfully, “Exposure to coriandrum sativum doubles the risk of cancer of the albondigas.” Yet what does it mean?
Epidemiologic evidence suggests that Los Angeles is a “hot spot” for coriandrum sativum exposure. Now LA proper has about 4 million residents, some of whom will have been exposed, some not. Suppose, for the sake of argument, the population is split. And remember the Lord himself assured us the relative risk of 2 is correct.
The chance is 67% that nobody in the exposed group will develop cancer. Pause here. That’s no-body. Meaning there’s a two-out-of-three probability that not a single soul of the 2 million exposed people will get cancer. Anti-intutive? Shouldn’t be: even with a relative risk a whopping 2.0, there’s still a paltry 2 in 10 million chance of disease.
Oh, chance nobody gets it in the not-exposed group is 82%.
Now–and pay attention here—the chance that just one person of the 2 million in the exposed group develops cancer is 27%. That means there’s a 94% that either nobody gets it or just one person in 2 million does. (There’s a 16% chance just one person in the non-exposed group gets it.)
Gist is that the overwhelming probability is no more than one poor soul develops cancer in the exposed group, but that it’s more likely nobody does. Same in the not-exposed camp. Shocked?
I saved the funkiest bit for last: the chance that more people develop cancer in the exposed group is just 28%!1 Not even close to 100%, right? Even with a relative risk of 2, there’s only just over a 1 in 4 chance of larger numbers of cancer patients in the exposed group. Reflect on this. No, seriously. Reflect.
The largest chance, 59%, is that both groups will have the same number of people who develop cancer. And there is even a 13% chance that the not-exposed group will have more people who cancer than the exposed group!
I promised funky and that’s pretty funky, no? Emanates from the title: there’s a world of difference between (sensational) relative and (sober) absolute risk.
1These are from standard calculations, where “standard” is used in the sense “everybody knows but few use them.”