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Relative Risks versus Changes in Absolute Risks

Absolute versus Relative Risks. In this video Professor Michael Spagat explains a nefarious technique for generating headline-grabbing health scares.
Just before I recorded this video, bacon hit the news like a story about royal marriage. The Sun of the UK had a tricky headline– Killer Rasher Pack of Bacon Increases Risk of Bowel Cancer by a Fifth, Study Shows. This was followed by a creative and even trickier sub-headline– Just 25 Grammes of Processed Meat Daily, Equal to One Rasher or 2/3 of a Banger, Fuels Cancer Twice as Much as Previously Thought. Wow, twice as much. But then, statistician David Spiegelhalter of Cambridge University pooh-poohed the whole story, saying, “The cancer risks of bacon are around double that reported before.
That means 100 people would need to eat a bacon sarnie every other day for their whole lives for one extra case of bowel cancer. Using previous estimates, it would have to be every day.” As you’re able to deduce from my accent, I’m a little hazy on rashers, bangers, and sarnies. But leave that aside. The point is that bacon will kill you before you finish watching this video. Watch out. That was a joke, actually. You’re not going to die. The actual point is that The Sun headlines incorporate two separate but similar layers of exaggeration. Layer one– the risks of colon cancer facing bacon eaters are 1/5, or in other words 20%, higher than the colon cancer risks to non-bacon eaters.
I won’t question this claim here. I merely want to point out that adding a fifth to something small produces, well, a pretty small increase. Here are some figures you can extract from the same Sun article. Slightly fewer than seven men out of 100 are eventually diagnosed with bowel cancer. For women, it’s more like six out of 100. 1/5 of seven is a bit more than one, so this means that a lifetime of bacon consumption converts a seven in 100 risks to roughly an eight in 100 risk. That’s pretty much where Professor Spiegelhalter’s one in 100 comes from. Think of it as follows. I go off to university and join a fraternity with 100 members.
As part of our wild initiation ritual, we all pledge to eat bacon every day for the rest of our lives. If we hadn’t all made this reckless pledge on that memorable night, then only seven of us would have eventually gotten bowel cancer. But since we were such idiots, one additional brother will get it. Layer two– The Sun also says that the risk has doubled. But this is compared to what we knew before. The recent research lowers by half the threshold for baking consumption that must be reached in order to produce that one extra case per 100 people. But the threshold is still very high, as Professor Siegelhalter points out.
Moral of the story– try to think in terms of 100 people and how their outcomes will change in response to some new risk. Also, always ask for a base from which to calculate percentage changes. A large percentage change can still be a small thing if it’s calculated off of a small base. And by the same token, half of a very large thing can still be very large.

Here are some sample calculations that, hopefully, help you to understand relative and absolute risks better.

We can extract the following pieces of information from this nice little article.

  1. 2.9 people out of 100 get bowel cancer by the age of 65 if they do not eat processed meat.
  2. The bowel cancer rate rises to 3.4 out of 100 for people who eat 50 grams of red meat per day.

Thus, meat eaters face a relative bowel cancer risk, compared to non-meat-eaters, of 1.17. The calculations is simply 3.4/2.9 = 1.17. Note, however, that relative risks are often rendered in percentage terms which, in this case, comes out to 17%. This is calculated as:

((3.4 – 2.9) / 2.9) x 100 = 17%

It’s better, however, to express the risks in absolute terms. Facts 1) and 2) together imply that, roughly 1 person out of 200 meat eaters will get bowel cancer because of their meat consumption. Here’s the calculation:

(3.4 out of 100) – (2.9 out of 100) = 0.5 out of 100 which is the same as 1 out of 200.

So the change in absolute risk caused by meat consumption is 1 case of bowel cancer per 200 people.

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