{"id":1145,"date":"2023-12-12T15:04:38","date_gmt":"2023-12-12T15:04:38","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=1145"},"modified":"2023-12-12T15:04:38","modified_gmt":"2023-12-12T15:04:38","slug":"smoking-causes-cancer-2","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2023\/12\/12\/smoking-causes-cancer-2\/","title":{"rendered":"Smoking Causes Cancer"},"content":{"rendered":"\n

In the preface of Probably Overthinking It<\/em>, I wrote:<\/p>\n\n\n\n

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Sometimes interpreting data is easy. For example, one of the reasons we know that smoking causes lung cancer is that when only 20% of the population smoked, 80% of people with lung cancer were smokers. If you are a doctor who treats patients with lung cancer, it does not take long to notice numbers like that.<\/p>\n<\/blockquote>\n\n\n\n

When I re-read that paragraph recently, it occurred to me that interpreting those number might not be as easy as I thought. To find out, I ran a Twitter poll<\/a>. Here are the results:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Some of the people who chose “other” said that there is not enough information — we need to know the absolute risk for one or both of the groups.<\/p>\n\n\n\n

I think that’s not right — with just these two numbers, we can compute the relative risk of the two groups. There are a few ways to do it, but a good way to get started is to check each of the multiple choice responses.<\/p>\n\n\n\n

Off the bat, “60 percentage points” is just wrong. If the lifetime risk of cancer was 20% in one group and 80% in the other, we could describe that difference in terms of percentage points. But those are not the numbers we were given, and the actual risks are much lower.<\/p>\n\n\n\n

But “a factor of 4” is at least plausible, so let’s check it. Suppose that the actual lifetime risk of lung cancer for non-smokers is 1% — in that case the risk for smokers would be 4%. In a group of 1000 people, we would expect 800 non-smokers and 8 cases among them, and we would expect 200 smokers and 8 cases among them. Under these assumptions 50% of people with lung cancer would be smokers, but the question says it should be 80%, so this check fails.<\/p>\n\n\n\n

Let’s try again with “a factor of 16”. If the risk for non-smokers is 1%, the risk for smokers would be 16%. Among 800 non-smokers, we expect 8 cases again, but among 200 smokers, now we expect 32. Under these assumptions, 80% of people with lung cancer are smokers, so 16 is the correct answer.<\/p>\n\n\n\n

Here are the same numbers in a table.<\/p>\n\n\n\n

<\/td>Number<\/td>Risk<\/td>Cases<\/td>Percent<\/td><\/tr>
Non-smoker<\/td>800<\/td>1%<\/td>8<\/td>20%<\/td><\/tr>
Smoker<\/td>200<\/td>16%<\/td>32<\/td>80%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

<\/p>\n\n\n\n

Now, you might object that I chose 1% and 16% arbitrarily, but as it turns out it doesn’t matter. To see why, let’s assume that the risk is x for non-smokers and 16x for smokers. Here’s the table with these unknown risks.<\/p>\n\n\n\n

<\/td>Number<\/td>Risk<\/td>Cases<\/td>Percent<\/td><\/tr>
Non-smoker<\/td>800<\/td>x<\/td>800x<\/td>20%<\/td><\/tr>
Smoker<\/td>200<\/td>16x<\/td>3200x<\/td>80%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

<\/p>\n\n\n\n

The percentage of cases among smokers is 80%, regardless of x.<\/p>\n\n\n\n

Now suppose you are not satisfied with this guess-and-check method. We can solve the problem more generally using Bayes’s rule. <\/p>\n\n\n\n