Why I don't believe in miracles

I mentioned yesterday that 'heart miracles' (studies that show the heart attack rate plunging by 10-40% after a smoking ban) are mathematically impossible. I would like to explain why.

The whole premise relies on a basic confusion between relative risk and absolute risk. Here is an example of a relative risk:

Drinking coffee whilst driving increases the risk of a fatal car crash by 80%

And here is an absolute risk:

U-boat crew in the Second World War had an 80% chance of dying at sea

The second example could also be phrased as '80% of U-boat crew died at sea during the Second World War'. It is the total risk for the individual, but it also tells us about the death rate amongst the whole group.

The first example is very different. It merely gives us the increased risk to the individual compared to that of the person who never drinks coffee whilst driving. 

A fallacious use of that first statistic would be to say that if motorists were banned from drinking coffee, the number of fatal car crashes would fall by 80%. A second's thought will tell you that that is obvious nonsense, because drinking coffee whilst driving is only one of many factors that increase risk whilst driving (and a fairly minor one, at that).

But it is precisely this fallacious reasoning that is used to justify the implausibly large drops in heart attacks reported in these smoking ban studies. The reasoning is:

Secondhand smoke exposure increases the risk of heart attack by 30%, therefore a smoking ban will reduce the number of heart attacks by 30%.

This could only be true if secondhand smoke was the only risk factor for heart disease and, of course, it isn't. According to the World Health Organisation there are 300 risk factors, including obesity, diabetes, physical inactivity, stress, high cholesterol, high blood pressure and a family history of heart disease - all of which increase risk more than passive smoking is said to.

It is a quite simple logical fallacy and yet it has been used from the very beginning with these heart miracle studies. Lest it be thought that I am putting words into anyone's mouth, here is how the authors of the Helena study (the first heart miracle study) explained a 40% drop in their 2004 paper (British Medical Journal).

[The study reported 40 heart attacks in the 6 months before the smoking ban and 24 after it]

The effect associated with the smoke-free law may seem large but is consistent with the observed effects of secondhand smoke on cardiac disease. 

Secondhand smoke increases the risk of a myocardial infarction by about 30%; if all this effect were to occur immediately, we would expect a fall of - 0.30 x 40.5 = - 12.2 in admissions during the six months the law was in effect, which is within the 95% confidence interval for the estimate of the effect (a drop of - 32.2 to - 0.8 admissions).

I have highlighted the key calculation. They are saying that 30% of heart attacks before the smoking ban were caused by secondhand smoke (12 of the 40 cases). Therefore, the smoking ban would be expected to save 12 lives. Their study actually found 16 fewer cases but, as they say, that is within the margin of error.

This is a fundamental error of logic. The 30% is the relative risk. It tells you nothing about the absolute risk to the population. 

So what is the absolute risk? I am not going to argue about the true effects of secondhand smoke here. For the purposes of this argument, let us go along with the Helena study authors and say that it increases risk by 30%. One of those authors was Stanton Glantz, a prominent and somewhat fanatical anti-smoking activist who has never been accused of underestimating the risks from secondhand smoke. I will use his figures so that I am not accused of cherry-picking. 

The US has around 630,000 deaths from heart disease each year. According to Stanton Glantz, 37,000 of these - or 5.9% - are caused by secondhand smoke. Even taking these figures, then, it should be self-evident that eliminating secondhand smoke could not possibly reduce heart attacks by more than 5.9%.

But smoking bans don't eliminate secondhand smoke. They eliminate it in bars, restaurants and other "public" places, but people continue to smoke in their homes and elsewhere. A smoking ban might perhaps halve secondhand smoke exposure, in which case it might - at the uppermost limit - reduce heart attacks by 3%.

I stress that I am not attempting to come up with my own estimate of how many lives are "saved" by smoking bans. Apply your own figures, by all means. The end result will be the same. It is mathematically impossible for a smoking ban to reduce the heart attack rate by 10-40% because even the most hyperbolic estimates indicate that secondhand smoke causes far fewer than 10% of heart attacks.

There is one other possibility - that smoking bans result in people giving up smoking and therefore suffer fewer heart attacks. On the face of it, this is a more plausible option. However, there is scant evidence that smoking bans actually do increase the quit rate, as Mark Wadsworth has been pointing out. 

It has been estimated that smoking is responsible for 11% of total cardiovascular deaths (PDF). Therefore, even if tobacco was banned and everybody stopped smoking, deaths from heart disease would not - could not - fall by more than 11%. The idea that banning smoking in bars and restaurants can reduce the heart attack rate more effectively than the total abolition of tobacco is plainly ludicrous.