Can selection bias shoot down an argument?

Manhattan in Motion

Urban policy is rich in opportunities for fallacious thinking – for example, surveys that purport to show huge latent demand for a particular mode of transport, but sample only users of that mode. So I’m always interested in new examples of where we can so easily go wrong.

Alex Tabarrok at Marginal Revolution recently provided a pointer to this great historical example by John D. Cook of the importance of selection bias. It isn’t specifically related to urban policy, but nevertheless provides a valuable lesson. Cook says:

During WWII, statistician Abraham Wald was asked to help the British decide where to add armour to their bombers. After analysing the records, he recommended adding more armour to the places where there was no damage!

According to Cook, while it seems backward at first, Wald realised his data came from bombers that hadn’t been shot down:

That is, the British were only able to analyse the bombers that returned to England; those that were shot down over enemy territory were not part of their sample. These (surviving) bombers’ wounds showed where they could afford to be hit. Said another way, the undamaged areas on the survivors showed where the lost planes must have been hit because the planes hit in those areas did not return from their missions.

Wald assumed that the bullets were fired randomly, that no one could accurately aim for a particular part of the bomber. Instead they aimed in the general direction of the plane and sometimes got lucky. So, for example, if Wald saw that more bombers in his sample had bullet holes in the middle of the wings, he did not conclude that Nazis liked to aim for the middle of wings. He assumed that there must have been about as many bombers with bullet holes in every other part of the plane but that those with holes elsewhere were not part of his sample because they had been shot down.

Here’s another example via Marc Gawley, who noted a BBC story implying that three quarters of those committing crimes during the London riots had previous convictions or cautions. Is that really true, he wonders, or:

is it that those with previous convictions have their details on a police database and it was therefore possible to identify (and find) those people based on images from photographs and recordings made last month?

And another example from Nassim Nicholas Taleb who recounts this story told by Francis Bacon on the effectiveness of prayer:

Bacon mentioned a man who, upon being shown the pictures of those worshippers who paid their vows then subsequently escaped shipwreck, wondered where were the pictures of those who happened to drown after their vows. The lack of effectiveness of their prayers did not seem to be taken into account by the supporters of the handy rewards of religious practice. “And such is the way of all superstition, whether in astrology, dreams, omens, divine judgments, or the like”, he wrote in his Novum Organum. That was written in 1620. This is a potent insight: the drowned worshippers, being dead, do not advertise their experiences. They are invisible and will be missed by the casual observer who will be led to believe in miracles.

4 Comments on “Can selection bias shoot down an argument?”

  1. Urt says:

    Likewise, Undergraduates want to do PhDs because everyone they meet at university who has a PhD also has a prestigious university teaching job! They don’t meet all the PhDs doing graduate programs in the Victorian Public Service alongside the BA grads until later, when for some, it’s too late.
    Betting on horses always sounds so profitable because people only talk about the nags that won.
    And people write novels because the unpublished ones are invisible.

  2. john says:

    Nice post Alan. All your posts are so good. The ones I read, anyway.

  3. David Walker says:

    +1 to the comment above.

  4. […] looked at interesting statistical “stories” before –e.g. see Can selection bias shoot down an argument. -37.781700 145.039432 Share […]

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