A recent news item (see here for the BBC report) covered research into the UK breast cancer screening. The problem with the screening is that it is not perfect: it can miss some genuine tumours but it can also 'over-diagnose', i.e. signal a tumour which is actually harmless.
These should be familiar as Type I and II errors. The null hypothesis is the absence of a tumour, so a Type I error is that of incorrectly diagnosing a tumour, when there isn't one. A Type II error is missing a genuine tumour. (One could look at this the other way round, with the null being the presence of a tumour, etc.)
According to the report, for every life saved, three women had unnecessary treatment for cancer. This seems quite a high ratio but partly reflects the fact that the incidence of cancer is actually quite low. The probability of a Type I error is given as 1% in the article. This would be consistent with something like the following: for every thousand women tested, 10 are incorrectly diagnosed and treated, while three are correctly diagnosed and treated (hence approximately three times as many false positives as genuine ones.)
As well as the probabilities, the costs of the errors should also be taken into account. The cost of missing a diagnosis is apparent to us, which is why there is a national system of screening. The costs of over-diagnosis are less obvious but can be substantial. The treatment is unpleasant, to say the least. The costs of over-diagnosis might also be masked because it is concluded that the treatment has worked, rather than that there never was a cancer.
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