The example given below of false positives and false negatives appeared originally in a 'reader' for senior students of psychology and of philosophy at the University of Malta. I later submitted the story to the Malta Skeptic's magazine under the pseudonym Bias.

Ignorans generalis

by Bias        

There is a human affliction called ignorans generalis, hereafter abbreviated to I.G. You're sure to have come across people suffering from it. It is usually not fatal; so much for evolution. On the contrary, some of those afflicted are known to have reached high positions. In spite of it or owing to it? It may well be a contributory circumstance that the people suffering from it don't suffer from it. (How's that again?).

By the way, if caught at an early stage it can be cured.

Now we know that the occurrence of I.G. is very rare, especially in Malta. In fact, DNA analysis has shown that it is limited to 1 in a thousand of the population as a whole. But DNA analysis is expensive and laborious.
Therefore, psychologists - never daunted - have developed a measuring instrument called the "Discover UNderdeveloped CErebrum" test (DUNCE for short). It can pinpoint I.G. at an early stage with a reliability of 90%, which is - for a psychological test - fantastically accurate.
Leaving moral considerations aside, should you submit yourself - or those dear to you - voluntarily to the DUNCE test for I.G. to improve chances in life? More in particular: how sure are you that a 'negative' test-result identifies you or yours correctly as at risk?

At first sight you may be inclined to answer: 90%. After all, there is a chance of 10% only that the test gives an incorrect answer, isn't it? So, by all means, take the test? Better first consider the following:
a reliability of 90% means that 10 % of those tested are diagnosed as sufferers; and let's further assume that the test at least does not let actual sufferers slip through the net (psychologically again an incredible feat).

For Malta this means that according to the test 40.000 people are suspected of suffering from I.G. But of these 40.000 unhappy Maltese only 400 (= 1 in a thousand of 400.000, remember?) are actual sufferers! A 'negative' test result then is in 99 out of 100 cases erroneous. In other words, after some further consideration you cannot be 90% sure but only 99% unsure . . .

And if you can't follow this reasoning, be on the alert for I.G.

Funny? Far-fetched? Theoretical drivel?

Read for ignorans generalis: breast-cancer. Read for DUNCE: mammography. The latter is more accurate than the 90% in the case of DUNCE, agreed. But it suffers from a more serious flaw: it does not pinpoint all cases of breast-cancer! Still funny? But for a long time the position in that medical field has been as described for I.G. Initially few doctors turned out to be aware of this pitfall. They are now, and testing for breast-cancer with mammography is a subject for controversy. Those who get a clean bill of health cannot be sure that there is nothing to worry about (a so-called false positive). And of those who get an invitation for a further test, many receive this message - and its accompanying trauma - unjustly (false negative).

In the past these considerations have in some cases led to a suspension of testing altogether. It was felt that early diagnosis of breast-cancer did not outweigh the negative aspect of frightening women unnecessarily. A very unsatisfactory state of affairs. Fortunately enough the quality of screening is steadily improving, though it is still far from perfect.

This is in other words not an appeal not to take part in tests if offered. On the contrary. And the customary follow-up of suspected cases further minimizes risks.

A word of warning is in order here however. The usual interval between screenings is a compromise between the discomfort of mammography plus the negative aspects of X-rays, and the chance of cancer developing during this period. Recent research has shown that exceeding the two-year period is unacceptably risky: the incidence of untreatable cases rises sharply.

In other words: once you start taking part in a population screening, insist on an interval between tests of a maximum of 2 years. Otherwise you exchange one sense of false security for another.