Letters

Risk measures expressed as frequencies may have a more rational response

A treatment that reduces deaths by 50% sounds better than one that reduces deaths by 5%, yet the latter treatment might be more valuable than the former: reducing a tiny risk by 50% might be trivial relative to reducing a large risk by 5%. The number needed to treat reflects the incidence and is more relevant for medical decision making.

Studies of medical decision making support use of this measure. Several studies show that experts --including medical clinicians--have great difficulty reasoning with probabilities.² For instance, Eddy asked 100 physicians questions of the following type. The prevalence of breast cancer is 1% (in a specified population). The probability that the result of mammography is positive if a woman has breast cancer is 79% and 9.6% if she does not. What is the probability that a woman with a positive result actually has breast cancer?³

Eddy reports that 95 physicians estimated the probability P (cancer and positive result) to be about 75%; the correct probability is only about 8%. Dawes reports a surgeon in the United States performing preventive mastectomies on the basis of this faulty logic.⁴

Nevertheless, the same problems presented by use of frequencies rather than probabilities are solved relatively easily. Gigerenzer reviewed several studies that show a dramatic improvement in reasoning with probabilities if they are converted into frequencies. For instance, we can change the example above as follows. Imagine 100 people (think of a 10x10 grid). We expect that one woman has cancer and a positive mammogram. Also we expect that there are 10 more women with positive mammograms but no cancer. Thus we expect 11 people with positive mammograms. How many women with positive mammograms will actually have cancer?

With frequencies you immediately "see" that only about one out of 11 women with a positive result will have cancer. Although staff of Harvard Medical School have difficulties with the probability version--most give wrong answers⁵--most undergraduates readily provide the correct answer to similar problems constructed with frequencies.⁵

Psychological research suggests that measures of risk communicated in terms of frequencies rather than probabilities will be more readily understood and rationally responded to.

Lecturer in psychology Department of Psychology, School of Social Sciences, City University, London EC1V 0HB

Peter Ayton 

1. Cook RJ, Sackett DL. The number needed to treat: a clinically useful measure of treatment effect. BMJ 1995;310:452-4. (18 February.) [Free Full Text]
2. Ayton P. On the competence and incompetence of experts. In: Wright G, Bolger F, eds. Expertise and decision support. London: Plenum, 1992:77-105.
3. Eddy DM. Probabilistic reasoning in clinical medicine: problems and opportunities. In: Kahneman D, Slovic P, Tversky A, eds. Judgement under uncertainty: heuristics and biases. Cambridge: Cambridge University Press, 1982:249-67.
4. Dawes RM. Rational choice in an uncertain world. Cambridge: Cambridge University Press, 1988.
5. Gigerenzer G. Why the distinction between single event probabilities and frequencies is important for psychology (and vice-versa). In: Wright G, Ayton P, eds. Subjective probability. Chichester: Wiley, 1994:129-61.