Intended for healthcare professionals


Risk measures expressed as frequencies may have a more rational response

BMJ 1995; 310 doi: (Published 13 May 1995) Cite this as: BMJ 1995;310:1269
  1. Peter Ayton
  1. Lecturer in psychology Department of Psychology, School of Social Sciences, City University, London EC1V 0HB

    EDITOR,—Richard J Cook and David L Sackett propose that the number needed to treat (to avoid an adverse event) should be used as a measure of the efficacy of treatment in reducing medical risks.1 This measure has advantages over probability measures (for example, reduction in relative risk) as it conveys both statistical and clinical significance.

    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.


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