Overcoming difficulties in Bayesian reasoning: A reply to Lewis and Keren (1999) and Mellers and McGraw (1999).

Abstract

Bayesian reasoning can be improved by representing information in formats rather than in probabilities. This thesis opens up applications in medicine, law, statistics education, and other fields. The beneficial effect is no longer in dispute, but rather its cause and its boundary conditions. C. Lewis and G. Keren (1999) argued that the effect of formats is due to rather than to frequency However, they overlooked the fact that our thesis is about formats, not just any kind of statements. We show that joint statements alone cannot account for the effect. B. A. Mellers and A. P. McGraw (1999) proposed a boundary condition under which the beneficial effect is reduced. In a reanalysis of our original data, we found this reduction for the problem they used but not for any other problem. We conclude by summarizing results indicating that teaching representations fosters insight into Bayesian reasoning. Degrees of uncertainty can be represented in various ways, including probability and formats. Let us first illustrate a format and how it improves Bayesian reasoning in medical experts. We asked a sample of 48 physicians with an average of 14 years of professional experience, including private practitioners, university professors, and clinic directors (Hoffrage & Gigerenzer, 1998), to make inferences about the presence of a disease given a positive result for four routinely used medical diagnostic tests. One was mammography. The relevant information (concerning a population of women aged 40 years) was presented to half of the physicians in a probability format, which can be summarized as follows: The probability of breast cancer is 1%; the probability of a positive test given breast cancer is 80%; and the probability of a positive test given no breast cancer is 10%. The question was What is the probability that a woman who tests positive actually has breast cancer? The other half of the physicians in the study received the same information in a format: 10 of every 1,000 women have breast cancer; 8 of those 10 women with breast cancer will test positive; and 99 of the 990 women without breast cancer will also test positive. The question was How many of those who test positive actually have breast cancer? When the information concerning mammography and breast cancer was presented in a probability format, only 8% of the physicians gave an estimate close to that yielded by Bayes's rule (i.e., .075). When the information was presented in a format, in contrast, 46% of them arrived at the Bayesian response. This beneficial effect of the format on physicians' judgments was obtained in each of the four diagnostic tasks. Thus, formats help to improve Bayesian reasoning not only in