The chance of a test detecting a disease is typically communicated in the form of a conditional probability, the sensitivity of the test: “If a woman has breast cancer the probability that she will have a positive result on mammography is 90%.” This statement is often confused with: “If a woman has a positive result on mammography the probability that she has breast cancer is 90%.” That is, the conditional probability of A given B is confused with that of B given A.4
Many doctors have trouble distinguishing between the sensitivity, the specificity, and the positive predictive value of test—three conditional probabilities. Again, the solution lies in the representation.
Consider the question “What is the probability that a woman with a positive mammography result actually has breast cancer?” The box shows two ways to represent the relevant statistical information: in terms of conditional probabilities and natural frequencies. The information is the same (apart from rounding), but with natural frequencies the answer is much easier to work out. Only seven of the 77 women who test positive actually have breast cancer, which is one in 11 (9%). Natural frequencies correspond to the way humans have encountered statistical information during most of their history. They are called “natural” because, unlike conditional probabilities or relative frequencies, they all refer to the same class of observations.5
For instance, the natural frequencies “seven women” (with a positive mammogram and cancer) and “70 women” (with a positive mammogram and no breast cancer) both refer to the same class of 1000 women. In contrast, the conditional probability 90% (the sensitivity) refers to the class of eight women with breast cancer, but the conditional probability 7% (the specificity) refers to a different class of 992 women without breast cancer. This switch of reference class can confuse the minds of doctors and patients alike.
shows the responses of 48 doctors, whose average professional experience was 14 years, to the information given in the box, except that the statistics were a base rate of cancer of 1%, a sensitivity of 80%, and a false positive rate of 10%.1,2
Half the doctors received the information in conditional probabilities and half in natural frequencies. When asked to estimate the probability that a woman with a positive result actually had breast cancer, doctors who received conditional probabilities gave answers that ranged from 1% to 90%, and very few gave the correct answer of about 8%. In contrast most doctors who were given natural frequencies gave the correct answer or were close to it. Simply stating the information in natural frequencies turned much of the doctors' innumeracy into insight, helping them understand the implications of a positive result as it would arise in practice. Presenting information in natural frequencies is a simple and effective mind tool to reduce the confusion resulting from conditional probabilities.6
This is not the end of the story regarding the communication of risk (which requires adequate exploration of the implications of the risk for the patient concerned, as described elsewhere in this issue7
), but it is an essential foundation.
Fig 1 Doctors' estimates of the probability of breast cancer in women with a positive result on mammography, according to whether the doctors were given the statistical information as conditional probabilities or natural frequencies (each point represents one (more ...)