Calculated Risks: How to know when numbers deceive you: Gerd Gigerenzer

Gerd Gigerenzer, Simon and Schuster, New York, 310 pp., ISBN 0-7432-0556-1, $25.00

Should healthy women get regular mammograms to screen for breast cancer?

The surprising answer, according to this wonderful new book by psychology professor Gerd Gigerenzer, is, at least for most women, probably not.

Deciding whether to have a mammogram or other medical screening (the book examines several) requires people to calculate the risk that is inherent is taking these tests.? This risk is usually poorly known or communicated and, because of this, people can make the wrong decisions and suffer unnecessarily.

What risk, you might ask, is there for an asymptommatic woman in having a mammogram? To answer that, look at what could happen.

The mammogram could correctly indicate no cancer, in which case the woman goes away happy. It could also correctly indicate true cancer, in which case the woman goes away sad and must consider treatment.

Are these all the possibilities? Not quite. The test could also indicate that no cancer is present when it is really there—the test could miss the cancer. This gives false hope and causes a delay in treatment.

But also scary and far more likely is that the test could indicate that cancer is present when it is not. This outcome is called a false positive, and it is Gigerenzer’s contention that the presence of these false positives are ignored or minimized by both the medical profession and by interest groups whose existence is predicated on advocating frequent mammograms (or other disease screenings, such as for prostate cancer or AIDS).

Doctors like to provide an “illusion of certainty” when, in fact, there is always uncertainty in any test. Doctors and test advocates seem to be unaware of this uncertainty, they have different goals than do the patients who will receive the tests, and they ignore the costs of false positives.

How is the uncertainty of a test calculated? Here is the standard example, given in every introductory statistics book, that does the job. This example, using numbers from Gigerenzer, might look confusing, but read through it because its complexity is central to understanding the his thesis.

If the base rate probability of breast cancer is 0.8% (the rate of cancer in women in the entire country), and the sensitivity (ability to diagnose the cancer when it is truly there) and specificity (ability to diagnose no cancer when it is truly not there) of the examination for cancer is 90% and 93%, then given that someone tests positive for cancer, what is the true probability that this person actually has cancer?

To answer the question requires a tool called Bayes Rule. Gigerenzer has shown here, and in other research, that this tool is unnatural and difficult to use and that people consistently poorly estimate the answer. Can you guess what the answer is?

Most people incorrectly guess 90% or higher, but the correct answer is only 9%, that is, only 1 woman out of every 11 who tests positive for breast cancer actually has the disease, while the remaining 10 do not.

If people instead get the same question with the background information in the form of frequencies instead of probabilities they do much better. The same example with frequencies is this: If out of every 1000 women 77 have breast cancer, and that 7 of these 77 who test positive actually have the disease, then given that someone tests positive for cancer what is the true probability that this person actually has cancer?

The answer now jumps out—7 out of 77—and is even obvious, which is Gigerenzer’s point. Providing diagnostic information in the form of frequencies benefits both patient and doctor because both will have a better understanding of the true risk.

What are the costs of false positives? For breast cancer, there are several. Emotional turmoil is the most obvious: testing positive for a dread disease can be debilitating and the increased stress can influence the health of the patient negatively. There is also the pain of undergoing unnecessary treatment, such as mastectomies and lumpectomies. Obviously, there is also a monetary cost.

Mammograms can show a noninvasive cancer called ductal carcinoma in situ, which is predominately nonfatal and needs no treatment, but is initially seen as a guess of cancer. There is also evidence that the radiation from the mammogram increases the risk of true breast cancer!

These costs are typically ignored and doctors and advocates usually do not acknowledge the fact the false positives are possible. Doctors suggest many tests to be on the safe side—but what is the safe side for them is not necessarily the safe side for you. Better for the doctor to have asked for a test and found nothing than to have not asked for the test and miss a tumor, thus risking malpractice.

This asymmetry shows that the goals of patients and doctors are not the same. The same is true for advocacy groups. Gigerenzer studies brochures from these (breast cancer awareness) groups in Germany and the U.S. and found that most do not mention the possibility of a false positive, nor the costs associated with one.

Ignoring the negative costs of testing makes it easier to frighten women into having mammograms, and he stresses that, “exaggerated fears of breast cancer may serve certain interest groups, but not the interests of women.”

Mammograms are only one topic explored in this book. Others include prostate screenings “where there is no evidence that screening reduces mortality”, AIDS counseling, wife battering, and DNA fingerprinting.

Studies of AIDS advocacy group’s brochures revealed the same as in the breast cancer case: the possibility of false positives for screenings and the costs associated with these mistakes were ignored or minimized.

Gigerenzer even shows how attorney Alan Dershowitz made fundamental mistakes calculating the probable guilt of O.J. Simpson, mistakes that would have been obvious had Dershowitz used frequencies instead of probabilities.

The book closes with tongue-in-cheek examples of how to cheat people by exploiting their probabilistic innumeracy, and includes several fun problems.

Gigerenzer stresses that students have a high motivation to learn statistics but that it is typically poorly taught. He shows that people’s difficulties with numbers can be overcome and that it is in our best interest to become numerate.

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  1. Joe Jensen

    You write:
    “If out of every 1000 women 77 have breast cancer, and that 7 of these 77 who test positive actually have the disease”.

    Isn’t that supposed to be something like:
    “If out of every 1000 women 77 are diagnosed with breast cancer, and that 7 of these 77 who test positive actually have the disease”.

    Or am I mistaken?

    Thank you for taking the time to spread an important message!

  2. Briggs


    I think it’s right here. That 77 out of 1000 represents the base rate, the approximate number of women out of 1000 who really do have cancer.

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