Understanding False Positives with Natural Frequencies

In a graduate course on healthcare economics a professor of mine had us think about drug testing student athletes. We ran through a few scenarios where we calculated how many true positive test results and how many false positive test results we should expect if we oversaw a university program to drug tests student athletes on a regular basis. The results were surprising, and a little confusing and hard to understand.


As it turns out, if you have a large student athlete population and very few of those students actually use any illicit drugs, then your testing program is likely to reveal more false positive tests than true positive tests. The big determining factors are the sensitivity of the test (how often it is actually correct) and the percentage of students using illicit drugs. A false positive occurs when the drug test indicates that a student who is not using illicit drugs is using them. A true positive occurs when the test correctly identifies a student who does indeed use drugs. The dilemma we discussed occurs if you have a test with some percentage of error and a large student athlete population with a minimal percentage of drug users. In this instance you cannot be confident that a positive test result is accurate. You will receive a number of positive tests, but most of the positive tests that you receive are actually false positives.


In class, our teacher walked us through this example verbally before creating some tables that we could use to multiply the percentages ourselves to see that the number of false positives will indeed exceed the number of true positives when you are dealing with a large population and a rare event that you are testing for. Our teacher continued to explain that this happens every day in the medical world with drug tests, cancer screenings, and other tests (including COVID-19 tests as we are learning today).  The challenge, as our professor explained, is that the math is complicated and it is hard to explain to person who just received a positive cancer test that they likely don’t have cancer, even though they just received a positive test. The statistics are hard to understand on their own.


However, Gerd Gigerenzer doesn’t think this is really a limiting problem for us to the extent that my professor had us work through. In Risk Savvy Gigerenzer writes that understanding false positives with natural frequencies is simple and accessible. What took nearly a full graduate course to go through and discuss, Gigerenzer suggests can be digested in simple charts using natural frequencies. Natural frequencies are numbers we can actually understand and multiply as opposed to fractions and percentages which are easy to mix up and hard to multiply and compare.


Rather than telling someone that the actual incidence of cancer in the population is only 1%, and that the chance of a false positive test is 9%, and trying to convince them that they still likely don’t have cancer is confusing. However, if you explain to an individual that for every 1,000 people who take a particular cancer test that only 10 actually have cancer and that 990 don’t, the path to comprehension begins to clear up. With the group of 10 true positives and true negatives 990, you can explain that of those 10 who do have cancer, the test correctly identifies 9 out of 10 of them, and provides 9 true positive results for every 1,000 test (or adjust according to the population and test sensitivity). The false positive number can then be explained by saying that for the 990 people who really don’t have cancer, the test will error and tell 89 of them (9% in this case) that they do have cancer. So, we see that 89 individuals will receive false positives while 9 people will receive true positives. 89 > 9, so the chance of actually having cancer with a positive test still isn’t a guarantee.


Gigernezer uses very helpful charts in his book to show us that the false positive problem can be understood more easily than we might think. Humans are not great at thinking statistically, but understanding false positives with natural frequencies is a way to get to better comprehension. With this background he writes, “For many years psychologists have argued that because of their limited cognitive capacities people are doomed to misunderstand problems like the probability of a disease given a positive test. This failure is taken as justification for paternalistic policymaking.” Gigerenzer shows that we don’t need to rely on the paternalistic nudges that Cass Sunstein and Richard Thaler encourage in their book Nudge. He suggest that in many instances where people have to make complex decisions what is really needed is better tools and aids to help with comprehension. Rather than developing paternalistic policies to nudge people toward certain behaviors that they don’t fully understand, Gigerenzer suggests that more work to help people understand problems will solve the dilemma of poor decision-making. The problem isn’t always that humans are incapable of understanding complexity and choosing the right option, the problem is often that we don’t present information in a clear and understandable way to begin with.

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