# Probabilities and false positives

In response to my post on random drug testing in Florida, commenter Scott mentioned the danger of false positives and said that a second test in his case cleared him of having taken drugs. My advice to anyone is that whenever you take a high-stakes test for anything that has a small incidence in the general population (drugs, diseases, whatever) and it comes out positive, always consider asking for a second test.

Here’s why.

Suppose the percentage of drug users in the general population is (say) about 1 in 100. You are told that there is a test that is pretty good in that it has a ‘false positive’ rate of only 5%, meaning that if a randomly selected group is tested, only 5% of the people who do not use drugs will have test results that come out positive. Also you are told that the false negative rate is negligible, meaning that if someone does take drugs, the test will almost certainly not come out negative.

Suppose someone takes part in this random testing and the result is positive. What do you think are the chances that the person has taken drugs? Most people would think that it is very high. They may put it as high as 95%, thinking that if there is a 5% false positive rate and 0% false negative rate, that means that the likelihood of someone testing positive actually having taken drugs is 95%. This sounds eminently reasonable but the actual chance is just 1 in 6 or less than 17%!

How come? This becomes easier to understand if we shift from talking in terms of probabilities (which are not intuitive) to talking about numbers. Suppose you are one of 1000 people being randomly tested. (Any size will do. I have chosen 1000 because it is a nice round number.) Then an incidence of drug use of 1 in 100 means that we expect 10 people to actually have taken drugs and thus test positive, and 990 to be drug-free.

But a 5% false positive rate will result in about 50 of the 990 people who do not take drugs also testing positive. So a total of 60 people will test positive, of whom only 10 will actually have taken drugs. So even with a false positive rate of only 5%, the chances are 5 out of 6 or 83% that a person who tests positive is actually drug-free.

What the positive test result has done is increase in the odds of the person having taken drugs from 1 in 100 (or 1%) to 1 in 6 (or slightly less than 17%). But there is still an 83% chance that the person has not taken drugs.

So always consider asking for data about the rates of incidence of drug use (or disease) and the rate of false positives to do your own calculations and consider a repeat test before making any major decision based on such tests.

1. Jeremy says

I love Bayesian analysis. Except in class…

2. Matthew says

I couldn’t agree more. I work in a research setting examining, among many other things, neuropsychological performance of those living with HIV. So we of course run toxicology screens of urine to determine if they are intoxicated or have been recently (which could obviously affect their performance on tests of memory, executive functions, motor skills, etc…). If we get a positive result with a participant who denies recent use, we send the sample out for confirmation.

The number of false positives found with a second screen isn’t alarmingly high, but it is definitely high enough for me to be concerned if I were in a position where I had a job offer, or a child custody hearing or something of that magnitude on the line, where the stakes of having a test come back falsely positive are extremely high and can have very lasting consequences or implications if you do not/cannot receive a confirmation test.

3. rukymoss says

If they are charging only \$30 for a drug test, it is probably the cheap, immune-reaction based tests. We use these a lot when screening for drug use in newly admitted psychiatric patients, so a drug reaction doesn’t get mistaken for psychosis, for example. Unfortunately, these tests can show false positives due to cross-reactions with common legal drugs. I have had patients show positive for phencyclidine (actually taking Zoloft); opioids (actually taking a rarely used totally nonopioid arthritis med); amphetamine (actually taking pseudoephedrine). Also, these tests do not reliably show many drugs that are commonly abused, such as clonazepam. Confirmatory tests, usually gas chromatography or mass spectrometry, are expensive and have a several-day turnaround time. So, there is lots of room for false positives, and who has money to pay for an expensive test to clear themselves of something that was a violation of their constitutional rights in the first place? Another reason to steer clear of Florida.

4. And using the same numbers, if all 60 people are retested, you’ll end up with the 10 actual drug users, but also 2 or 3 people with a second false positive. So we’ve gone from a 1 in 6 chance of being a drug user to about 5 in 6, which is much higher, but still not 100%.