Read risk figures correctly
"Doubles your risk" can mean almost nothing, and a positive test can still mean you probably don't have the disease. Turn percentages into counts of people to see what a risk figure really says.
PsychologyThink clearlyAbout 5 min: 3 reading, 2 practiceUpdated
Educational, not medical or mental-health advice.
- Tell relative risk from absolute risk, and ask for the absolute figure
- Use the base rate when judging a test result
- Turn percentages into natural frequencies (counts out of 1,000) to work out what a result means
Builds toward: Think clearly. Catch common thinking errors in yourself and in other people's arguments, and make better calls when you can't be sure.
Think clearly: thinking errors and better decisions: lesson 4 of 7
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Pairs with Gigerenzer and colleagues (2007), "Helping doctors and patients make sense of health statistics"; Gigerenzer and Hoffrage (1995)
What it is
Two ideas fix most mistakes with risk figures.
Relative and absolute risk. Relative risk says how much a risk changes ("doubles", "cuts by half"). Absolute risk says how big it is to start with and after ("from 1 in 10,000 to 2 in 10,000"). "Doubles your risk" sounds alarming, but doubling a tiny risk still leaves a tiny risk. Always ask: out of how many?
Quick check
Tap an answer to check it.
A headline says a habit "doubles your risk" of a rare illness, from 1 in 50,000 to 2 in 50,000. How worried should you be?
- Doubling a tiny risk still leaves it tiny
Show answer
Not very: the risk rises by 1 in 50,000. The absolute change, 1 extra case in 50,000 people, is what tells you how much it matters.
Why it matters
A real scare. In 1995 British women were warned that some newer contraceptive pills doubled the risk of a dangerous blood clot. In absolute terms, the risk went from about 1 in 7,000 to about 2 in 7,000. Many women stopped taking the pill, and the following year there were an estimated 13,000 more abortions in England and Wales (Gigerenzer and colleagues, 2007). The relative figure was correct, and badly misleading on its own.
Doctors find it hard too. Gigerenzer and colleagues (2007) gave about 160 gynaecologists this problem: a screening test for a disease that 1% of women have; the test finds 90% of those who have it, and wrongly flags 9% of those who don't. A woman tests positive. How likely is it she has the disease? Only about one in five chose the right answer: about 1 in 10. Most chose 8 or 9 in 10.
Worked example
Work out that screening result by turning percentages into people (natural frequencies). Imagine 1,000 women.
| Have the disease | Don't have it | |
|---|---|---|
| Out of 1,000 | 10 (1%) | 990 |
| Test positive | 9 (90% of 10) | about 89 (9% of 990) |
- 10 of the 1,000 have the disease; the test finds 9 of them.
- 990 don't have it; the test wrongly flags 9% of them: about 89.
- Altogether 98 women test positive. Only 9 of them have the disease.
- 9 ÷ 98 ≈ 9%, about 1 in 11.
Because the disease is rare, the false alarms from the large healthy group outnumber the true cases. That is why a positive screening result usually leads to a second, more accurate test, not straight to a diagnosis. Gigerenzer and Hoffrage (1995) found that people solve problems like this far more often when the numbers are given as counts like these rather than as percentages.
Common mistake
Confusing "the test finds 90% of people who have the disease" with "90% of people who test positive have the disease". They sound alike but are different questions. The first is about the test; the second, the one you care about, depends heavily on the base rate.
Key terms
- relative risk
- how much a risk changes, as a ratio ("doubles", "halves")
- absolute risk
- how big a risk is, as a count out of a group ("2 in 10,000")
- base rate
- how common something is in the group to begin with
- base-rate neglect
- judging likelihood from the evidence alone and ignoring how common the thing is
- natural frequencies
- counts of people out of a round number, used instead of percentages
- false positive
- a test result that says yes when the true answer is no
Practise the key terms
Tap a card to turn it over. 6 cards.
| Front | Back |
|---|---|
| relative risk | how much a risk changes, as a ratio ("doubles", "halves") |
| absolute risk | how big a risk is, as a count out of a group ("2 in 10,000") |
| base rate | how common something is in the group to begin with |
| base-rate neglect | judging likelihood from the evidence alone and ignoring how common the thing is |
| natural frequencies | counts of people out of a round number, used instead of percentages |
| false positive | a test result that says yes when the true answer is no |
Practice
Work it through
A condition affects 2% of people. A test finds 95% of people who have it and wrongly flags 10% of people who don't. Imagine 1,000 people.
- 2% of 1,000
- 95% of 20
- 10% of 980
- 19 ÷ (19 + 98)
Quick quiz
Tap an answer to check it.
A medicine "cuts your risk of a stroke by 50%". What do you most need to know?
- A 50% cut of a tiny risk is a tiny change
Show answer
The risk with and without the medicine, as counts out of a group. Halving 2 in 1,000 to 1 in 1,000 saves 1 person in 1,000; halving 20 in 100 to 10 in 100 saves 10 in 100.
In the 1995 pill scare, what was the absolute change in risk?
- That would be a huge risk; the real one was small
Show answer
From about 1 in 7,000 to about 2 in 7,000. "Doubled" was true, but the absolute increase was about 1 woman in 7,000.
Why can a positive result on a good test still mean you probably don't have a rare disease?
- The test can be quite accurate; the problem is the low base rate
Show answer
The healthy group is so large that its false positives outnumber the true cases. With a rare condition, even a small false-positive rate applies to many more people.
"The test detects 90% of people with the disease." What does that tell you directly?
- That also depends on the base rate
Show answer
How often the test finds the disease in people who have it. The chance after a positive result needs the base rate and the false-positive rate too.
In short
- Relative risk ("doubles") says how much a risk changes; absolute risk ("1 in 7,000 to 2 in 7,000") says how big it is. Always ask: out of how many?
- The base rate, how common something is to begin with, matters as much as the evidence.
- With a rare condition, most positive screening results can be false alarms.
- Turn percentages into counts out of 1,000 (natural frequencies) to work it out.
- "The test finds 90% of cases" is not "90% of positives are cases".
Mastery check
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Check yourself
Answer in your head first, then open the answer. Nothing is saved.
What is the difference between relative and absolute risk?
Show answer
Relative risk says how much a risk changes (for example "doubles"); absolute risk says how big it is, as a count out of a group (for example "from 1 in 10,000 to 2 in 10,000").
In the screening example, about how many of the 98 women who tested positive had the disease?
Show answer
9, so about 1 in 11.
A friend tests positive on a screening test for a rare condition and is terrified. What would you say?
Show answer
That a positive screening result for a rare condition is often a false alarm, and to ask the doctor how likely a positive result is to be correct and what the follow-up test is.
Further reading
- Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8(2), 53–96.
- Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704.
- Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.
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