Statistics From Zero · Lesson 29 of 31
Odds and Odds Ratios: Why "Nearly Triples Your Chances" Is Often Wrong
In one sentence: Odds compare the chance something happens with the chance it doesn't; an odds ratio is close to the risk ratio when the outcome is rare, but exaggerates it a lot when the outcome is common.
The idea
There are two ways to describe how likely something is. Probability compares it with all the possibilities: 1 chance in 4. Odds compare it with the chance it doesn't happen: 1 to 3. They carry the same information, but they're different numbers, and mixing them up produces some of the most common exaggerations in health and science news.
Uncle Gary at the bookmaker
Uncle Gary has put $10 on a horse called Linda's Revenge at "3 to 1". "Three to one," he explains to Aunt Linda, "means it wins three times out of four."
Linda, who checked, says it means the opposite.
Linda is right. "3 to 1 against" means for every 1 way the horse wins there are 3 ways it loses: 1 win in 4 races, a probability of 25%. The conversions are simple:
- From probability to odds: odds = p ÷ (1 - p). A 25% chance is 0.25 ÷ 0.75 = 1 to 3, or odds of 0.33.
- From odds to probability: p = odds ÷ (1 + odds). Odds of 4 (4 to 1 on) mean 4 ÷ 5 = 80%.
For rare events the two are almost the same: a 2% chance is odds of 0.0204. For common events they drift far apart: an 80% chance is odds of 4.
Comparing two groups: risk ratio and odds ratio
To compare two groups, you can divide their probabilities, giving a risk ratio (relative risk, met in Lesson 10), or divide their odds, giving an odds ratio. Here's a flu jab, with 1,000 people in each group:
| Got flu | Risk | Odds | |
|---|---|---|---|
| Vaccinated | 20 of 1,000 | 2% | 20 to 980 = 0.0204 |
| Not vaccinated | 50 of 1,000 | 5% | 50 to 950 = 0.0526 |
| Ratio | 0.40 | 0.39 |
Flu is rare in both groups, so the two ratios agree: the jab cuts the risk to about 40% of what it was. Now a common outcome. Cousin Kevin's college says students with a tutor pass the exam more often:
| Passed | Risk (chance) | Odds | |
|---|---|---|---|
| With a tutor | 80 of 100 | 80% | 80 to 20 = 4.0 |
| Without | 60 of 100 | 60% | 60 to 40 = 1.5 |
| Ratio | 1.33 | 2.67 |
The chance of passing is a third higher with a tutor. But the odds ratio is 2.67, and a headline writer will happily turn that into "a tutor nearly triples your chances of passing". It doesn't. It nearly triples your odds, which is a much less impressive thing when the chance was already 60%.
Why use odds ratios at all?
- Some studies can only produce them. In a case-control study, researchers pick people who have a disease and people who don't, then look back at their habits. Because they chose how many of each to include, they can't compute risks, but the odds ratio still works.
- Logistic regression, the standard way to model yes-or-no outcomes with several factors (the cousin of Lesson 27), naturally reports odds ratios.
- They're symmetric. The odds ratio for passing is the reverse of the one for failing, which makes the maths tidy.
The rule for reading them: when the outcome is rare (under about 10%), an odds ratio is a fair stand-in for a risk ratio. When it's common, convert back to actual chances before saying anything about how much more likely something is.
Try it yourself
A news story says: "People who walk to work have twice the odds of reporting good health (odds ratio 2.0)." Among people who don't walk, 75% report good health.
1. What are the odds of good health for non-walkers, and for walkers? 2. What share of walkers report good health? Is "twice as likely" a fair summary?
Show the answer
1. Non-walkers: 75 to 25, odds of 3. Walkers: twice that, odds of 6.
2. Odds of 6 mean 6 ÷ 7, about 86%. So 86% versus 75%: walkers are about 14% more likely to report good health, not twice as likely. (And it's an observational finding, Lesson 25: healthier people may simply be the ones able to walk.)
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Odds | Chance it happens divided by chance it doesn't: p ÷ (1 - p) |
| Risk ratio (relative risk, RR) | One group's probability divided by another's |
| Odds ratio (OR) | One group's odds divided by another's |
| Rare disease assumption | When outcomes are rare, the odds ratio is close to the risk ratio |
| Case-control study | Comparing people with and without an outcome and looking back at causes; gives odds ratios |
| Logistic regression | Regression for yes-or-no outcomes; its coefficients become odds ratios |
| Absolute risk difference | The plain gap in chances, such as 80% minus 60% = 20 points |
Three things to remember
- Odds compare happens with doesn't happen; probability compares happens with everything. "3 to 1 against" is 25%.
- For rare outcomes, odds ratio and risk ratio agree. For common ones, the odds ratio can be double the risk ratio or more.
- When you see an odds ratio, convert it back to actual chances before believing "twice as likely".
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.