7. Odds Ratio vs Risk Ratio, Attributable Fraction, NNT Explained
After this lesson you can
- Convert a probability to odds and compute the odds ratio from a two-by-two table
- State when the odds ratio approximates the risk ratio and why a case-control study yields only an odds ratio
- Compute the attributable fraction among the exposed and the population attributable fraction, and name whose cases each describes
- Compute the number needed to treat and the number needed to harm from the risk difference
- Compute the preventive fraction as one minus the risk ratio and tell vaccine efficacy from effectiveness
- Write attributable sentences that begin with the causal condition
Video coming soon
Scheduled for release on November 4, 2026.
This lesson completes the measures of association and impact. It converts probability to odds, computes the odds ratio by the cross-product, and shows with two tables why the odds ratio approximates the risk ratio only when the outcome is uncommon. It then builds the impact measures from the risk difference: the attributable fractions, the population risk difference, the number needed to treat or harm, and the preventive fraction, each stated with the condition that the association is causal.
Questions and answers
Questions a learner would ask about this lesson. Think through your own answer before you open each one.
- What is the difference between odds and risk?
Show answer
Risk is cases over everyone at risk, while odds are cases over non-cases. Risk lies between 0 and 1, but odds run from 0 upward with no ceiling. The two are nearly equal when the outcome is uncommon and drift apart as it becomes common: a probability of 0.25 gives odds of 0.33, and a probability of 0.50 gives odds of 1.0.
- If a report says the odds ratio was 3.0, do the exposed have three times the risk?
Show answer
Not necessarily. The odds ratio approximates the risk ratio only when the outcome is uncommon in both exposure groups; with a common outcome it overstates the risk ratio. In the lesson, two tables share a risk ratio of 2.0, but the odds ratio is 2.01 when the outcome is uncommon and 3.0 when it is common. If the design produced odds, report odds: "the odds of the condition were three times higher among the exposed."
- Why does a case-control study give an odds ratio and not a risk ratio?
Show answer
In a case-control study the investigator chose the number of cases and controls, so the column totals are fixed and no risk exists. The odds ratio ignores the column totals: in the lesson's study of a rare neurological condition, doubling the controls leaves the odds ratio at 3.0, while the row proportion falls from 75 of 125 to 75 of 175.
- What is the difference between the attributable fraction among the exposed and the population attributable fraction?
Show answer
Both come from the same excess risk but use different denominators. The attributable fraction among the exposed is the share of the exposed group's cases due to the exposure, 50 percent in the factory. The population attributable fraction is the share of the whole population's cases, 20 percent in Oakhaven where one in four residents is exposed. Both carry the condition "if the association is causal."
- Why must the number needed to treat come from the risk difference and not the risk ratio?
Show answer
The number needed is one over the risk difference, so it counts people. The same risk ratio of 2.0 can sit on a large gap or a tiny one: in the factory the risk difference of 0.125 gives a number needed to harm of 8, while a rare outcome with the same ratio would need far more people. Dividing one by a ratio gives a number that means nothing.
Practice quiz
- Question 1A probability of 0.25 corresponds to what odds?
- Question 2A two-by-two table has a = 100, b = 300, c = 75 and d = 525. What is the odds ratio?
- Question 3A case-control study has 75 exposed and 25 unexposed cases, with 50 exposed and 50 unexposed controls. The odds ratio is 3.0. If the number of controls is doubled to 100 exposed and 100 unexposed, what happens to the odds ratio?
- Question 4When does the odds ratio provide a reasonable approximation of the risk ratio?
- Question 5Risks are 25 percent among the exposed and 12.5 percent among the unexposed, and one in four residents is exposed. What is the population attributable fraction?
- Question 6The factory risk difference is 12.5 per 100. What is the number needed to harm?
- Question 7In a trial, the outcome risk is 20 percent with usual care and 15 percent with a program. What is the number needed to treat?
- Question 8In the field, the attack rate is 10 percent among the unvaccinated and 4 percent among the vaccinated. What is the vaccine effectiveness?
- Question 9In the population attributable fraction formula, what does p stand for?
- Question 10How should every attributable statement begin?
Saved in your browser only: no account, no server.
Principles of Epidemiology is education in epidemiologic reasoning, not clinical, screening, or policy guidance. Every example is set in the fictional Oakhaven Health District, and every number is invented for teaching.