13 Research methods, statistics, critical review and evidence-based practice13.8 Descriptive statistics
13.8.3 Ratios, odds and rates
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Written by Resident Doctor Committee · Published
New in the 2027 syllabus
The College added this topic for exams from January 2027.
Four denominators
These measures differ mainly in what goes on the bottom of the fraction (the denominator).
Measure | What it is | Possible values |
|---|---|---|
Ratio | One quantity divided by another | Depends on the quantities |
Proportion | Part of a group divided by the whole group it comes from | 0 to 1 |
Odds | Number with the event divided by number without it | 0 upwards |
Incidence rate | New events divided by the person-time at risk | 0 upwards, with time units |
Risk is the probability of an event over a stated period. You estimate it as the proportion with the event in a suitable population that you follow up.
A percentage is a proportion multiplied by 100. Calling it a rate does not give it a person-time denominator.
Odds equal p / (1 minus p), where p is the probability of the event. To convert back, probability equals odds / (1 + odds).
Worked example: risk and odds over a fixed period
Two hypothetical groups are each followed for one year. Each person can have at most one recorded event.
Group | Event | No event | Total |
|---|---|---|---|
A | 20 | 80 | 100 |
B | 10 | 90 | 100 |
Group A risk = 20 / 100 = 0.20. Its event odds = 20 / 80 = 0.25.
Group B risk = 10 / 100 = 0.10. Its event odds = 10 / 90 = 1/9.
Risk ratio = 0.20 / 0.10 = 2.
Odds ratio = 0.25 / (1/9) = 2.25, not 2.
The event is not rare in either group, so the two ratios differ.
Person-time and incidence rates
Person-time adds up the time each person spends under follow-up while still at risk.
Hypothetical observation | Value |
|---|---|
Events | 12 |
Total follow-up at risk | 150 person-years |
Incidence rate = 12 / 150 = 0.08 per person-year.
Multiply by 100 to give the same rate as 8 per 100 person-years.
Comparing groups
A rate ratio compares two rates measured in the same time units. It is not the same as a ratio of fixed-period risks.
When an outcome is rare, odds and risks are close, so an odds ratio can approximate a risk ratio. The approximation gets worse as risks rise.