Incidence rates and person-time
When people are followed for different lengths of time, as in cohort studies, clinical registries or workplace safety records, counting events per person is misleading. An incidence rate divides the number of events by the total person-time at risk, such as person-years:
Rate = events ÷ person-time
Ten events among 500 people followed for two years each is 10 ÷ 1,000 person-years, or 10 per 1,000 person-years.
The incidence rate ratio
To compare two groups, such as exposed against unexposed or treated against control, divide their rates:
IRR = (a ÷ T₁) ÷ (b ÷ T₂)
An IRR of 1 means equal rates; 2 means the event happens twice as often per unit of time in group 1.
Confidence interval and p-value
- CI of the IRR: computed on the log scale, with standard error √(1/a + 1/b), then transformed back.
- Exact CI of each rate: from the chi-square distribution, the exact Poisson method.
- Exact p-value: conditional on the total number of events, a follows a binomial distribution with probability T₁ ÷ (T₁ + T₂) under equal rates.
Example
Group 1: 30 events over 1,000 person-years. Group 2: 15 events over 1,200 person-years.
- rates: 30 and 12.5 per 1,000 person-years;
- IRR = 30 ÷ 12.5 = 2.4;
- 95% CI: 2.4 × e^(±1.96 × 0.316), about 1.29 to 4.46;
- exact p ≈ 0.007: the rates differ significantly;
- rate difference: 17.5 extra events per 1,000 person-years;
- exact 95% CI of the group 1 rate: 20.2 to 42.8 per 1,000.
Rate ratio, risk ratio and odds ratio
- The risk ratio compares the proportions of people with the event over a fixed period. It suits trials with complete, equal follow-up.
- The rate ratio uses person-time and handles unequal follow-up, dropouts and recurrent events.
- For rare events, all three measures are close.
Assumptions
- Events occur independently at a constant rate within each group (Poisson model).
- Overdispersion, where some people have many events, makes the CI too narrow. Negative binomial regression handles it.
- Confounders such as age are not adjusted for here. Stratified (Mantel–Haenszel) or Poisson regression methods do that.