Incidence Rate Calculator: Person-Years, Cumulative Incidence and Rate Ratio
Calculate the incidence rate per 1,000 or 100,000 person-years with an exact Poisson confidence interval, the cumulative incidence (risk) over follow-up, risk from the rate, implied prevalence from disease duration, and an incidence rate ratio and rate difference against a comparison group.
How it works
Incidence rate = new cases ÷ person-time · person-time (estimate) = N × t − ½ × cases × t · exact CI = [χ²(α/2, 2c) ÷ 2, χ²(1 − α/2, 2c + 2) ÷ 2] ÷ person-time · cumulative incidence = cases ÷ population at risk · risk = 1 − e^(−rate × t) · prevalence ≈ I × D ÷ (1 + I × D) · IRR = rate₁ ÷ rate₂, SE(ln IRR) = √(1/c₁ + 1/c₂)
Incidence measures how fast new disease appears. The incidence rate (incidence density) divides new cases by the person-time at risk, the sum of each person's disease-free time under observation, so it handles people who join late, drop out or die of something else, and is expressed per 1,000 or 100,000 person-years. The cumulative incidence, or risk, is the share of a fixed population at risk that develops the disease over a stated period. When you do not have individual follow-up times, the calculator estimates person-time the way CDC teaches it: everyone contributes the whole period except the cases, who are assumed to fall ill halfway through. The default confidence interval is the exact Poisson (Garwood) interval from chi-square quantiles, which keeps its coverage even with a handful of cases; Byar's approximation, the log-normal interval and the normal (Wald) interval are shown for comparison. Over short periods with a rare outcome the rate and the risk agree; over long follow-up the risk saturates, and risk = 1 − e^(−rate × time) converts between them. Prevalence at steady state is roughly incidence times average duration, P ÷ (1 − P) = I × D. Turn on the comparison group to get the incidence rate ratio with a log-scale interval and an exact conditional mid-p value, and the rate difference.
Worked example
A cohort of 12,000 people is followed for 2.5 years and 48 new cases appear. Person-time is 12,000 × 2.5 − ½ × 48 × 2.5 = 29,940 person-years, so the incidence rate is 48 ÷ 29,940 = 0.001603 per person-year, or 1.60 per 1,000 person-years. The exact 95% Poisson interval for 48 cases is 35.39 to 63.64 cases, which divided by 29,940 gives 1.18 to 2.13 per 1,000 person-years. The cumulative incidence is 48 ÷ 12,000 = 0.40% over 2.5 years, and the rate predicts 1 − e^(−0.001603 × 2.5) = 0.40%, so the two agree. If the disease lasts 3 years on average, the steady-state prevalence is 0.00481 ÷ 1.00481 = 0.48%.
Sources: CDC, Principles of Epidemiology, Lesson 3: incidence rate and person-time · MetricGate, person-time rate: exact Poisson CI, IRR and rate difference formulas · MetricGate, cohort incidence rate (worked example with exact and normal CIs) · fmsb IRCIPois: exact Poisson CI for an incidence rate
Frequently asked questions
How do you calculate an incidence rate?
Divide the number of new cases by the total person-time at risk and multiply by a base such as 1,000 or 100,000. CDC's example: 16 new cases in 6,400 person-years is 16 ÷ 6,400 = 0.0025, or 2.5 per 1,000 person-years.
What is the difference between incidence rate and cumulative incidence?
The incidence rate uses person-time in the denominator and is a speed (cases per person-year). Cumulative incidence uses the number of people at risk at the start and is a probability over a stated period. In CDC's example the same 16 cases give 2.5 per 1,000 person-years but 7.6 per 1,000 women over four years.
When should I use a rate instead of a risk?
When follow-up time varies between people: open populations, cohorts with drop-outs or staggered entry, or long studies. Risk is fine for closed groups over a short, complete follow-up, such as an outbreak or a surgical complication.
Why subtract half the cases' time?
A person who becomes a case stops being at risk. If cases occur evenly through the period, each contributes about half the follow-up. With exact dates, sum the actual time at risk and enter it as person-years observed.
Which confidence interval should I report?
The exact Poisson interval is the standard choice and is valid for small counts. With more than about 30 cases the Byar and log-normal intervals are almost identical to it; the normal (Wald) interval can go below zero and under-covers when cases are few.
How do I compare incidence between two groups?
Divide one rate by the other to get the incidence rate ratio. Its 95% interval is IRR × e^(±1.96 × √(1/c₁ + 1/c₂)). An interval that excludes 1, and a small mid-p value, indicate a real difference.
Related calculators
Standard epidemiological formulas (CDC Principles of Epidemiology, Rothman) with exact and large-sample confidence intervals. Results depend on case definitions, ascertainment and study design; they inform, and do not replace, formal analysis by an epidemiologist or public-health authority.