No events observed. The exact two-sided 95% upper limit is 3.688879 per 1,000 person-years — not zero, and not the 3 per 1,000 the rule of three suggests, because that rule is the ONE-sided bound.
0 events in 1,000 person-years
0.0000 per 1,000
Exact 95% Poisson interval 0.0000 to 3.6889. With no events the point estimate is zero and the upper limit is the only informative number — it is 3.688879 divided by the person-time, which is the exact two-sided bound rather than the 2.995732 the one-sided "rule of three" gives.
Incidence rate
0.00000
per 1,000 person-years
Lower limit
0.00000
exact Poisson, 95%
Upper limit
3.68888
exact Poisson, 95%
Rate per unit time
0.00000000
events per person-year
Exact Poisson interval, not a Wald approximation A rate has units; a risk does not Zero events still gives an upper limit
What this tool shows
With zero events observed, the exact two-sided 95% upper limit on the rate is 3.688879 ÷ person-time — not the 3 ÷ person-time the “rule of three” gives. That rule is exactly right, but it is the ONE-sided bound (2.995732, which is why it rounds to 3). Quoting it beside two-sided intervals everywhere else understates the upper limit by 23%. This tool computes the exact Poisson interval throughout.
Incidence rate per person-time, reported per 1, 100, 1,000, 10,000 or 100,000
The exact Poisson (Garwood) confidence interval, not a Wald approximation
A meaningful upper limit when zero events were observed
The same rate at seven follow-up lengths, with interval width as a ratio
Person-time explained as the sum of individual time at risk
The rule of three placed correctly as a one-sided bound
Rate per person-time Exact Poisson CI Zero-event case Precision sweep
Updated 12 September 2026 · Works in any browser, no installation
An incidence rate is new events divided by the person-time at risk, so it has units of “per year” and can exceed 1. That is what distinguishes it from a risk, which is a proportion of people and is bounded by 1. Person-time is the sum of how long each participant was actually at risk, so ten people followed two years and twenty followed one contribute the same twenty person-years.
At a glance
Formula shown
Rate = events ÷ person-time. The exact 95% interval comes from the chi-square relation to the Poisson: lower = χ²(0.975, 2e)/2 ÷ T and upper = χ²(0.025, 2(e+1))/2 ÷ T, where e is the event count. At e = 0 the lower limit is 0 and the upper is χ²(0.025, 2)/2 = −ln(0.025) = 3.688879, divided by T. The one-sided version uses −ln(0.05) = 2.995732, which is the rule of three.
Scenario support
Cohort studies and registries, adverse-event rates in trials and pharmacovigilance, occupational injury and workplace safety reporting, infection and readmission rates, equipment failure rates, and any setting where participants were followed for different lengths of time.
Educational estimate
Planning support from the values you enter — not professional advice.
Zero events is not zero risk, and the rule of three is one-sided
“No adverse events were observed” is the most common result in a small safety study and the easiest to over-read. The upper limit is the whole finding.
The exact two-sided 95% upper limit at zero events is 3.688879 ÷ T, which is −ln(0.025). With 1,000 person-years that is 3.688879 per 1,000 — an event rate up to about 1 in 271 person-years is entirely consistent with having seen none.
The familiar “rule of three” gives 3 ÷ T, and it is exactly right for a ONE-sided 95% bound: −ln(0.05) = 2.995732, which is why it is quoted as 3.
The two are 23% apart, and papers routinely mix them. A study reporting two-sided 95% intervals for everything else and then invoking the rule of three for a zero count has quietly switched conventions at the point where the reader is least able to notice.
Neither bound depends on how many people were enrolled, only on the person-time. Which is why a short study of many people and a long study of few give the same upper limit when the person-time matches.
Precision comes from events, not from follow-up
The tool’s sweep scales events and person-time together, holding the rate constant. The interval narrows, and it narrows on a schedule set entirely by the event count.
Five events spans a factor of 7.19 from lower to upper limit. Fifty events spans 1.78; a hundred spans 1.49. The point estimate is identical in all three.
The standard error of a Poisson count is √e, so the relative precision is 1/√e — which depends on e alone. Person-time appears in the rate but not in that ratio.
Which is why extending follow-up in a cohort where nothing is happening does not help. More person-time with no additional events lowers the point estimate and the upper limit together; it does not produce a precise estimate of a small number.
And it is why studies are powered on expected events rather than on enrolment. “We will recruit 5,000 people” says nothing until it is paired with how many events those people are expected to produce.
A rate is not a risk, and they answer different questions
The two get used interchangeably and they have different units, different bounds and different behaviour when follow-up is uneven.
Risk is a proportion of people: cases ÷ people, bounded by 1, dimensionless. It answers “what fraction of this group developed the condition?” and needs everyone followed for the same period to be meaningful.
Rate is events ÷ person-time, unbounded, with units of inverse time. A rate of 1.5 per person-year is perfectly possible for a recurrent event.
Person-time is what handles unequal follow-up, which is the ordinary situation: people enrol at different times, withdraw, or are lost. A risk computed on such a cohort silently treats a one-month participant as equivalent to a five-year one.
For a rare outcome over a short period the two are numerically close, which is why the distinction is easy to ignore and why it matters most exactly when follow-up is long.
A ratio of two rates is an incidence rate ratio, which is a different quantity from a risk ratio even though both are often called “relative risk”.
Why the exact interval rather than the usual approximation
The standard Wald interval is rate ± 1.96√e/T. It is easy, and it fails in exactly the situations where incidence rates are most often reported.
At zero events the Wald interval is zero to zero. It claims certainty that the rate is exactly zero, which is the single worst answer available.
At small counts it can go negative, and software that clamps the lower limit to zero produces an interval with the wrong coverage rather than a fixed one.
The exact interval inverts the Poisson test through its relation to the chi-square distribution, so it is guaranteed to be inside [0, ∞) and to have at least the nominal coverage at every count.
It is conservative, which is the price. Actual coverage exceeds 95% because the Poisson is discrete, and mid-p variants trade a little of that guarantee for a shorter interval.
Above about 100 events the two agree closely, so the exact method costs nothing where the approximation works and fixes it where it does not.
What a constant rate assumes about your cohort
Summing person-time across people and periods is only valid if the rate is the same across all of it. That assumption is usually the weakest part of the analysis.
The rate must not change over follow-up time. Post-surgical complications concentrate in the first weeks, so a single rate over five years describes a period that never existed.
It must not differ between the people being pooled. Combining a high-risk and a low-risk group gives a weighted average that describes neither, which is the reason for stratum-specific rates.
Events must be independent. Recurrent events in the same person are not, and counting them all against that person’s time overstates precision badly.
And people must leave the risk set when they should. Time after the outcome, after withdrawal, or after the person is no longer susceptible does not belong in the denominator.
When the rate genuinely varies, stratify or model it — a Poisson regression with time as a covariate, or separate rates per period, both say more than one number can.
Reporting an incidence rate
Four things, and the second is what lets anyone recompute the interval.
Give the rate with its units. “5 per 1,000 person-years” is a rate; “0.5%” is not, because it omits the time the events occurred over.
Give the event count and the person-time separately. They are the two numbers the whole analysis rests on, and a reader can check everything from them.
Give the confidence interval and say which method produced it. Exact and Wald intervals differ materially below about 30 events, which is where most reported rates live.
And say how person-time was accrued. When follow-up started, when it stopped, and what removed a person from the risk set are the assumptions that make the denominator meaningful.
Sources and methodology
References for incidence rates and exact Poisson intervals.
Method. The interval is the exact Garwood interval, obtained by inverting the Poisson test through its chi-square relation, with the chi-square quantiles found by bisection on the survival function to 200 iterations rather than from a table. That is what makes the zero-event claim exact rather than quoted: the upper limit is χ²(0.025, 2)/2 = −ln(0.025) = 3.688879 divided by the person-time, against the one-sided −ln(0.05) = 2.995732 that the rule of three rounds to 3. The precision sweep scales events and person-time together so the point estimate is held constant and only the interval moves, which demonstrates that relative precision depends on the event count alone. Zero or negative person-time, negative counts and a non-positive multiplier all return no result. That engine is verified on every change against 115 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Poisson DistributionPoisson probabilities with a dispersion test against your own variance — because a Poisson forces variance to equal the mean, and real count data usually does not, which is exactly where the tail goes wrong.
Standardized Mortality RatioSMR with the expected count built stratum by stratum, and the small-count disagreement shown: 5 observed against 2.07 expected gives p = 0.0417 and an exact interval of 0.7843 to 5.6369.
Relative RiskRisk ratio and odds ratio from one table with the divergence between them plotted: they agree to half a percent at a 1% baseline, and at an 80% baseline the odds ratio is exactly half the risk ratio.
Attributable RiskRisk difference, attributable fraction and population attributable fraction, which rank risk factors in opposite orders: RR 9 at 1% exposure gives a PAF of 7.41%, RR 1.2 at 80% gives 13.79%.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
Wilson Score IntervalWilson, Wald, Agresti-Coull and Clopper-Pearson on one set of counts, with the EXACT coverage each delivers at your sample size — a "95%" Wald interval covers the truth 80.85% of the time at n = 30, p = 0.1.
An educational tool, not medical or safety advice. A single incidence rate assumes the underlying rate is constant across everyone and every period whose person-time is pooled into it, and zero observed events is consistent with a rate up to roughly 3.69 divided by the person-time — not with a rate of zero.
Published a rate tool that uses the exact Garwood Poisson interval throughout, obtained by inverting the Poisson test through its chi-square relation with the quantiles found by bisection rather than from a table.
Placed the 'rule of three' correctly, which most sources do not: it is the ONE-sided 95% bound, -ln(0.05) = 2.995732, which is why it rounds to 3. The exact TWO-sided upper limit at zero events is -ln(0.025) = 3.688879, 23% higher — and papers that report two-sided intervals everywhere else routinely switch conventions at the zero count without saying so.
Made the zero-event case informative rather than degenerate: the Wald interval there is zero to zero, claiming certainty that the rate is exactly zero, which is the worst available answer.
Added a sweep scaling events and person-time together, which shows precision is driven by the EVENT count alone — five events span a factor of 7.19 from limit to limit, fifty span 1.78, a hundred span 1.49, with the point estimate identical throughout.
Verified that the reporting multiplier scales the rate and both limits by exactly the same factor, and that the interval always contains the point estimate and widens monotonically as events fall.
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