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Uncommon events in populations, such as the occurrence of specific diseases, are usefully modelled using a Poisson distribution. A common application of Poisson confidence intervals is to incidence rates of diseases (Gail and Benichou, 2000; Rothman and Greenland, 1998; Selvin, 1996).

The incidence rate is estimated as the number of events observed divided by the time at risk of event during the observation period.

__Technical validation__

Exact Poisson confidence limits for the estimated rate are found as the Poisson means, for distributions with the observed number of events and probabilities relevant to the chosen confidence level, divided by time at risk. The relationship between the Poisson and chi-square distributions is employed here (Ulm, 1990):

- where Y is the observed number of events, Y_{l} and Y_{u} are lower and upper confidence limits for Y respectively, χ^{2}_{ν,a} is the chi-square quantile for upper tail probability on ν degrees of freedom.

__Example__

Say that 14 events are observed in 200 people studied for 1 year and 100 people studies for 2 years.

The person time at risk is 200 + 100 x 2 = 400 person years

For this example:

Events observed = 14

Time at risk of event = 400

Poisson (e.g. incidence) rate estimate = 0.035

Exact 95% confidence interval = 0.019135 to 0.058724

Here we can say with 95% confidence that the true population incidence rate for this event lies between 0.02 and 0.06 events per person year.

See also incidence rate comparisons

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