The random number of cases in an open population during given period of time D*t* will follow a
Poisson distribution with unknown parameter µ. Suppose, for example, we observe 3 cases in a
population consisting of 298.5 person-years. Given biological variation and random sources of
variation, the number of cases is assumed to follow a Poisson distribution with expected value µ.

A **confidence interval **for µ, the expected number of cases, can be determined with the Web
calculator found at http://members.aol.com/johnp71/confint.html#Poisson. Using this calculator,
a 95% based on an observed value of 3 is equal to (0.6197, 8.7673).

The **incidence rate **for the illustrative data = observed no. of cases / sum of person time. For the
illustrative example, the incidence rate = 3 cases / 298.5 person-years = .01005 cases per
person-year. The denominator of this rate is assumed to be constant, so a 95% confidence
interval for the rate parameter is equal to (0.6197 cases / 298.5 person-years, 8.7673 cases / 298.5
person-years) = (.00208, .029371) cases per person-year. This, of course, is equivalent to (0.2,
2.9) cases per 100 person-years.

We may wish to **test **whether the observed count is significantly greater than expected. To do
this, let µ_{0} represent the expected number of cases under the null hypothesis. The expected
number under the null hypothesis is µ_{0} = (*T*)(*I*_{0}), where

*T*represents the observed person-time and*I*_{0}represents the expected incidence rate.

The expected incidence rate *I*_{0} comes from an external source, such as might be provided by
national vital statistics. For illustrative purposes, suppose *I*_{0} = 0.003667 cases per person- year.
Then, during 298.5 person-years of observation, µ_{0} =(298.5)(0.003667) @ 1.1. A one-sided test
now addresses the alternative *H*_{1}: µ > 1.1. An exact one-sided *p *value is derived from the area
under the curve in the right tail of a Poisson distribution with µ = 1.1 (as specified under *H*_{0}).
For example, to test *H*_{1}: µ > 1.1 when 3 cases are observed, the probability of seeing *at least* 3
cases is equal to 0.0996 (i.e., *p* = .0996) Assuming a = .05, *H*_{0} is retained.

Poisson probabilities can be calculated with `EpiTable | Probability | Poisson` or
with a Web-based probability calculator such as the one found at http://www.cytel.com/statable/.