# A Count of Cases in an Open Population

The random number of cases in an open population during given period of time Dt will follow a Poisson distribution with unknown parameter �. 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 �.

A confidence interval for �, 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 �0 represent the expected number of cases under the null hypothesis. The expected number under the null hypothesis is �0 = (T)(I0), where

• T represents the observed person-time and
• I0 represents the expected incidence rate.

The expected incidence rate I0 comes from an external source, such as might be provided by national vital statistics. For illustrative purposes, suppose I0 = 0.003667 cases per person- year. Then, during 298.5 person-years of observation, �0 =(298.5)(0.003667) @ 1.1. A one-sided test now addresses the alternative H1: � > 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 � = 1.1 (as specified under H0). For example, to test H1: � > 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, H0 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/.