# 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 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 µ0 represent the expected number of cases under the null hypothesis. The expected number under the null hypothesis is µ0 = (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, µ0 =(298.5)(0.003667) @ 1.1. A one-sided test now addresses the alternative H1: 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 H0). For example, to test H1: 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, 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/.