Significance tables for the exact variance test for the Poisson distribution with alternative underdispersion

Abstract

Fisher (1950) introduced the variance or dispersion index test statistic to test deviations of the Poisson distribution. For this test approximate critical values exist for large sample sizes. If the number of observations is small this approximation can lead to a wrong conclusion. For small samples, the exact critical values can only be derived by enumeration of all possibilities. Tables of critical values for overdispersion already exist (e.g., Rao and Chakravarti, 1956) However, in many biological situations underdispersion, a more-regular-than-Poisson distribution, is a common phenomenon. Therefore, we have tabulated in this paper the one-tailed critical values for a small number of observations under the null hypothesis (H₀) that the random variable is Poisson distributed against the alternative hypothesis of underdispersion. With the help of this table, the hypothesis that the observations in a data set are Poisson distributed, can be tested easily with the variance test. The tables are illustrated with examples from the literature and some observations from our own research. In general, the χ² approximation gives a smaller significance level than the exact variance test.