Homogeneity tests for several Poisson populations

In this paper we compare the size distortions and powers for Pearson’s χ²-statistic, likelihood ratio statistic LR, score statistic SC and two statistics, which we call UT and VT here, proposed by Potthoff, R.F., Whittinghill, M., 1966. Testing for homogeneity: II. The Poisson distribution. Biometrika 53, 183–190] for testing the equality of the rates of K Poisson processes. Asymptotic tests and parametric bootstrap tests are considered. It is found that the asymptotic UT test is too conservative to be recommended, while the other four asymptotic tests perform similarly and their powers are close to those of their parametric bootstrap counterparts when the observed counts are large enough. When the observed counts are not large, Monte Carlo simulation suggested that the asymptotic test using SC, LR and UT statistics are discouraged; none of the parametric bootstrap tests with the five statistics considered here is uniformly best or worst, and the asymptotic tests using Pearson’s χ² and VT always have similar powers to their bootstrap counterparts. Thus, the asymptotic Pearson’s χ² and VT tests have an advantage over all five parametric bootstrap tests in terms of their computational simplicity and convenience, and over the other four asymptotic tests in terms of their powers and size distortions.