Some goodness-of-fit methods for the Poisson plus added zeros distribution

Abstract

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatibility of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W.G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C.R. Rao and I.M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.