Fitting of Certain Contagious Distributions to Some Available Data by the Maximum Likelihood Method

Several distributions have been proposed for fitting to a large class of biological data which have been characterized as being contagious. Two of the earliest distributions used in fitting this type of data are the Negative Binomial and the Neyman Type A (cf. Greenwood and Yule [1920] and Neyman [1939]). More recently the Poisson Binomial and the Poisson with zeroes have been suggested (cf. McGuire et al. [1957] and A. C. Cohen [1960]). The selection of the best distribution for a particular set of data is complicated by the interrelations of these distributions and their occasionally ambiguous relationship to the models used in formulating them. Some of the distributions are limiting forms of the others for extreme parameter values. For example, the Neyman Type A and the Negative Binomial are limits of the Poisson Binomial as n -co and n -0 respectively, while both the Poisson and Poisson with zeroes are limiting forms of the Neyman Type A. Feller [1943] and Gurland [1958] have shown that different models can be used in deriving the same distributions. Apparently because of the complexity of the N.A. distribution, it is often fitted by the method of moments, and then compared with the negative binomial distribution which is fitted by the maximum likelihood method (c.f. Bliss [1953] and McGuire, et al. [1957]). This results in the confounding of the fitting process with the comparison of the distributions. The same is true about the fitting of the Poisson Bionomial. We note for reference that certain simplifications have been obtained by Douglas [1955] for the Neyman Type A fitting and by Sprott [1958] for the Poisson Binomial fitting. In spite of the simplification of various methods of fitting some of the contagious distributions the task of setting up computer routines or of using desk calculators to fit these distributions is relatively tedious. This is particularly true when the investigator is not directly concerned with the distributions fitted in this paper but wishes to use them for a comparison with some new distribution. In view of this, it was decided to set up programs