A Generalization of Neyman's Contagious Distributions

Neyman (1939) presented a new class of contagious distributions applicable to situations where individuals or items are supposed initially dispersed in randomly scattered groups-egg masses in the case of insects or clumps in the case of bacteria-which are subject to a chance fluctuation in size. It may be supposed that there occurs subsequently some spacial dispersion from the initial groups. Under such a scheme, the distribution of individuals, or items, cannot be random. Since the probability of an individual occurring in a given unit area is related to the probability of other individuals occurring in that unit area, the distributions may be called contagious. Upon the previous general grounds, Neyman has suggested three types of theoretical frequency distribution called Types A, B, and C. These types all correspond well to observed data for cases where the mechanism of dispersion is roughly that previously set forth. They proved better than a Poisson distribution and superior to the negative binomials, often used under such circumstances. The novel character of the new distributions may be seen, in the extreme, from the fact that they can be multimodal. This character is, of course, in violent contrast to that of the Poisson or negative binomial but yet corresponds very well with the biological reality. These conclusions were reached on various biological applications, as in the original work by Neyman (1939) and in work by one of the writers (Beall 1940) and by Archibald (1948). Often in this experience, Types A, B and C fit data progressively better although sometimes the sequence of types seems not to go far enough. For illustration, there is the case of Pyrausta nubilalis Hubn. in 1937 (Table VII, Beall 1940), where the types give values of x2 with progressively the probabilities of .011, .064 and .110. As a second illustration there is the case of P. nubilalis in the same source but which has been reproduced in the present discussion as Table II. It can be seen the values of x2 again fall and have progressively the probabilities of .05, .48 and .57.