154 



FISHERY BULLETIN OF THE FISH AND "WILDLIFE SERVICE 



I 2. 3 4 5 6 7 8 9 10 II 12 13 14 15 16 



TERM OF DISTRIBUTION 



Figure 7. — The negative binomial distributions compared to the limiting form, the Poisson. Each distribution has a 



mean of 10 and consists of 1,000 observations. 



Type A distribution, the number of colonies or 

 groups per sample has a Poisson distribution with 

 mean m,, while the number of individuals per group 

 has a Poisson distribution with mean m 2 . The 

 Thomas distribution is similar, the number of indi- 

 viduals per group assumed to be one plus an obser- 

 vation from a Poisson distribution with mean m 2 . 

 The Polya-Aeppli distribution, however, supposes 

 that the number of individuals per colony has a 

 geometric distribution; it has, therefore, some 

 interest in connection with sampling from growing 

 populations (Anscombe 1950). 



In the absence of a series of equally spaced 

 modes, Anscombe (1950) points out, one may 

 reasonably feel reluctant to use the Neyman or 

 Thomas distributions. Such modes are not 

 evident in the trawl data. 



The choice between the negative binomial and 

 the Polya-Aeppli distributions lies less in consid- 

 eration of the goodness of fit obtained in applying 

 them than in the various theoretical considerations 

 which might justify their use. Kendall (1948) 

 shows that colonies established simultaneously 



from single ancestors and unaffected by immigra- 

 tion will, after a fixed lapse of time, have a dis- 

 tribution of population sizes resulting from growth 

 at geometric rates equivalent to the difference 

 between constant reproduction and mortality 

 rates. This is the model for the Polya-Aeppli dis- 

 tribution provided that the distribution of the 

 ancestors establishing each colony was random. 

 If, on the other hand, the distribution of ancestors 

 does not occur simultaneously throughout the 

 area of observation but is uniform in time, the 

 distribution of colony sizes resulting will be that 

 of the negative binomial, provided that reproduc- 

 tion and mortality rates are constant. Kendall's 

 analysis indicates that the index of the resulting 

 negative binomial, k, is intrinsically associated 

 with the reproductive power of the species. 



These models for the Polya-Aeppli and the 

 negative binomial distributions appear over- 

 simplified in considering the application of them 

 to fish populations. The origin of schools of fish 

 can hardly be supposed to occur simultaneously 

 throughout the area of observation or to arise from 



