158 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



Equation 6 is the same as the (r-f l)th term in 

 a negative binomial series with parameter x and 

 index a m. 



It is thus seen that the two basic assumptions 

 necessary to obtain the observed distributions are 

 (1) the species are distributed at random and move 

 independently of one another, and (2) the indi- 

 viduals within each species tend to aggregate so 

 that the mean m is a Eulerian variable from 

 sample to sample. 



APPENDIX B 



Characteristics and fitting of the negative binomial 

 distribution 



The negative binomial distribution is described 

 by two parameters, the mean m and the index k. 

 The variance of the distribution is 



m-\-m 2 /k (1) 



The expected frequency of zeros is 



P =(l+m/fc)-» (2) 



and the chance of observing any positive count r is 



Pr _ *ft-1) .  . ft-r+lfr (1 _ p)t _, (3) 



or, more conveniently, letting p' — —p t k'=—k, 

 Pr= W + l) • • . V+r-1) p ,, (1+pT „-, (4) 



Haldane (1941) shows that the log-likelihood 

 equation for (4) is 



Mlog 8 (fc' + m)-log,Z-'l 



_ w 1 +n 2 + . . . +n R 

 k' + l 



n R 



k'+R-l 



• ■(5) 



where n T is the observed frequency of r, and R is 

 the maximum value of r. 



Equation 5 gives fully efficient estimates of 

 k and p. The equation may be solved by elemen- 

 tary methods (Haldane 1941) but the procedure 

 is tedious and not ordinarily necessary. The 

 procedure is to guess a value for k', evaluate 

 both sides, and, by successive approximation 

 and interpolation, carry the value of k' to the 



required number of decimal places, usually not 

 more than four. It is convenient to make an 

 initial approximation of k' by use of equation 2, 

 where P is estimated as the ratio of observed 

 zero observations to total observations. For 

 further details, as well as a simple numerical 

 example, the reader is referred to Haldane (1941). 



A less tedious but not fully efficient procedure 

 is to estimate k by successive approximation to 

 satisfy equation 2. The method and its effi- 

 ciency as compared to the maximum-likelihood 

 method are discussed by Anscombe (1949, 1950). 



The negative-binomial distributions in this 

 paper were first fitted using the log-likelihood 

 equation 5 to estimate k. Some tabulation errors 

 were later discovered in the data and all of the 

 distributions were recalculated by estimating k 

 from equation 2, with the exception of the distri- 

 butions for haddock in depth zones II and III 

 (table 5) and for ocean perch (table 9). The 

 values of k were not materially changed by using 

 the latter method which is, for these data, about 

 90-percent efficient. Since the first term of the 

 expected series, using equation 2, is determined, 

 there is one less degree of freedom than in using 

 equation 5. 



As pointed out in the text, the negative bi- 

 nomial approaches the Poisson as k increases. 

 Some examples of the shape which the negative 

 binomial may take as k increases with the mean m 

 fixed are shown in figure 7. A value of k of 

 about 0.1 would produce a curve similar in shape 

 to the types found in this paper. 



APPENDIX C 



The logarithmic series 



Fisher (1943) shows that the logarithmic series 

 arises from the negative binomial distribution 

 when, in equation 2, appendix A, we let £=0, 

 write x for pl(p-\-\), and replace the constant 

 factor (k— 1)! in the denominator by a new 

 constant factor, a, in the numerator. The ex- 

 pression for the expected number of species with 

 r individuals is then 



oaf/r. 



(1) 



where r cannot now be zero. The successive 

 terms of equation 2 are, of course, the required 

 series 



ax, ax72, ar73, (2) 



