VARIABILITY EST TRAWL CATCHES 



153 



and mean m, the appropriate forms of the 

 transformation are 



40 80 120 160 



MEAN 



200 



240 



Figure 6. — The mean and standard deviation of whiting 

 catches for the subareas and depth zones of Georges 

 Bank. 



Equation 7 indicates a logarithmic transforma- 

 tion of the catch data. This transformation 

 has been frequently employed in the past in the 

 treatment of both plankton and trawl data 

 (Winsor and Clarke 1940; Silliman 1946; Barnes, 

 1949a, 1949b, 1951). 



It has been demonstrated above, however, that 

 the hypothesis of a negative binomial distribution 

 graduates the trawl data satisfactorily. The 

 relation between the mean and variance for a 

 negative binomial distribution is 



<r x 2 =m-\-m?/k. 

 Applying equation 6 we have 



Kdm 



(8) 



g(m)=j 



■)jm-\-m 2 lk 



leading to either a sinh -1 or logarithmic integral. 

 Anscombe (1948) shows that for a negative bi- 

 nomial distiibution with variable x, exponent k, 



v^VG^ 



and 



y=log (x+V 2 k) (10) 



If m is large and k greater than 2, the optimum 

 value of C (equation 9) is rougldy %. With 

 k less than 2 but greater than %, and m large, 

 the simpler transformation (10) may be used. 



The negative binomial distributions which have 

 been fitted to the catch-per-tow data have, 

 generally, a fairly large m, but the value of k is, 

 in all instances, less than %. Under these cir- 

 cumstances, it is apparent that the difference 

 between the transformatians indicated by equa- 

 tions 7 and 10 is slight. In applying equation 7, 

 it is customary to use the empirical transforma- 

 tion log (1 + x) to avoid difficulty with zero 

 observations. Equation 10 has the advantage of 

 assigning a numerical value to zero observations, 

 based on properties of the observations themselves 

 and may be preferable in some cases. 



DISCUSSION 



The distribution theory 



It has been shown that the distribution of catch 

 per tow for various species of fish caught by the 

 research vessel Albatross III over a 3-year period 

 on Georges Bank conforms to the negative bi- 

 nomial distribution. On the other hand, the dis- 

 tribution of species and numbers of individuals 

 represented by each conforms satisfactorily to 

 Fisher's logarithmic series distribution. 



Anscombe (1950) lists seven 2-parameter dis- 

 tributions constructed on models postulating 

 various types of heterogeneous Poisson sampling. 

 Considering only the catch-per-tow data, there 

 appear to be three of these distributions, in addi- 

 tion to the negative binomial, which might be 

 applied: the Neyman Type A, the Polya-Aeppli, 

 and the Thomas, all three being of the so-called 

 contagious type. These distributions are ex- 

 amined, since the trawl catches may represent 

 samples from populations so distributed. The 

 three distributions are similar in supposing a 

 Poisson distribution of colonies or groups, but 

 they differ in the assumptions concerning the dis- 

 tribution of items within groups. In the Neyman 



