FISHERY BULLETIN: VOL. 78. NO. 3 



discrete function, catches are measured in num- 

 bers of fish instead of weight as in the preceding 

 section. 



Fitting data from a systematically designed 

 survey to the negative binomial distribution is a 

 common practice (Moyle and Lound 1960; Taft 

 1960; Roessler 1965; Clark 1974). Hairston et al. 

 (1971) made one of the few studies of sampling 

 design for measuring spatial pattern. They found 

 that estimates based on a grid (systematic) pat- 

 tern were superior to those made from sampling at 

 random. The grid pattern correctly reflected the 

 spatial patterns of 17 of 22 species, while random 

 sampling with the same number of samples cor- 

 rectly reflected only 12 of 22 species. 



The standard negative binomial model requires 

 a constant element size. This leads to two prob- 

 lems. The first is that comparisons can only be 

 made at one sampling element size. The second 

 problem is the negative binomial model cannot be 

 fit to data with variation in sample element size. 



To specifically deal with these problems, Bissell 

 (1970, 1972) derived a negative binomial model 

 that can be used when sample element size is 

 variable and/or to predict the distribution of 

 events for element sizes which differ from those 

 on which the observations were based. The prob- 

 ability of observing Xi events on the i element 

 which has a size of wi is 



Table 6. — Estimates of mean densities (d) (in numbers per 

 kilometer), k, standard errors, with chi-square goodness of 

 fit tests for trawl catches made in all depths in the Queen 

 Charlotte survey. 



then compared with the observed value using a 

 chi-square test. Values of k from trawls in all 

 depths ranged from 0.2 to 0.6 for the most abun- 

 dant species. The low values of k indicate that the 

 more abundant species are highly aggregated. 

 The estimates for k are close to the value of k 

 (0.27) that we estimated for S. marinus, an abun- 

 dant species of rockfish, from Georges Bank from 

 data in the paper by Taylor (1953). 



We next divided the trawls into three depth 

 intervals: 91-145 m, 146-181 m, and >181 m. 



Estimates of mean densities, k, and goodness of 

 fit tests by depth strata are presented in Table 7. 



P{x.lWi) = {klimw^ + k)}" {mw-limw, + k)}^i n {(A- +7 - 1)/;] (7) 



where m = mean value of jc, for element size of 

 unit size 

 k = parameter representing the degree 

 of aggregation (Note that k in this 

 section has a different meaning than 

 in the section on sampling schemes) 

 Wi = element size (distance towed). 



Iterative maximum likelihood solutions (Bissell 

 1972) gave estimates of values of m, k, and their 

 standard errors relative to the average distance 

 towed. The values were converted to densities id) 

 with units of numbers per kilometer. Estimates of 

 d, k, and chi-square goodness of fit tests are given 

 in Table 6. These tests were made by calculating 

 the probability of a given number of fish occurring 

 in a trawl of a given length from the probability 

 density function given by Bissell (1972). This 

 probability was cumulated over all trawls and 



The chi-square tests show that the data combined 

 over all depths are not well represented by the 

 negative binomial model. However, when the data 

 are divided up by depth strata, the agreement is 

 quite good. When the data from low density and 

 high density depth strata are combined, the result- 

 ing frequency distribution has too many zero 

 elements and too many high abundance elements. 

 This results in the high chi-square values from 

 trawls at all depths. In comparing the results in 

 Tables 6 and 7, it is obvious that depth stratifica- 

 tion is important. The differences between densi- 

 ties of species among depth strata were tested at 

 the 10% level of significance. Of 43 possible 

 comparisons, 27 (or 63%) were significantly 

 different. This can be tested against what would 

 have occurred randomly as a binomial proportion 

 (Hollander and Wolfe 1973). The proportion is 

 significantly different than random {z = 6.77, 



668 



