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Fishery Bulletin 94(2). 1996 



Table 2 



Parameter estimates from a fit of the negative binomial with added zeros to tuna 

 discard data from the U.S. tuna purse-seine fleet fishing in the eastern tropical 

 Pacific Ocean, 1989-92. p is the mixing parameter, and p and a are the mean 

 and shape parameters of the conditional negative binomial. Standard errors are 

 in parentheses. Geographic areas are defined according to Federal Register! 1989) 

 (Fig. 1). 



School sets 



Log sets 



Parameters Dolphin sets Area 1 



Area :: 



Area 1 



Area 3 



0.982(0.015) 0.715(0.182) 0.825(0.111) (0) (0) 



3.53 (3.21) 



3.87 (5.36) 



5.53 (3.60) 



7.20(6.28) 



15.4 (1.31 7.09(0.55 



2.34(0.19)3.93(0.25) 



At the other extreme, nonzero discard observations 

 were very frequent for log sets (650 out of 998 sets. 

 Table 1). Using fishing area as a covariate for both 

 the mean and shape parameters p and a, we im- 

 proved the fit significantly (p-value <0.001 ) over sim- 

 pler models. However, the numerical optimization 

 failed to converge to a positive value for p in either 

 area, producing estimates of zero forp in both areas 

 (Table 2). Thus, the estimated probability distribu- 

 tions effectively collapsed to unmodified NB's. Be- 

 cause positive observations were so abundant, esti- 

 mated standard errors for the mean and shape pa- 

 rameters (Table 2) were small (CV's <8.5%, Fig. 3). 



Discard from school sets presented an intermedi- 

 ate case in which we selected a model which included 

 marginally different estimates for the mixing prob- 

 ability p in areas 1 and 3, but no geographic depen- 

 dence for a or /j (Table 2). Because there were consid- 

 erably fewer nonzero observations (80 out of 960 sets, 

 Table 1) for school sets than for log sets, parameter 

 estimates were much less precise. Likelihood-ratio tests 

 indicated that fishing area should be included as a 

 covariate for either the shape parameter a or the mix- 

 ing probability p, but that including areal dependence 

 for p and a simultaneously, or for /i, did not further 

 improve the fit. Because the approximate p-values for 

 adding areal dependence to the two parameters were 

 fairly similar (0.04 for a, 0.12 forp) and the two pa- 

 rameters have similar effects in the model, 1 there was 

 no clear basis for selecting one parameter over the other. 

 We subsequently decided to include areal dependence 

 only forp for two reasons. First, the small number of 



positive observations for school sets 

 limits the precision of the shape es- 

 timate. Second, the difference in the 

 estimated shape between areas was 

 due mainly to two unusually large 

 observations in area 3. Without 

 these two observations, the differ- 

 ence in estimated shapes was re- 

 duced, and the significance levels of 

 the two different models were 

 nearly equal (approximate p-values 

 of 0.09). As was the case for dolphin 

 sets, the predominance of zeros in 

 the school set discard data led to 

 small estimated standard errors for 

 the mixing probability p (CV's 

 <13.5%, Fig. 3) but to large esti- 

 mated standard errors for the mean 

 and shape parameters (CVs > 65%, Fig. 3). 



In our model, p may be interpreted as the prob- 

 ability of exactly zero discard, as opposed to small 

 amounts of discard that have been rounded down to 

 zero in the data. The estimates of p for the three set 

 types imply that essentially all dolphin sets (98%) 

 involve no discard, whereas log sets always involve 

 at least some discard. Observer experience 2 indicates 

 that this result is consistent with generally observed 

 patterns for dolphin and log sets. 



The estimated shape parameters varied widely 

 between the three set types (Table 2), but the large 

 standard error estimates for the school and dolphin 

 shape parameters prevent us from making any strong 

 statements about shape as a function of set type. As 

 mentioned above, the estimated shape parameter for 

 school sets was strongly affected by the presence of 

 two unusually large observations (100 and 125 tons 

 of discard) in area 3. Repeating the analysis without 

 these two observations led to a shape estimate of 3.75 

 (SE=2.10), which is more similar to the shape esti- 

 mates for log sets (2.94 for area 1, 3.93 for area 3). 



We could not use bootstrap methods to compute 

 standard errors for dolphin sets because there were 

 so few sets observed with positive discard recorded 

 (Table 1). In resampling for the bootstrap, approxi- 

 mately one-third of the samples contained too few 

 positive observations for the maximum likelihood 

 algorithm to converge. Therefore, Table 2 includes 

 only the standard errors computed from the analytic- 

 approximation formulae. 



p and a are similar in the effect they have on the estimated 

 distribution. Increasing either one increases the probability of 

 ;i zero observation, although increasing a also increases the 

 probability of a large observation. 



2 Jackson, A. 1994. Southwest Fisheries Science Center, Natl. 

 Mar. Fish. Serv., P.O Box 271, La Jolla, CA92038. Personal 



commun. 



