Perkins and Edwards A mixture model for estimating discarded bycatch 



337 



Estimating mean discard per set 



Because the model we fitted for discard from log fish- 

 ing reduced to a simple NB distribution (with p =0), 

 the estimates of mean discard per log set in each fish- 

 ing area are just the corresponding mean parameters 

 H . Mean discard per school or dolphin set was esti- 

 mated with Equation 4. 



Estimates of mean discard per log set were an or- 

 der of magnitude larger than those for school sets 

 and two orders of magnitude higher than those for 

 dolphin sets (Fig. 4). Most of this difference is due to 

 the wide range in the estimated proportion of sets 

 with zero discard. By comparison (Table 2), estimated 

 mean parameters for the NB component of the model 

 differ by less than a factor of five. Thus, the model 

 that we fitted indicates that, on average, there is a 

 considerable difference among set types in per-set dis- 

 card, although for sets in which discard actually oc- 

 curs, there is comparatively less difference in the 

 amount. 



Mean discard for log sets was estimated at 10.5 t 

 per set pooled over areas, ranging from 7.1 t per set 



Dolphin sets School sets 



Log sets 



Figure 4 



Estimated mean tuna discard per set for the U.S. tuna 

 purse-seine fleet fishing in the eastern tropical Pacific 

 Ocean, 1989-92. Geographic areas are defined in Federal 

 Register (1989) (Fig. 1). Pooled estimates are fisherywide, 

 across all areas. Standard errors are indicated by error 

 bars. 



in area 1 to more than double that value ( 15.4 t per 

 set) in area 3 (Fig. 4). Mean discard for school sets 

 was estimated at 1.16 t per set pooled over areas, 

 ranging from 1.57 1 in area 1 to 0.97 1 in area 3. Mean 

 discard per set for dolphin sets was estimated at 0.06 

 t per set fisherywide. Implications of these results 

 for the fishery are discussed in another study 

 (Edwards and Perkins, in prep.). 



The coefficients of variation (CVs) for the estimates 

 of mean discard per school and dolphin sets (21% 

 and 33%, respectively) are much smaller than those 

 for the individual parameter estimates of a and ja 

 (Fig. 3). As noted in Methods, this is because esti- 

 mating mean discard per set (i.e. (\-p)ji) is a more 

 robust procedure than estimating the individual pa- 

 rameters. In the case of log sets, the CVs for the es- 

 timates of E[Y] and ji differ (Fig. 3), even though in 

 this case the model reduced to a NB distribution 

 where E[Y] = ,u. The CVs differ because in estimat- 

 ing variances for the individual parameter estimates 

 we used analytic approximations, while in estimat- 

 ing variances for mean discard, we used bootstrap 

 methods (see Methods). 



Where possible (i.e. log and dolphin sets), we esti- 

 mated variances using the analytic expression in Equa- 

 tion 10 (see Discussion) and found that the results 

 agreed with bootstrap estimates to within about 5%. 



Note that the fisherywide estimates for log and 

 school sets are not simply the average of the esti- 

 mates in each fishing area. This is because the num- 

 ber of sets in each area for which discard was re- 

 corded was not proportional to the actual number of 

 sets made in that area. This imbalance was an im- 

 portant reason for including geographic area in the 

 analysis. Nonproportional sampling was not a fac- 

 tor for dolphin sets, because the estimated discard 

 in that case was the same for all fishing areas. 



Discussion 



Estimating model parameters 



Our approach differs somewhat from that of Mangel 

 and Smith ( 1990), who used the NBAZ to estimate 

 the total biomass of a fish stock. In their analysis, 

 observations of catch from within a stock's geographic 

 range were modelled with the NB component, while 

 the probability mass at zero accounted for observa- 

 tions from outside the range. The sole parameter of 

 interest was the mean p of the NB component, and 

 fixed values were assumed for the mixing and shape 

 parameters p and a. They reduced the count data to 

 "presence-absence" and derived a likelihood for fj in 

 terms of that reduction. In contrast, we were inter- 



