NOTE Millar: Incorporation of between-haul variation 



569 



Fits of the complimentary log-log, log-log, and 

 Richards curve were also used on the individual haul 

 data. The complimentary log-log curve fits were mar- 

 ginally better than the logistic fits but were still clearly 

 inadequate. The log-log curve fits displayed worse re- 

 sidual structure than the logistic fits. This is because 

 the log-log curve has a longer tail to the right of Z 50 , 

 whereas the data suggest a longer tail to the left. The 

 three parameter Richards curve fits provided a big 

 improvement and, though very hard to judge, fits to 

 about half of the individual hauls appeared to be 

 adequate. 



Figure 2 shows the combined hauls data fits of the 

 logistic, complimentary log-log, log-log, and Richards 

 curves. The Richards curve fit is the only one that 

 could possibly be considered adequate, though there is 

 an obvious clustering of negative residuals for shell 

 heights between 61mm and 71mm. Since this group 

 of residuals contains the estimated value of Z 50 

 (66.8 mm) it is of some concern, and the fit was deemed 

 to be inadequate. (One might consider performing a 

 run's test (say) for independence of the residuals. How- 

 ever, the run's test would be very approximate since it 

 assumes that residuals will be positive or negative 

 with equal probability 0.5. This is not the case for 

 these data, especially for the very small and very large 

 size classes, even when the model is correct.) 



Nonparametric analysis 



The nonparametric selection curve fits to the individual 

 haul and combined hauls data are overlaid on propor- 

 tion-retained plots in Figure 3 and the corresponding 

 estimated sizes of 25%, 50%, and 75% retention are 

 given in Table 1. Note that the flat portions of the 

 curves (Fig. 3) correspond to size classes that were 

 pooled. Considerable variability in the estimated Z 50 's 

 is evident, the smallest being 45.3 mm (haul 7) and 

 the largest 80.1mm (haul 3). Figure 3 suggests that 

 the estimated Z 50 for haul 7 may be very unreliable — 

 there were relatively few scallops less than 70 mm in 

 this haul and the observed retention proportions of 

 the smaller scallops are extremely variable because of 

 the low numbers caught. 



The estimates of Z 25 , Z 50 , and Z 76 from the combined 

 hauls fit were 50.5, 69.4, and 77.3 mm, respectively. 

 The percentile method (Efron 1982, p. 78), was used to 

 determine approximate confidence intervals from the 

 bootstrap. This gave 95% confidence intervals for Z 25 , 

 l m , and Z 75 of 21.0-53.8 mm, 66.1-72.5 mm and 73.6- 

 80.5 mm, respectively. The extremely large confidence 

 interval on Z 2S reflects the paucity of data for the smaller 

 scallops. Retention by meat weight of commercial sized 

 scallops was estimated to be 73%- with a 95% confi- 

 dence interval of 63%-82%.. 



Discussion 



Bootstrapping (resampling) the experimental units (se- 

 lectivity tows) is a natural way to emulate the effect of 

 between-haul variability. In doing so, one requires an 

 automated procedure for fitting a selectivity curve to 

 the bootstrapped combined hauls data. Isotonic regres- 

 sion is well suited to this task because the selection 

 curve for the bootstrapped combined hauls will always 

 satisfy assumption Al. In contrast, parametric selec- 

 tion curves may not be sufficiently flexible to adequately 

 fit all the possible bootstrapped combined hauls data 

 sets. 



One Referee of this paper made the interesting sug- 

 gestion that it may not matter that parametric fits to 

 the combined hauls data or bootstrapped combined 

 hauls could be inadequate, because the bootstrap pro- 

 cedure should nonetheless be applicable and any prob- 

 lems with the fits would be indicated by wide confi- 

 dence intervals or indications of bias (e.g., Efron, 1982, 

 p. 33). To investigate this, Richards curves were fitted 

 to the same bootstrapped combined hauls used in the 

 nonparametric analysis. The combined hauls fit had 

 Z 25 , Z 50 , and Z 75 of 48.5 mm, 66.8 mm, and 77.6 mm, re- 

 spectively, and the 95 f # confidence intervals obtained 

 from the bootstrap were 33.3-56.6 mm, 59.4-71.5 mm 

 and 74. 1-80.3 mm, respectively. These confidence in- 

 tervals have widths of 23.3, 12.1, and 6.2 mm, respec- 

 tively, compared with 32.8, 6.4, and 6.9 mm from the 

 nonparametric fits. The percent retention value and 

 its confidence interval were the same as for the non- 

 parametric analysis. However, the more subtle and pos- 

 sibly more relevant consequence of bootstrapping with 

 a parametric curve is that the bootstrap indicates the 

 ability of the combined hauls parametric fit as an esti- 

 mator of the parametric fit to the entire hypothetical 

 population of tows on the fishery. The latter may not 

 be adequately modelled by a parametric curve — 

 but the bootstrap will not consider this. The isotonic 

 curve can not suffer this deficiency since the selection 

 curve for the entire hypothetical fishery will be non- 

 decreasing. 



It was assumed that the selectivity tows were repre- 

 sentative of survey tows on the scallop fishery (as- 

 sumption A2). The selectivity gear used here was a 

 covered survey dredge and it was deployed under sur- 

 vey conditions on a random subsample of survey loca- 

 tions. Assumption A2 will therefore by reasonable pro- 

 vided that the covers on the dredge did not have 

 significant impact on its selectivity. To address this 

 question Millar and Naidu (1991) compared the catch 

 in the covered dredge with that in an uncovered dredge 

 that was towed simultaneously. There was evidence to 

 suggest a possible cover effect for scallops below about 

 58-mm shell height. This is unlikely to affect the 



