NOTE Millar: Incorporation of between-haul variation 



565 



variation is binomial. When between-haul variation is 

 present the approximate chi-square distribution no 

 longer holds because the replicates across different size 

 classes are then not independent. Nonetheless, the es- 

 timator provides a correction to the standard errors 

 that incorporates both between-haul variation and 

 within-haul variation. 



If between-haul variation is of specific interest then 

 fits to individual haul data are required. Fryer (1991) 

 and Reeves et al. (1992) modelled between-haul vari- 

 ability by permitting parameters a and b of the logis- 

 tic curve (1) to vary between hauls according to a bi- 

 variate normal distribution which can be estimated 

 from the individual haul fits. Suuronen et al. (1991) 

 and Suuronen and Millar (1992) regressed the esti- 

 mated /,,,'s for individual hauls against their catch sizes, 

 and in four of five separate selectivity trials a decrease 

 in / 50 with catch size was indicated, though only one of 

 these was statistically significant at the 5% level. These 

 regressions used weights given by the inverse of the 

 estimated variance of the individual haul / 50 's. 



In the next section it is shown that neither the indi- 

 vidual haul or combined hauls data from a scallop 

 dredge selectivity study could be adequately modelled 

 by any of the above mentioned approaches and that 

 extreme between haul variation was present. A non- 

 parametric analysis of the combined hauls data was 

 implemented and between-haul variability was incor- 

 porated into the estimates of reliability through 

 bootstrapping. The approach assumed 



Al) the selection curve r(l) is a nondecreasing func- 

 tion of/. 



In addition, being a combined hauls approach, it was 

 also assumed that 



A2) the selectivity tows were representative of tows 

 on the fishery. 



Material and methods 



Selectivity trials 



Selectivity trials were performed onboard the 82-m 

 stern trawler Gadus Atlantica during the last week of 

 August 1991 as part of an Iceland scallop iChlamys 

 islandica) biomass survey for the St Pierre Bank (off 

 the South coast of Newfoundland). The objectives of 

 the study were 1) to summarize the retention proper- 

 ties of the survey dredge by estimating the shell heights 

 Z 25 , / 50 and l n corresponding to 25%, 50% and 75% re- 

 tention, and 2) to estimate the survey dredge's percent 

 retention (by meat weight) of commercial sized (>60 mm 

 shell height) scallops. 



The survey dredge was a 3.66-m (12-ft) wide off- 

 shore scallop dredge with belly constructed from 3 inch 

 (inside-diameter) metal rings joined together with metal 

 links. One selectivity tow was performed at each of 

 ten locations randomly chosen within the survey area. 

 For these tows, shrimp netting covers (35 mm inside 

 mesh opening) were attached behind the dredge, and 

 chafing gear was used under the bottom cover. The 

 covered dredge was towed over the distance (1.0 nauti- 

 cal mile) and at the speed (3.0kn) used in regular 

 biomass survey tows. The contents of the dredge and 

 covers were separately dumped, carefully picked over, 

 and all Iceland scallops were removed. The scallop catch 

 was weighed and a representative sample of between 

 20 and 40kg (200-400 scallops), or the entire scallop 

 catch if less than 20 kg, was taken for measurement. 

 Each scallop in the sample was measured to the near- 

 est millimetre in shell height. The catch weights were 

 then used to estimate the size frequencies for the en- 

 tire catch in the dredge and covers. 



Selectivity analysis 



We had planned to perform parametric analyses of the 

 individual haul and combined hauls data using the 

 standard maximum likelihood (McCullagh and Nelder, 

 1989) theory of the binomial model to choose the most 

 appropriate form of the selection curve from those dis- 

 cussed above. However, as seen in the Results section, 

 neither the individual haul nor combined haul data 

 were amenable to parametric analysis. 



Although it may not be possible to specify a parsi- 

 monious parametric form for the selection curves, it is 

 at least reasonable to insist that they be nondecreasing. 

 That is, the larger a scallop, the greater its chances of 

 being retained in the dredge. The nonparametric sta- 

 tistical technique of isotonic regression fits non- 

 decreasing curves to data. When the data are binomi- 

 ally distributed then the isotonic regression curve is 

 the maximum likelihood fit to the data (Barlow et al., 

 1972, p. 38). 



Isotonic regression curves are piecewise linear and 

 can be fitted in an intuitive way using the PAV (pool 

 adjacent violators) algorithm (Barlow et al. 1972, 

 p. 13). In this application, the essence of the PAV 

 algorithm is to pool adjacent size classes whenever 

 their observed retention proportions violate the non- 

 decreasing constraint. Isotonic regression views this 

 violation as an artifact due to insufficient numbers in 

 the "offending" size classes and so the pooling results 

 in a block of size classes having a common observed 

 retention proportion. 



Barlow et al. (1972) show that the isotonic regres- 

 sion curve is unique and does not depend on the order 



