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Fishery Bulletin 91(3), 1993 



in which violators are pooled. The PAV algorithm can 

 be implemented on computer as follows: Initially, treat 

 all size classes as blocks of size 1. At each step of the 

 algorithm there is an active block which is compared 

 with the adjacent block in the active direction. The 

 latter will be denoted L or R for active directions left 

 and right, respectively. The initial active block and 

 active direction are the smallest size class and R, re- 

 spectively. The smallest size class is deemed to satisfy 

 the nondecreasing constraint in active direction L. The 

 PAV algorithm proceeds as follows: 



Bl) If comparison in the active direction results in 

 violation of the nondecreasing constraint, then 

 the two blocks are pooled to form a larger active 

 block and the active direction becomes (or re- 

 mains) L. 



B2) If comparison in the active direction does not 

 violate the nondecreasing constraint then the ac- 

 tive direction becomes (or remains) R. In addi- 

 tion, 



• if the active direction was L then the active 

 block remains the active block. 



• if the active direction was R then the active 

 block becomes the next block on the right. 



After a finite number of steps, the algorithm termi- 

 nates when the rightmost block is active and R is the 

 active direction. 



If the observed retention proportions are non- 

 decreasing for increasing /, then the isotonic regres- 

 sion curve is simply given by connecting all the pro- 

 portions together with straight line segments. If the 

 isotonic regression curve fitted to the observed reten- 

 tion proportions is flat (corresponding to a pooled block) 

 at 0.25, 0.50 or 0.75, then the estimated / 2S , l- M , or l ls is 

 given by the shell size that is the midpoint of that 

 pooled block. 



For this study, published FORTRAN code (Cran, 

 1980) for implementation of the PAV algorithm (Barlow 

 et al., 1972) was interfaced to the Splus statistical 

 package. 



Isotonic regression does not provide an estimate of 

 the standard errors of the estimated / 2S , l 50 , and l 7S . 

 These were obtained by bootstrapping the data (Efron, 

 1982). To this end, the individual selectivity hauls were 

 used to define a "population" of hauls. To include be- 

 tween-haul variability, the bootstrap resamples (with 

 replacement) from this population. Within each 

 resampled haul the retention proportions were also 

 bootstrapped to include within-haul variability. That 

 is, for each size class, the bootstrapped retention pro- 

 portion was the proportion of dredge caught scallops 

 in a sample taken with replacement from the captured 



(in dredge and covers) scallops of that size. The boot- 

 strap samples were the same size as those represented 

 in the data. For example, in haul 1 there were 48 

 scallops of 70-mm shell height, of which 28 were caught 

 in the dredge. For this size class, a bootstrap sample 

 of 48 scallops was taken by sampling with replace- 

 ment from the 48 scallops whenever haul 1 was se- 

 lected for the bootstrapped combined hauls. 



The above resampling scheme was performed 200 



times and on each occasion / 25 , l 5 



and l 7S were esti- 



mated from isotonic regression fits to the combined 

 hauls data, and percent retention (by meat weight) of 

 commercial sized scallops was calculated by using the 

 shell height to meat weight relationship given in Naidu 

 (1991). 



Results 



The first four tows were taken over a relatively smooth 

 bottom consisting mainly of small stones and pebbles. 

 The remaining six tows were taken over a rougher 

 bottom consisting of larger stones, rocks and boulders. 

 The data from tow 5 were discarded owing to a torn 

 cover. The proportion of commercial-sized scallops was 

 lower in hauls 1-4 (58%) than in hauls 6-10 (81%). 

 The weight of trash (rocks, sea cucumbers, starfish, 

 etc.) exceeded the weight of scallops in every haul, 

 particularly so in hauls 6-10. A complete summary of 

 the hauls can be found in Millar and Naidu ( 1991 ). 



The replication estimate of dispersion, calculated over 

 size classes with a total combined catch of at least 10 

 scallops, was 828 on 480 degrees of freedom. Under 

 the null hypothesis, H : (No between-haul variation 

 and binomial within-haul variation! the estimator has 

 an approximate chi-square distribution, hence H n is 

 rejected with P-value <10 6 . Binomial variation within 

 hauls should be a reasonable assumption for scallops, 

 so rejection of H„ suggests significant between-haul 

 variation. 



Results of parametric analysis 



Figure 1 shows, for each successful haul and combined- 

 over hauls, the proportions of the covered dredge's catch 

 of scallops that were in the dredge. These retention 

 proportions can be extremely variable, especially for 

 the smaller scallops, because of the low numbers en- 

 countered. Logistic curves fitted to these retention pro- 

 portions are shown as dashed lines. The residual plots 

 show that the fitted logistic curves are inadequate. 

 This is particularly true for the combined hauls data. 

 (The residuals plotted in Fig. 1 are deviance residuals, 

 as defined by McCullagh and Nelder [1989, p.39].) 



