Incorporation of between-haul 

 variation using bootstrapping and 

 nonparametric estimation of 

 selection curves. 



Russell B. Millar 



Department of Mathematics and Statistics. University of Otago 

 PO. Box 56, Dunedin, New Zealand 



The most frequently used paramet- 

 ric description of trawl selectivity 

 is the logistic curve. If r(l) denotes 

 the retention probability of a length 

 / individual, the logistic selection 

 curve is specified as 



nh- exp(q + 6/) (1) 



1 + exp(a + bl) 



where a and b are parameters to be 

 estimated. Under this formulation 

 it can be seen that 6>0, because this 

 is a requirement for r(l) to increase 

 with /. Also, c<0 because we require 

 the retention probability of a length- 

 individual to be (effectively) zero. 

 A similar curve is provided by the 

 probit function (the cumulative 

 distribution function of the Normal 

 distribution) which has slightly 

 shorter tails than the logistic curve 

 (McCullagh and Nelder, 1989). Both 

 of these curves are symmetric about 

 the length at which retention is 

 50 f /r, which will be denoted by / so . 

 More generally, l x will denote the 

 length at which retention is x%. 



Some selectivity data suggest an 

 asymmetric selection curve. The 

 log-log curve (also known as the 

 Gompertz curve) and complimen- 

 tary log-log curve are two param- 

 eter asymmetric curves that can be 

 used in the analysis of count data 

 (McCullagh and Nelder, 1989). Al- 

 though Pope et al. (1975) mention 

 the log-log curve as a potential se- 

 lection curve, neither of these asym- 

 metric curves appears to have been 



used in published selectivity stud- 

 ies prior to this current study. 



Richards curves (Richards, 1959) 

 are three parameter curves that 

 generalize the logistic in the form 



r(l) = 



( exp(q + bl) 

 \l + exp(a + bl) 



(2) 



Parameter 8 controls the amount of 

 asymmetry with 8>1 or 0<8<1 giv- 

 ing longer tail to the left or right of 

 / 50 respectively, and 8=1 giving the 

 symmetric logistic curve. The au- 

 thor (Millar, 1991) has found that 

 the Richards curve will often pro- 

 vide an adequate fit to data in cases 

 where the logistic curve is clearly 

 inappropriate. 



Selectivity data are count data, 

 and in fitting selection curves to 

 these data it is usual to assume that 

 the counts are binomially distrib- 

 uted (McCullagh and Nelder 1989). 

 Within a single selectivity haul, the 

 binomial assumption is appropriate 

 if the fish encountering the gear be- 

 have independently. It is common 

 practice to fit a selection curve to 

 the data combined over all success- 

 ful hauls, and for the binomial as- 

 sumption to remain valid it is then 

 also necessary to assume that se- 

 lectivity does not vary between 

 hauls. This assumption is not valid 

 in general, owing in part to vari- 

 ables such as catch size and haul 

 duration. Gear saturation may oc- 

 cur for high catch sizes because of 



reduced selectivity in the latter part 

 of the tow caused by meshes becom- 

 ing clogged with fish (e.g., Suuronen 

 and Millar, 1992) or distorted by the 

 strain upon the gear. Hauls of 

 longer duration may increase selec- 

 tivity (e.g., Clark 1957) by allowing 

 fish more time to escape, notwith- 

 standing that the effect will be con- 

 founded with catch size. 



If between-haul variation is not 

 of primary interest, then fitting a 

 selection curve to the combined 

 hauls data remains a reasonable 

 approach because the estimated se- 

 lection curve parameters are quite 

 insensitive to violation of the bino- 

 mial assumption (McCullagh and 

 Nelder, 1989). The combined hauls 

 approach can be viewed as model- 

 ling the "average" r(l), where the 

 average is over the population of 

 all hauls that could be made on that 

 fishery. The selectivity hauls must 

 therefore be a representative 

 sample from this hypothetical popu- 

 lation. 



Between-haul variation does, 

 however, invalidate the estimates of 

 variability for the parameters of the 

 combined hauls fit. To correct for 

 this it is common to apply a good- 

 ness-of-fit based correction to the es- 

 timated standard errors (McCullagh 

 and Nelder, 1989), but Fryer (1991) 

 has demonstrated that this can un- 

 derestimate the effect of between- 

 haul variation. Suuronen and Millar 

 ( 1992) corrected the standard errors 

 by using the replication estimator 

 of dispersion (McCullagh and 

 Nelder, 1989, p. 127). This is a non- 

 parametric estimator that is analo- 

 gous to the pure error sums of 

 squares estimator of linear regres- 

 sion analysis (Myers, 1990). The 

 replication estimate of dispersion 

 has an approximate chi-squared dis- 

 tribution when there is no between- 

 haul variation and within-haul 



Manuscript accepted 27 April 1993. 

 Fishery Bulletin: 91:564-572 (1993). 



564 



