Somerton and Munro: Bridle efficiency of a survey trawl for flatfish 
645 
For each size class, fitting Equation 5 to the herding da- 
ta required estimation of N+l parameters, where N is the 
number of blocks sampled (i.e. a unique value of k for each 
block and a common value of h for all blocks). Because the 
model is nonlinear, because of the product of k and h, it 
was fitted to the catch and trawl measurement data by us- 
ing nonlinear regression (Venables and Ripley, 1994). Fish 
length classes used in the calculations were chosen such 
that the number of length observations was approximate- 
ly equal among classes. Ten length classes were usually 
chosen, but for species with narrow length ranges, as few 
as eight were used. After fitting the model, estimates of 
the variances of the parameters were obtained from the 
inverse of the information matrix (Seber and Wild, 1989). 
Bridle efficiency ( k h ) for each size class was calculated 
from the estimated value of the herding coefficient (h) by 
using the relationship 
One implicit assumption of the herding model is that h 
is constant for all three bridle lengths. We tested the va- 
lidity of this assumption by comparing the goodness of fit 
of a modified version of the herding model to that of the 
original. In the modified model, the herding coefficient was 
represented as 
h t = h+ cLL, 
where h l = the herding coefficient at bridle length L; 
h = the mean herding coefficient for all bridle 
lengths; 
d = a free parameter; and 
L = a coded value of bridle length equal to — 1, 0, 1 
for the short, medium, and long bridles (with 
this notation, negative values of d indicate 
that h t declines with increasing bridle length). 
K 
Woff ) 
(W rf -w„)J- 
( 6 ) 
Although up to this point h was treated as a constant for 
all bridle lengths, k b is not a constant because the propor- 
tion of the bridle path in contact with the bottom will vary 
with bridle length. Therefore k b was evaluated only for the 
standard bridle length. Variance of k b was approximated 
as 
Var(k b ) = 
i- f 
(W d -W n )J 
Var(h). 
(7) 
Although the model that we used allows for length- 
dependent herding, herding may not be a length-depen- 
dent process in all species. We performed a simple test 
of linear length-dependency by regressing the estimated 
values of k b on the midpoints of the associated length in- 
tervals. For species in which the slope was not significant- 
ly different from zero, the herding model was refitted to 
the data with all length intervals combined. For species in 
which the slope differed significantly from zero, we calcu- 
lated a functional relationship between k b and fish length. 
Among the possible relationships examined, the best fit 
was obtained with the exponential model 
k b = a + b exp {cL), (8) 
where a, b, c are parameters; and 
L = the midpoints of the length classes. 
Model fitting was done by using nonlinear regression. 
Variance of the predicted value of k b was estimated with 
bootstrapping (Efron and Tibshirani, 1993), where the 
bootstrap samples were obtained by sampling the resid- 
uals from the regression, with replacement, and adding 
them to the predicted values of k b . The functional relation- 
ship between the standard deviation of k h and fish length 
was approximated by using a parabolic model fitted by 
linear regression. 
Because this is a comparison of two nonlinear models with 
differing numbers of parameters, we used as a measure 
of fit the penalized sum of squares; that is, RSS/(n-2p), 
where RSS is the residual sum of squares, n is the number 
of data values, and p is the number of parameters (Hilborn 
and Mangel, 1997, page 115). A linear change in h t is con- 
sidered significant when the penalized sum of squares for 
the modified model is less than that for the original model. 
Results 
Herding experiments 
Fifteen blocks were completed in the Bering Sea herding 
experiment; 19 blocks were completed in the West Coast 
herding experiment. Although the intention of the experi- 
ments was to maintain a constant trawl geometry at all 
bridle lengths, in both experiments wing spread decreased 
0.9 m and bridle angle decreased an average of 2.5% from 
the shortest to the longest bridles (Table 1). The experi- 
mental change in bridle length had a pronounced effect on 
the area swept by the bridles. In the Bering Sea experi- 
ment, for example, total bridle length (i.e. bridle plus tail- 
chain length) was increased by 143% from the shortest to 
the longest bridle, which, in turn, produced a 64% increase 
in door path width ( W d ) and a 113% increase in total bridle 
path width (W d - W n ). However, this increase in bridle 
length also produced a 930% increase in the bridle contact 
path width (W d - W n - W o ^; Table 1) because only a small 
portion of the bridle remained in contact with the bottom 
at the shortest bridle length. 
Catch per area swept by the net increased significantly 
with increasing bridle length for six of the seven species 
(rock sole, yellowfin sole, flathead sole, Dover sole, Eng- 
lish sole, rex sole; weighted linear regression, P<0.010) 
indicating strong herding by the bridles (Fig. 2). For Pa- 
cific sanddab, however, the increase was not significant 
(P-0.096). 
Fish length changed significantly (P<0.05) with increas- 
ing bridle length for only two species (English sole and Pa- 
cific sanddab; Table 2) and in both cases smaller fish were 
