Munro: Correcting relative fishing power differences in trawl survey data 
539 
follows, “vessel” refers to the entire system that com- 
prises the sampling instrument, from the bow of the 
fishing vessel to the codend.) 
A fishing power difference forces a choice between 
two estimators of mean CPUE, one that incorporates 
a fishing power correction and one that does not. 
Correcting a fishing power difference amounts to 
changing an observation to an estimate. The follow- 
ing model is used to estimate what a standard ves- 
sel would have caught had it executed exactly the 
same tow as was done by a nonstandard vessel: 
x t = y t FPC, 
where x, = estimated CPUE at station i by the stan- 
dard vessel; 
y . = observed CPUE at station i by the non- 
standard vessel; and 
FPC = estimated fishing power correction factor. 
Each original observation, y L , has no error beyond 
measurement error. Every x, has error due to the 
variance of the estimate of FPC. Consequently the 
mean CPUE estimated from the x t has at least two 
components of variation, one stemming from the 
usual sampling variance in the observations (y ( ), the 
other due to uncertainty in the estimate of FPC. This 
added component of estimation error has been rec- 
ognized as the cost of correcting systematic error in 
the observations, but only in passing, (Sissenwine 
and Bowman, 1978; Byrne et al., 1981; Koeller and 
Smith, 1983; Fanning, 1984). 
Estimators are often chosen on the basis of their 
relative error, yet most researchers have ignored this 
in deciding whether or not to correct a fishing power 
difference. Investigations have been focused on esti- 
mating the difference but have not evaluated it in 
terms of estimating mean CPUE. Very few decision 
rules have been explicitly stated, most having been 
implied by testing the statistical significance of the 
fishing power difference itself. Early CPUE calibra- 
tions (Gulland, 1956; Robson, 1966) were based on 
multiplicative models of different sources of variabil- 
ity in CPUE data, including vessel effects. Log trans- 
formation of the data produced linear models with 
coefficients that could be estimated with regressions 
and for which classical hypotheses could be formu- 
lated and tested. Sissenwine and Bowman (1978), 
Kimura (1981), and Gavaris (1980) have followed this 
strategy in their decisions to apply FPCs. Gavaris 
(1980) estimated the FPC using the method of Bradu 
and Mundlak (1970) and reported approximate con- 
fidence intervals, without explicitly stating a deci- 
sion rule. Fanning (1984) proposed an explicit deci- 
sion rule using a beta-distributed index for the fish- 
ing power difference in paired observations. If the 
confidence interval included the value that repre- 
sented identical fishing power, he recommended that 
the estimated FPC not be applied. Byrne and Fogarty 
( 1985) tested the significance of fishing power differ- 
ences using Hotelling’s Usquared test when several 
species were considered simultaneously, or the non- 
parametric Friedman’s test for a single species. They 
offered no interpretation of a significant fishing 
power difference, in particular, whether or not it 
should be corrected. The Utest was used by Byrne et 
al. (1991) to determine the significance of a fishing 
power difference. In response to a significant differ- 
ence, they estimated an FPC using the method of 
Bradu and Mundlak ( 1970). They then produced con- 
fidence intervals for that estimate using a bootstrap 
approximation. However, they did not state an ex- 
plicit decision rule based on those intervals. 
Correcting a fishing power difference would be 
worthwhile only when it reduces the error in the es- 
timate of mean CPUE. Statistical significance of a 
fishing power difference is not a compelling justifi- 
cation because the cost of the added uncertainty may 
out weigh the benefit of removing bias that entered 
through systematic error in CPUE data. If the esti- 
mate of a correction factor has a lot of uncertainty, 
then the error of the estimate of mean CPUE could 
actually become worse by correcting data, even for a 
statistically significant fishing power difference. A 
decision rule for correcting a fishing power differ- 
ence must avoid this mistake by accounting for the 
cost of correcting as well as the benefit. Such a rule 
would permit choosing the estimate of mean CPUE 
that yields the lower total error. 
Methods 
The notion of the mean square error (MSE) lends a 
useful structure for defining such a decision rule. The 
MSE is a widely recognized measure of error between 
an estimator and its parameter (Mood et al., 1974). 
The MSE is defined as 
MSE [C] = E 
( c-cr 
which is the expectation of the squared difference 
between the estimator of mean CPUE, C, and the 
parameter being estimated, true CPUE or, C. The 
MSE can be rewritten as 
MSE [C] = Var [C] + b 2 [C] , 
or the sum of the variance and the squared bias of 
the estimator. By defining the following estimators, 
