16 
Fishery Bulletin 108(1 ) 
Table 1 
Definitions of variables and acronyms for models used to estimate relative efficiency from comparative fishing data. 
R tJ Random variable for catches obtained at the i’th paired-tow station by survey protocol j=c (control) or j=t (test) 
r |; Observation of R tJ 
R i R ic +R it 
R ljk Catches at the i’th tow station and &’th length class by survey protocol j 
R ik Total catch (from both vessels) at length class k from set i, R lch +R ltl; 
n Total number of paired-tow stations 
n l Number of length classes caught in the i’th pair of tows 
n* Total number of sets and length classes, n * = n , 
A Fish densities encountered at station i and tow j 
d, log(A lc /A, ; ) 
q ■ Probability an encountered fish is captured, j=c, t 
p Relative efficiency, p=q c /q t 
P log(p) 
p Probability a captured fish was caught by the control protocol 
D lt Tow duration at the z’’th paired-tow station by vessel j 
F ijt Subsampling fraction for length l fish 
z u Logit offset, Z u =\og(D ic F icl I D it F itl ) 
<j> Binomial over-dispersion 
o 2 Random effect variance 
CIs Confidence intervals 
GLIM Generalized linear model 
MLE Maximum likelihood estimation 
GLMM Generalized linear mixed model 
PQLE Penalized quasi-likelihood estimation 
CV Coefficient of variation 
VO Vessel-effect over-dispersed binomial model estimation 
VM Vessel-effect binomial model with random intercept for each set; marginal MLE 
VP Vessel-effect binomial model with random intercept for each set; PQLE 
VLO Vessel- and length-effects over-dispersed binomial model estimation 
VLM ( Vessel- and length-effects binomial model with random intercept for each set; marginal MLE 
VLP 1 Vessel- and length-effects binomial model with random intercept for each set; PQLE 
VLM 1S Vessel- and length-effects binomial model with random intercept and slope for each set; marginal MLE 
VLP is Vessel- and length-effects binomial model with random intercept and slope for each set; PQLE 
simultaneous survey approach. This is analogous to the 
common paired versus unpaired experiment situation 
(e.g., Devore, 1991). Pelletier (1998) reviewed estimation 
methods used in many vessel calibration experiments. 
The basic data obtained from paired-tow calibration 
studies are the catches R lJ obtained at the ith paired- 
tow station (i=l, ... ,n ) by survey protocols j=c (control) 
or j=t (test). Let A„ denote the fish densities encoun- 
tered at station i and tow j. These densities may be 
different because of small-scale spatial heterogeneity in 
stock densities. We assume that each tow catches fish 
with probabilities q c and q t which are the same from 
site to site (i.e., i), and that catches are Poisson random 
variables with means 
E(R lt ) = q t X lt = p,,and E(R ic ) = q c l u . = pp, exp(<5, ), (2) 
where 6 i = \og( X ic / A it ) . 
If both vessels encounter exactly the same stock densi- 
ties at each tow station, then <5 ( =0, i= 1, ... , n. 
When there is no spatial heterogeneity in stock den- 
sities, p can be estimated by using a Poisson general- 
ized linear model (GLIM; e.g., McCullagh and Nelder, 
1989). This is essentially the approach used by Benoit 
and Swain (2003), although they adjusted for extra- 
Poisson variability in the catches. There are 2 n ob- 
servations that can be used to estimate the n density 
parameters ( ju. ) and p. Pelletier (1998) used a similar 
approach, with a negative binomial mean-variance as- 
sumption, which is a type of Poisson over-dispersion. 
These approaches are complicated because the number 
of ju parameters can be large if many tow stations are 
sampled, and the situation is worse if there are length 
effects. 
A better approach for inferences about p (see section 
4.5 in Cox and Snell, 1989, and example 3.1 in Reid, 
1995) when catches are Poisson random variables is 
to use the conditional distribution of R ; , c , given R t = 
R ic +R it . Let r- be the observed value of i?-. The condi- 
tional distribution is binomial with a probability mass 
function 
