18 
Fishery Bulletin 108(1 ) 
Materials and methods 
We focus on statistical inferences for p (i.e., Eq. 1) based 
on data obtained from paired-tow vessel calibration 
studies like those described in Cadigan et al. 3 . Briefly, 
in their study, data from paired-tows were collected 
to quantify potential differences in the catchabilities 
of two research survey vessels fishing with the same 
trawl and other protocols. Ranges of catch sizes, fish 
sizes in the catch, and tow depths were sought for the 
distributions of the species likely to be encountered. 
Tow stations were selected randomly as part of research 
surveys. High density aggregations were not specifically 
targeted because information was required on differ- 
ences in catchability when stock densities were both 
high and low — a variability in densities that typically 
occurs in research surveys. The full details of this cali- 
bration study are given in Cadigan et al. 3 . We use their 
data on witch flounder ( Glyptocephalus cynoglossus) as 
a case study to illustrate methods. 
The focus in Cadigan et al. 3 was on the relative ef- 
ficiency of two vessels fishing with otherwise identical 
protocols (gears, speed, tow duration, etc.). Hence, in 
this article we refer to vessel effects, but more generally 
the effects relate to differences in fishing protocols. 
The first step in analyzing calibration data is to ex- 
amine whether there is an effect on total catch per set. 
In the next section we describe a model for this pur- 
pose. Effects on the length compositions of the catches 
are considered later in this article. 
log 
( \ 
Pi 
1 ~Pi 
= P + 8 it 
(5) 
Usually it is reasonable to assume that the ratio of stock 
densities varies randomly from site to site. Earlier we 
claimed it was reasonable to assume 8 { ~N( 0, a 2 ), i= 1, 
... n. In this case equation 5 defines a standard GLMM 
and there are many approaches and software packages 
available to estimate (3 and o 2 (e.g., Bolker et al., 2009). 
We examined the robustness of statistical inferences 
about p to the normal approximation for d (see Simula- 
tions section) when 8 is actually a log ratio of gamma 
random variables. 
We used two different packages to estimate the 
GLMMs. The first was SAS/STAT PROC NLMIXED, 
which fits nonlinear mixed models, including binomial 
logistic regression, using marginal MLE. We refer to 
this model and estimation procedure as the VM (vessel 
effect and random set-effects binomial model with mar- 
ginal MLE) approach. The second was the more flexible 
SAS/STAT PROC GLIMMIX, which fits GLMMs using 
PQLE. We refer to this as the VP (vessel effect and 
random set-effects binomial model with PQL estima- 
tion) approach. We used the default estimation method 
in PROC GLIMMIX, which is a restricted pseudolikeli- 
hood estimation with an expansion around the current 
estimate of the best linear unbiased predictors of the 
random effects. Both packages provide Wald-type CIs 
for fixed-effect parameters such as /3. 
Vessel effect 
Vessel and fish-length effects 
A common approach used for analyzing comparative 
fishing data is binomial regression with an adjust- 
ment for over-dispersion. This is one of the options we 
considered. In the conditional binomial model defined 
by Equation 3, the logit function of the binomial prob- 
ability ip) is 
log 
= log ip) = P, 
(4) 
and j 6 can be estimated as the intercept with a logit 
link function by using software for binomial regres- 
sion. The range of f3 is (-oo, oo). We derived CIs for p by 
exponentiating intervals for /3 and therefore CIs for p 
should have better coverage properties, and they at least 
would never include infeasible values. We used version 
9.1.3 of SAS/STAT (SAS, Cary, NC.) PROC GENMOD 
software to estimate this model, and we used the option 
(dscale) that estimates $ as the deviance divided by the 
degrees of freedom. We also selected the option (lrci) that 
provides two-sided CIs based on the profile likelihood 
function. We refer to this GLIM model and estimation 
approach as the VO (vessel-effect binomial model with 
over-dispersion) approach. 
If there is spatial heterogeneity in stock densities, 
then the model for the logit proportion of catch taken 
by the control vessel at station i is 
Length effects are expected if there is a change in the 
survey trawl, but they could also occur with only a 
change in the survey vessel. Length-based models for 
paired-tow comparative fishing data are straightforward 
extensions of the models in the previous section. The data 
are extended to include the paired catches at length, R ljk , 
i=l, ... , n;j= 1,2; k=l, ... n ; , where n t is the number of 
length classes caught in the fith pair of tows. 
If it is reasonable to assume that there are no be- 
tween-pair differences in the length distributions of 
fish encountered by both vessels, then binomial logistic 
regression models are appropriate. Usually the effect 
of length will be such that relative efficiency changes 
monotonically with length, l. If the change is linear 
in fi=log(p), then a binomial GLIM with a logistic link 
can be used to estimate the intercept and slope; that is, 
13=13(1) in Equation 4, where is taken to be a 
function of length, /3(Z). If the length effect is more com- 
plicated, then alternative models may be required (see 
Fryer et al., 2003; Holst and Revill, 2008); however, in 
this article we focus only on linear models. 
If there is spatial heterogeneity in stock densities, 
then the situation is more complicated. If the heteroge- 
neity is such that one vessel encounters more fish than 
the other, but otherwise the length distributions are 
the same, then the use of a random intercept binomial 
GLMM is appropriate: 
