Cadigan and Dowden: Statistical inference about the relative efficiency of a new survey protocol 
17 
probi R IC = x\R, =r l )~ 
p x (l-p) r ' 
(3) 
where p = p/(l+p) is the probability a captured fish is 
taken by the control vessel. 
The only unknown parameter in this distribution is p. 
The n nuisance p parameters are eliminated in Equation 
3. There are n conditional observations that can be used 
to estimate p. For the binomial distribution E(R lc )=r i p 
and Var(R ic )=r j p(l-p). This approach is commonly used 
in commercial fishing gear size-selectivity studies (e.g., 
Millar 1992). 
Paired-tow experiments do not eliminate spatial het- 
erogeneity between the stock densities fished by each 
vessel. This heterogeneity leads to Poisson over-disper- 
sion which has to be properly accounted for to provide 
reliable statistical inferences. Similarly, the relative 
efficiency may vary somewhat from site to site and this 
must also be accounted for. It is well-known in fishing 
gear selectivity studies that not accounting for over- 
dispersion and correlation leads to confidence intervals 
that are too narrow and spurious statistical significance 
(Fryer, 1991; Millar et al., 2004). 
An approach to deal with over-dispersion is to use 
quasi-likelihood (e.g., McCullagh and Nelder, 1989) 
with a Poisson over-dispersion parameter <j>, VarfR y )= 
0 EiRjj), or a binomial over-dispersion parameter, 
Var(i?- e li? ! =r J )= <p r^pd-p). Confidence intervals (CIs) 
are adjusted based on an estimate of <j>. This was the 
approach used by Benoit and Swain (2003) to account 
for extra-Poisson variation, and Lewy et. al. (2004) to 
account for extra-binomial variation. Benoit 1 observed 
that using an over-dispersion parameter did not com- 
pletely account for the true variability in the data and 
too often led to the false statistical conclusion that p 
yd. Benoit used randomization approaches to test for 
statistical significance of vessel effects. We consider 
this approach further in the Discussion section. 
A reasonable assumption for spatial heterogeneity 
in stock densities is that k ie and k it in Equation 2 are 
independent and identically distributed gamma random 
variables with means A- and variances tA 2 ; . If R jc \k ic 
and R it \k it are Poisson distributed, then the marginal 
distributions of R ic and R it are negative binomials (e.g., 
see Cameron and Trivedi, 1998). This distribution is 
often suggested to be appropriate for modeling the vari- 
ability of measurement error in trawl survey catches 
(e.g., Gunderson, 1993). This implicitly provides a ra- 
tionale for assuming stock densities are gamma dis- 
tributed. Dowden 2 showed that r=0.049, 0.223, 0.372 
1 Benoit, H. P. 2006. Standardizing the southern Gulf of 
St. Lawrence bottom trawl survey time series: Results of 
the 2004-2005 comparative fishing experiments and other 
recommendations for the analysis of the survey data. DFO 
Can. Sci. Advisory Secretariat Res. Doc. 2006/008. [Available 
from http://www.dfo-mpo.gc.ca/csas/csas/publications/res- 
docs-docrech/2006/2006_008_e.htm, accessed April 2009.] 
corresponds to Var(6 ; ) = 0.1, 0.5, and 0.9, and that the 
distribution of 6 (see Eq. 2) is well approximated by 
a normal distribution with o 2 =Var(6 ( ). In this case a 
generalized linear mixed-effects model (GLMM; e.g., 
McCulloch and Searle, 2001) can be used to estimate p 
and account for spatial heterogeneity in stock densities. 
GLMMs contain both fixed and random effects, and 
usually the random effects are assumed to have normal 
distributions. We refer to models with no random ef- 
fects as fixed effects models (e.g., GLIMs). 
GLMMs are frequently used to account for heteroge- 
neity in fishing gear size-selectivity data (e.g., Fryer, 
1991; Fryer et al., 2003; Millar et al., 2004). Fryer et 
al. (2003), Cadigan et al. 3 and Holst and Revill (2008) 
used GLMMs with paired-tow calibration data. Cadigan 
et al. 3 compared models with and without random ef- 
fects for vessel calibration data for seven species, and 
suggested that GLMMs provided results that were more 
reliable. Cadigan et al. 3 concluded that vessel effects 
were not significantly different from zero; however, dif- 
ferent conclusions could be drawn from some of their 
GLIM results. 
There are a variety of approaches available for fitting 
GLMMs. A common approach is marginal maximum- 
likelihood estimation (MLE), which is limited in the 
complexity of random effects that can be accommodated. 
A more flexible approach is penalized quasi-likelihood 
estimation (PQLE). Bolker et al. (2009) provided some 
advantages and disadvantages of these methods. They 
also provided many references, including some for soft- 
ware packages. In some situations, PQLE is known to 
produce biased estimates of fixed-effect parameters 
like p. 
In this article we extend the analyses for one of 
the stocks considered by Cadigan et al. 3 . By means 
of simulation studies we examine which of the ap- 
proaches — the GLIM, GLMM with marginal MLE, or 
GLMM with PQLE — provides more reliable statistical 
inferences about p. We focus on the bias in estimates 
of p, on the accuracy of CIs, and on the power to de- 
tect if pA 1 (i.e., a true difference in catchabilities 
between vessels). Our purpose is to recommend the 
most reliable approach, at least for paired-tow sur- 
vey calibration studies similar to those in Cadigan et 
al. 3 . We focus on methods to accommodate within-pair 
variations in stock densities, but our methods are also 
applicable when there is between-set variations in rela- 
tive efficiency. 
2 Dowden, J. J. Generalized linear mixed effects models with 
application to fishery data. M.A.S. practicum report, 128 
p. Memorial Univ. Newfoundland. St. John’s, Newfoundland 
and Labrador, Canada. 
3 Cadigan, N. G., S. J. Walsh, and W. Brodie. Relative 
efficiency of the Wilfred Templeman and Alfred Needier 
research vessels using a Campelen 1800 shrimp trawl in 
NAFO Subdivision 3Ps and Divisions 3LN. DFO Can. Sci. 
Advisory Secretariat Res. Doc. 2006/085. [Available from 
http://www.dfo-mpo.gc.ca/csas/Csas/Publications/ResDocs- 
DocRech/2006/2006_085_e.htm, accessed April 2009.] 
