Cadigan and Dowden: Statistical inference about the relative efficiency of a new survey protocol 
19 
log 
Pik 
1 ~ Pik 
- A) + A + P\hk > 
( 6 ) 
iid 
8 ■ ~ N( 0,cr ), i = 1 k = 1 
where p lk = the probability that at site i a 
length l [k captured fish came 
from the control vessel. 
Holst and Revill (2008) used a random 
intercepts model, although their models 
for fixed lengths effects were more com- 
plicated than what we consider. However, 
if there are differences in the length dis- 
tributions encountered by both vessels 
then Equation 6 will not be appropriate. 
The differences will usually be such that 
d=log(A c /A ( ) varies smoothly with length. 
Several hypothetical examples are also 
shown in Figure 1. This type of spatial 
heterogeneity can be approximated by lin- 
ear functions, whose slopes (5j) and inter- 
cepts (6 0 ) vary randomly from set to set. 
A GLMM for this model is 
log 
Pik 
Pik 
iid 
- A)+Ao +( A+AiA*> 
~ N(0,Oj ), i = l,...,n; 
j - 0,1; A = 1, 
(7) 
This is a common GLMM used in fishing 
gear selectivity studies. If the means of the 
densities are the same and the only dif- 
ference is the height of the distributions, 
then the b log ratio would be a horizontal 
line in Figure 1, which is the type of effect accounted 
for in Equation 6. 
We used the same SAS software to estimate the 
length-based GLIMs and GLMMs. We denote the 
length-based model with no random effects as VLO. 
Mixed-effects random intercept models are denoted as 
VLj, (i.e., Eq. 6), and models with random intercepts 
and slopes are denoted as VL is . Models and estimation 
methods are denoted as VLMj, VLP ; , VLM 1S , and VLP is 
(see Table 1) depending on whether marginal MLE or 
PQLE is used. 
Standard errors for /3(Z) = /3 0 +/3 1 Z=log|p(Z)( can be con- 
structed from the estimates of /3 0 and /3 l5 and their 
estimated covariances: 
SE|j8(Z)J = |var(^ 0 ) + 2cov(^ 0 ,4i)Z + var(/3 1 )Z 2 | . (8) 
These standard errors can be used to produce approxi- 
mate 95% pointwise CIs for p(l): 
Figure t 
(A) Hypothetical length distributions of fish encountered by the control 
(c) and test (t) fishing protocols from paired-tow fishing experiments, 
to illustrate the impact of within-pair variations in stock densities 
on the relative efficiency of the test compared to the control protocol. 
Line types correspond to pairs of tows. Thin vertical lines indicate 
mean length per tow and coefficients of variation equal to 0.5. (B) Cor- 
responding log density ratios (i.e., <5’s) are shown for ranges of lengths 
where the average density (in A) was greater than 0.01. 
Cl = exp \ji(l) ± 1.96 xSE j/3(Z)|J. 
(9) 
Occasionally in comparative fishing the duration (D-) 
of the tows may differ somewhat between vessels. Also, 
because of operational time constraints the catches 
may have to be subsampled for some species. The sub- 
sampling fraction (F ;;7 ) may depend on size as well. 
To account for these effects we added an offset (Z) to 
Equations 6 and 7, Z ;/ =log(H ic F lcJ /D jt F itl ). For length- 
pooled analyses of vessel effects we added the offset 
Z L =\og(D ic F lc /D lt F it ) to Equation 5, where F lf is the total 
subsampling fraction. 
Simulations 
Vessel effects The design of the simulation experiments 
mimicked the design for the data analyzed by Cadigan 
et al. 3 , that is, we simulated data for the seven species 
that they considered, and for the same number of tows 
and total catches for both paired-tows (R). This design 
is summarized in Table 2. Therefore, for example, in 
