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Statistical inference about the relative efficiency of a new survey protocol, based on paired-tow survey calibration data.

Abstract--Paired-tow calibration studies provide information on changes in survey catchability that may occur because of some necessary change in protocols (e.g., change in vessel or vessel gear) in a fish stock survey. This information is important to ensure the continuity of annual time-series of survey indices of stock size that provide the basis for fish stock assessments. There are several statistical models used to analyze the paired-catch data from calibration studies. Our main contributions are results from simulation experiments designed to measure the accuracy of statistical inferences derived from some of these models. Our results show that a model commonly used to analyze calibration data can provide unreliable statistical results when there is between-tow spatial variation in the stock densities at each paired-tow site. However, a generalized linear mixed-effects model gave very reliable results over a wide range of spatial variations in densities and we recommend it for the analysis of paired-tow survey calibration data. This conclusion also applies if there is between-tow variation in catchability.

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Surveys are an important source of information for most fish stock assessments. They provide indices of abundance often used in mathematical models of the population to estimate absolute stock size and to provide future catch advice or to evaluate catch options for fishery managers (Kimura and Somerton, 2006). Survey indices are measures that we expect to be proportional to, or to indicate, stock size. The expected value of a random index [R.sub.y] available for year y is related to stock size ([S.sub.y]) by the model E([R.sub.y])=q[S.sub.y]. The constant of proportionality, q, is usually referred to as the catchability of the index. Although we cannot directly infer stock size from a time series of indices [R.sub.1], ..., [R.sub.Y], we can infer trends in stock size when q is the same each year. The survey observation is commonly referred to as a set (as in set the gear), or a tow when a trawl is used. The average survey catch for all sets provides an index of stock size. If the same survey protocols are used from year to year then the catchability of the index should remain relatively constant. The catchability may depend on length or age of fish, and we consider such extensions later in this article.

There are many stock assessment models that are based on survey indices (e.g., see Quinn and Deriso, 1999) and information on fishing and natural mortality, to estimate absolute stock size. For most models it is necessary to have a fairly long time-series of survey indices, often 10 years or more. Over such time frames it may be necessary to change survey protocols. This could be due to a need to replace the survey vessel, or to change gears for new priority species, etc. When such changes occur, it is useful to have information about how the catchability of the new survey protocol compares to the old protocol.

Performing simultaneous paired-tow surveys using both protocols (e.g., old and new vessels, old and new fishing gears) provides direct data on how the catchabilities compare (e.g., Kimura and Zenger, 1997) Another approach is to simply fish side by side using both protocols and use the paired-catch data to estimate the ratio of catchabilities. We refer to this ratio as the relative efficiency,

p = [q.sub.c]/[q.sub.t], (1)

where [q.sub.c] and [q.sub.t] = the catchabilities of the old (control, c) and new (test, t) survey protocols.

Notations are given in Table 1. If the fish densities entering both trawls are the same, or similar, and densities at different tow sites vary considerably, then for the same number of tows a paired-tow calibration study should produce better results than the 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.sub.ij] obtained at the ith paired-tow station (i=1, ... , n) by survey protocols j=c (control) or j=t (test). Let [[lambda].sub.ij] denote the fish densities encountered 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.sub.c] and [q.sub.t] which are the same from site to site (i.e., i), and that catches are Poisson random variables with means

E([R.sub.it]) = [q.sub.t] [[lambda].sub.it] = [[mu].sub.i], and E([R.sub.ic]) = [q.sub.c] [[lambda].sub.ic] = p[[mu].sub.i]exp([[delta].sub.i]), (2)

where ([[delta].sub.i] = log([[lambda].sub.ic]/[[lambda].sub.it]).

If both vessels encounter exactly the same stock densities at each tow station, then [[delta].sub.i] = 0, i = 1, ... , n.

When there is no spatial heterogeneity in stock densities, p can be estimated by using a Poisson generalized 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 2n observations that can be used to estimate the n density parameters ([micro]) and p. Pelletier (1998) used a similar approach, with a negative binomial mean-variance assumption, which is a type of Poisson over-dispersion. These approaches are complicated because the number of [micro] 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.sub.i,c], given [R.sub.i] = [R.sub.ic] + [R.sub.it]. Let [r.sub.i] be the observed value of [R.sub.i]. The conditional distribution is binomial with a probability mass function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where p = [rho]/(1+[rho]) is the probability a captured fish is taken by the control vessel.

The only unknown parameter in this distribution is p.

The n nuisance [mu] parameters are eliminated in Equation 3. There are n conditional observations that can be used to estimate [rho]. For the binomial distribution E([R.sub.ic])=[r.sub.i]p and Var([R.sub.ic])=[r.sub.i]p(1-p). This approach is commonly used in commercial fishing gear size-selectivity studies (e.g., Millar 1992).

Paired-tow experiments do not eliminate spatial heterogeneity between the stock densities fished by each vessel. This heterogeneity leads to Poisson over-dispersion 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 [phi], Var([R.sub.ij])= [phi] E([R.sub.ij]), or a binomial over-dispersion parameter, Var([R.sub.ic]|[R.sub.i]=[r.sub.i])= [phi] [r.sub.i]p(1-p). Confidence intervals (CIs) are adjusted based on an estimate of [phi]. 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 completely account for the true variability in the data and too often led to the false statistical conclusion that p [not equal to]1. 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 [[lambda].sub.ic] and [lambda].sub.it] in Equation 2 are independent and identically distributed gamma random variables with means [[lambda].sub.i] and variances [tau][[lambda].sup.2.sub.i]. If [R.sub.ic]|[[lambda].sub.ic] and [R.sub.it]|[[lambda].sub.it] are Poisson distributed, then the marginal distributions of [R.sub.ic] and [R.sub.it] are negative binomials (e.g., see Cameron and Trivedi, 1998). This distribution is often suggested to be appropriate for modeling the variability of measurement error in trawl survey catches (e.g., Gunderson, 1993). This implicitly provides a rationale for assuming stock densities are gamma distributed. Dowden (2) showed that [tau]=0.049, 0.223, 0.372 corresponds to Var([[delta].sub.i])=0.1, 0.5, and 0.9, and that the distribution of [delta] (see Eq. 2) is well approximated by a normal distribution with [[sigma].sup.2]=Var([[delta].sub.i]). 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 effects as fixed effects models (e.g., GLIMs).

GLMMs are frequently used to account for heterogeneity 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 effects 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, different 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 software packages. In some situations, PQLE is known to produce biased estimates of fixed-effect parameters like [rho].

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 approaches--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 detect if [rho] [not equal to] l (i.e., a true difference in catchabilities between vessels). Our purpose is to recommend the most reliable approach, at least for paired-tow survey 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 relative efficiency.

Materials and methods

We focus on statistical inferences for [rho] (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 differences 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 calibration 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 efficiency 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 examine whether there is an effect on total catch per set. In the next section we describe a model for this purpose. Effects on the length compositions of the catches are considered later in this article.

Vessel effect

A common approach used for analyzing comparative fishing data is binomial regression with an adjustment 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 probability (p) is

log (p / 1 - p) = log(rho)=[beta], (4)

and [beta] can be estimated as the intercept with a logit link function by using software for binomial regression. The range of [beta] is (-[infinity], [infinity]). We derived CIs for [rho] by exponentiating intervals for [beta] and therefore CIs for [rho] 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 [phi] 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

log ([p.sub.i] / 1 - [p.sub.i]) = [beta] + [[delta].sub.i], (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 [[delta].sub.i]~N(0, [[sigma].sup.2]), i=1, ... n. In this case equation 5 defines a standard GLMM and there are many approaches and software packages available to estimate [beta] and [[sigma].sup.2] (e.g., Bolker et al., 2009). We examined the robustness of statistical inferences about [rho] to the normal approximation for [delta] (see Simulations section) when [delta] 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 marginal 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 estimation) approach. We used the default estimation method in PROC GLIMMIX, which is a restricted pseudo-likelihood 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 [beta].

Vessel and fish-length effects

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.sub.ijk], i=1, ..., n; j=1,2; k=1, ... [n.sub.i], where [n.sub.i] is the number of length classes caught in the i'th pair of tows.

If it is reasonable to assume that there are no between-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 [beta]=log([rho]), then a binomial GLIM with a logistic link can be used to estimate the intercept and slope; that is, [beta]-=[beta](l) =[[beta].sub.0]+[[beta].sub.1]l in Equation 4, where [beta] is taken to be a function of length, [beta](l). If the length effect is more complicated, 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 heterogeneity 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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [p.sub.ik] = the probability that at site i a length [l.sub.ik] captured fish came from the control vessel.

[FIGURE 1 OMITTED]

Holst and Revill (2008) used a random intercepts model, although their models for fixed lengths effects were more complicated than what we consider. However, if there are differences in the length distributions encountered by both vessels then Equation 6 will not be appropriate. The differences will usually be such that [delta]=log([[lambda].sub.c]/[[lambda].sub.t]) varies smoothly with length. Several hypothetical examples are also shown in Figure 1. This type of spatial heterogeneity can be approximated by linear functions, whose slopes (61) and intercepts ([[delta].sub.0]) vary randomly from set to set. A GLMM for this model is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

This is a common GLMM used in fishing gear selectivity studies. If the means of the densities are the same and the only difference is the height of the distributions, then the [delta] 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 VLi, (i.e., Eq. 6), and models with random intercepts and slopes are denoted as [VL.sub.is]. Models and estimation methods are denoted as [VLM.sub.i], [VLP.sub.i], [VLM.sub.is], and [VLP.sub.is] (see Table 1) depending on whether marginal MLE or PQLE is used.

Standard errors for [beta](l)= [[beta].sub.0]+[[beta].sub.1]l=log{[rho](l)} can be constructed from the estimates of rio and ill, and their estimated covariances:

SE{[??](l)}=[{var([[??].sub.0]) + 2cov([[??].sub.0], [[??].sub.1])l + var([[??].sub.1][l.sup.2])}.sup.1/2]. (8)

These standard errors can be used to produce approximate 95% pointwise CIs for [rho](l):

CI=exp[[??](l) [+ or -] 1.96 x SE{[??](l)}]. (9)

Occasionally in comparative fishing the duration ([D.sub.ij]) 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.sub.ijl]) may depend on size as well. To account for these effects we added an offset (Z) to Equations 6 and 7, [Z.sub.il]=log([D.sub.ic][F.sub.icl] /[D.sub.it][F.sub.itl]). For length-pooled analyses of vessel effects we added the offset [Z.sub.i]=log([D.sub.ic][F.sub.ic]/[D.sub.it][F.sub.it]) to Equation 5, where [F.sub.ij] 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 the simulation based on American plaice (Hippoglossoides platessoides), data were generated for 105 paired-tows with an average R of 109.5. Twenty-five percent of the sets had R [less than or equal to] 9, and 25% of sets had R [greater than or equal to] 169. The number of paired-tows in the seven sets of simulations varied from 25 to 105 which is a practical range consistent with many comparative fishing studies (e.g., Table 2 in Pelletier, 1998). The catches ranged from low (Atlantic cod, Gadus morhua) to high (yellowtail flounder, Limanda ferruginea). Some of the stocks had very skewed catches; for example, Atlantic cod and thorny skate (Raja radiata) had mean catches that exceeded their 75th percentiles.

The simulations were similar to a parametric bootstrap procedure; however we varied [beta] and [[sigma].sup.2] to examine how the accuracy of statistical inferences varied with changes in these parameters. The values of [beta] ranged from 0 to 2 by increments of 0.25, with [beta]=0, or [rho]=l, representing no vessel effect, and [beta]=2, or [rho]=7.4, representing a test vessel catchability that was 14% of the control vessel. Note that this range in [beta] is much larger than the results in Cadigan et al. (Table 7 in Cadigan et al. (3)). Their largest absolute estimate was 0.08; however, these simulations were designed to examine the accuracy of statistical inferences for small and large vessel effects. The levels of [[sigma].sup.2]=0, 0.1, 0.5, and 0.9 represented no to high spatial heterogeneity and broadly reflected the range of estimates in Cadigan et al. (Table 7 in Cadigan et al. (3)). The lowest estimate of [[sigma].sup.2] in Cadigan et al. (3) was 0.10 for Greenland halibut (Reinhardtius hippoglossoides), and the highest estimate was 0.99 for Atlantic cod.

Simulated catches for the control vessel (Re) were generated as binomial random numbers; the number of trials was equal to the observed R in Cadigan et al. (3) and probability was based on Equation 5. The [delta]s were generated randomly from a normal distribution with mean zero and variance [[sigma].sup.2]. The simulated test vessel catch was [R.sub.t]=R-[R.sub.c]. Note that the total catches for each paired-tow, R, ..., [R.sub.n], were the same in each simulation; hence, our simulation results were conditioned on these values.

We examined the robustness of the GLMM results to the assumption of a normal distribution for the random effects, [delta], when in fact [delta] was the log of a ratio of two independent and identically distributed gamma random variables with Var([dekta])= [[sigma].sup.2].

Estimates and 95% CIs for [beta] were obtained from 2000 simulations. We approximated the estimation bias as the median [beta] from the simulations minus the true simulation value. Bias results based on means were very similar. The coverage accuracy of the CIs was measured as the proportion of simulations in which the CI contained the true value of [beta]. If the 95% CIs are accurate, then the simulation proportion should be close to 0.95. We also computed the proportion of simulations in which [beta] was less than the lower CI, and the proportion in which [beta] was greater than the upper CI. If the CIs are two-sided accurate, then these proportions should both equal 0.025.

We performed other simulations using a much finer scale for [beta] to examine the power of detecting a vessel effect based on the proportion of simulations whose CIs did not cover zero when the true [beta] was greater than zero.

Vessel and fish-length effects These simulations were similar to those described in the last section. We simulated data for the seven species and with the same number of tows and total catches-at-length for both paired-tows ([R.sub.l]). However, there was an additional simulation factor for the slope of the length effect. We standardized lengths in the data from Cadigan et al. (3):

[l.sub.std] = l - [l.sub.50] / [l.sub.75] - [l.sub.25] (10)

where [l.sub.[alpha]] = the [apha]th percentile of the lengths from all sets, weighted by total catch, for each species.

[FIGURE 2 OMITTED]

This standardization allowed us to use the same slopes in simulations for different species; we considered [beta]=0, 0.5, 1.0 to represent no, medium, and large length effects. This scale increased the number of simulations three-fold. The length-based models were also slower to estimate because of the larger size of the data sets (see [n.sup.]* in Table 2), and to save time we reduced the number of simulations to 1000.

We simulated data from Equation 7. We set [[sigma].sup.2.sub.1]=[[sigma].sup.2.sub.2]=[[sigma].sup.2] and used the same values as before, [[sigma].sup.2]=0, 0.1, 0.5, and 0.9. However, we fitted both Equation 6 and Equation 7 to the simulated data. This procedure allowed us to examine the accuracy of statistical inferences from the random intercept model, which is a common mixed effects model, when slopes were random as well. We summarized the simulations for [beta]([l.sub.std])=[[beta].sub.0]+[[beta].sub.1][l.sub.std] at three points, [l.sub.std]=0, 0.5, and 1.0, which reflects relative efficiency at median to large lengths.

Results

Case study

We illustrate methods using data for witch flounder from Cadigan et al. (3). There is some evidence that the catchability of the control vessel was less than the test vessel. For example, there were five sets (solid circles in Fig. 2 where the test vessel caught more than 100 fish but the control vessel caught fewer than 100 fish. Rarely were catches by the control vessel much larger than the test vessel. However, in most paired-tows the catches by both vessels, when adjusted for tow distance and subsampling, were similar. The length distributions over all sets (Fig. 2, bottom panel) did not indicate that potential differences in catchabilities were length dependent because catch differences were approximately equally distributed over a broad range of sizes.

The GLIM estimate of [beta]=log([rho]) (Table 3; VO model) was significantly less than zero indicating that the control vessel had a catchability that was significantly less than the test vessel. However, both GLMM estimates (VM, VP) were somewhat larger and not significant, indicating that the test and control vessels catchabilities were not significantly different. The VM and VP estimates of [beta] and [[sigma].sup.2] were very similar. The PQL software (PROC GLIMMIX) we used did not provide standard errors for the estimate of [[sigma].sup.2], but the marginal MLE software (PROC NLMIXED) did.

The length-based models provided similar results, with the GLIM model (VLO) producing significant differences whereas the mixed models did not. Note that the length-based estimates are very different from those in Cadigan et al. (3) because we standardized lengths (i.e., Eq. 10) but Cadigan et al. (3) did not. The 25th, 50th, and 75th length percentiles were 22, 29, and 34 cm. CIs for p(l) based on Equation 9 were derived for [l.sub.std]=0, ..., 2 (Fig. 3). The VLO model suggested [rho](l) decreased with l and was significantly different from one over the range of lengths. The four mixed models all indicated a slight increase in p(l) with l but were not significantly different from one for any length. The CIs for the random intercept model (Eq. 6) were shorter than those for the random intercept and slope models, especially when [l.sub.std]>l. The marginal MLE CIs ([VLM.sub.i], [VLM.sub.is]) were slightly wider than PQLE CIs ([VLP.sub.i], [VLP.sub.is]).

Simulations

Vessel effect Simulation results were very similar for the seven species. We present the best and worst cases in Figures 4 and 5. The VO model performed poorly even when there was small spatial heterogeneity in stock densities (i.e., [[sigma].sup.2]=0.1). The likelihood ratio CIs had poor coverage properties and the probability that they contained the true value of the vessel effect ([beta]) was much less than the 95% nominal value. When [[sigma].sup.2] and [beta] were large this method produced biased estimates of[beta] and very inaccurate CIs. Note that to facilitate comparison of the methods the y-axis was fixed to be less than the range of some of the GLIM results, particularly in Figure 5. The VP CIs were more accurate, except when [[sigma].sup.2] [greater than or equal to] 0.5 for the thorny skate simulation (Fig. 5). For larger values of[beta] the CIs from this method covered less than 95%, about 80% for [[sigma].sup.2]=0.9 and [beta] =2. The bias was negative which meant that the lower and upper bounds were too small. The VM CIs were quite reliable across the range of values for [[sigma].sup.2] and [beta] and for all seven simulation scenarios (i.e., species). The log gamma ratio simulation results were almost identical to the normal simulations results and are not presented.

We performed simulations at a finer scale of [beta] to determine the size of a vessel effect that could be detected with a power of 0.8 or 0.95, based on the VM model. The power was computed from the proportion of CIs that did not cover zero. The results are shown in Table 4, expressed in terms of percent change, 100x([rho]-1). For example, when [[sigma].sup.2]=0.5 there was a 95% chance of detecting a 44% increase in catchability with data like that for American plaice.

Vessel and length effects The VLO model performed poorly compared to the mixed models and those results are not presented. We examined statistical inferences for [beta]([l.sub.std]) based on the [VLM.sub.i], [VLP.sub.i], [VLM.sub.is], and [VLP.sub.is] models. Note that simulated data were generated by using Equation 7 but fitted with both Equation 6 and Equation 7; hence, the results based on Equation 6, i.e. [VLM.sub.i] and [VLP.sub.i], will reflect model mis-specification biases. Results were similar for each simulation scenario (i.e., species). We present results for [beta]([l.sub.std]) only for the Atlantic cod scenario, small and medium spatial variability ([sigma] = 0.1, 0.5), no or large lengths effects ([[beta].sub.1]=0, 1), and at the center of the length distribution ([l.sub.std]=0; Fig. 6 or at a larger value ([l.sub.std]=1; Fig. 7).

The random intercept models gave unreliable results especially when [l.sub.std]=1. The total random effect variance based on Equation 7 increased with length and this was not accounted for by the [VLM.sub.i] or [VLP.sub.i] models. The poor performance of CIs for [beta]([l.sub.std]) derived from the [VLM.sub.i] and [VLP.sub.i] models was caused by both bias in estimates of [beta]([l.sub.std]) and bias in estimates of the variance of the estimator for [beta]([l.sub.std]). These biases are a complicated function of [[beta].sub.0], [[beta].sub.1], [[sigma].sup.2] and [l[l.sub.std].

The PQL estimation bias was similar to the results of the vessel effects only simulation (not shown for Atlantic cod) when [l.sub.std]=0, which is not surprising because the conditional distributions based on Equation 5 and Equation 7 are essentially the same in this case. However, when [l.sub.std]=1 the [VLP.sub.is] model gave less accurate results compared to those when [l.sub.std]=0. The [VLM.sub.is] model gave reliable results in all simulation settings.

The worst case [VLM.sub.is] result for the lower 95% CI coverage was for the deepwater redfish (Sebastes mentella) scenario in which the simulation coverage was 0.057 (nominal value is 0.025) when [[beta].sub.0]=0.5, [[beta].sub.1]=1, [[sigma].sup.2]=0.5, and [l.sub.std]=0. The worst case result for the upper interval was 0.055 when [[beta].sub.0]=2, [[beta].sub.1]=0, [[sigma].sup.2]=0.9, and [l.sub.std]=0 for Atlantic cod. In 95% of the simulations for all species the absolute error for the lower CI was less than 0.013, and for the upper interval it was less than 0.016. This demonstrates that CIs from the [VLM.sub.is] model were almost always very accurate.

[FIGURE 3 OMITTED]

Discussion

Our simulation results demonstrated that the commonly used over-dispersed binomial logistic regression model did not provide accurate statistical inferences for paired-trawl calibration data when there was spatial variation in stock densities. In practice, such variations will occur and therefore this approach is not recommended. Fortunately, our simulations showed that a binomial logistic regression model that included random site effects in addition to fixed vessel effects did provide accurate inferences for a wide range of spatial variations in stock densities. This conclusion also applied to pooled or length-based analyses. We recommend this binomial generalized linear mixed-effects model (GLMM) for analyzing comparative fishing data. When assessing for length effects, we recommend using a binomial GLMM with between-site random variation in both the vessel and length effects.

Between-set variability in catchability is commonly observed in covered-codend experiments (e.g., Fryer 1991, Millar et. al. 2004) that directly measure catchability. This will also produce between-site variability in p. Trenkel and Skaug (2005) assumed that the Poisson density for fish abundance was spatially constant on a small scale (~1000 [km.sup.2]), and that between-haul variation in catchability caused all additional Poisson over-dispersion in bottom-trawl survey catches. Cadigan et al. (2) demonstrated that this type of variability has a similar effect to that of spatial heterogeneity in stock densities encountered by both vessels at a trawl station. Hence, these two sources of variability are confounded in paired-trawl experiments and the random effects represent the cumulative impacts of both types of variability. For reasons outlined in the previous paragraph, we recommend the GLMM approach when there is between-set variability in catchability.

[FIGURE 4 OMITTED]

Our power analyses indicated that 50% changes in catchability could not be detected with high probability (i.e., 0.95) for some species. For example, with data like that obtained for Atlantic cod (see Table 2, and Cadigan et al. (3)) the power was fairly low. For this stock, the estimate of [[sigma].sup.2] was 0.99 and our power analysis indicated that we could detect only large changes in catchability (>90%) in this situation. Estimates of [[sigma].sup.2] were closer to 0.5 for most other species, in which case the power to detect a 50% change in catchability would be between 0.8 and 0.95. The exception was for yellowtail flounder which would have even lower power when [[sigma].sup.2]=0.5 because of the smaller number of positive sets (n=24) for this species. Changes in catchability between 20% and 50% would be important in stock assessment, and our simulation results indicated that more sets would be necessary to detect such changes in a comparison of paired-tow fishing data when the amount of spatial heterogeneity is similar to the levels in Cadigan et al. (3). If spatial heterogeneity could somehow be removed or kept low, then 50% changes in catchability could be detected with high power.

[FIGURE 5 OMITTED]

Another common approach to analyze comparative fishing data is to log transform catches and use normal linear models for analysis; however, this approach does not often adequately account for the stochastic nature of the data (e.g., counts) and involves arbitrary choices to deal with zero catches. However, the lognormal approach may be reasonable and appropriate in some situations, or when the focus is on catch weights (e.g., Kingsley et al., 2008).

We studied two methods to estimate GLMMs. One was based on maximizing the marginal likelihood, integrated over the random effects. The other approach was penalized quasi-likelihood estimation based on a linearization of the model and a double optimization procedure. The advantage of the latter approach was its ability to accommodate more complicated types of random effects like those with autocorrelation. However, our simulations results indicated that estimates and CIs from the linearization method were less reliable than those from the marginal likelihood approach. We recommend the marginal approach to estimate GLMMs for comparative fishing data.

[FIGURE 6 OMITTED]

We demonstrated that statistical inferences from GLMMs based on normal distribution random effects were equally as accurate when the random effects were actually the log of a ratio of two independent and identically distributed gamma random variables, which we hypothesize is a real and important source of over-dispersion in vessel calibration studies. This is good because otherwise we could not recommend the standard GLMM approach. However, this does not mean that GLMMs are always robust to mis-specifications of random effects (e.g., Heagerty and Kurland, 2001). We demonstrated this in our length-based simulations.

[FIGURE 7 OMITTED]

We demonstrated that the random intercept model gave inaccurate statistical inferences (bias and CIs) if there was between-set random variations in length effects. This is a type of model mis-specification. However, we did not fully examine the other side of this result, which is bias caused by assuming between-set random variation in length effects when in fact none exists. Results from simulations with [[sigma].sup.2]=0 for both vessel and length effects showed that the random intercept and slope model (i.e., [VLM.sub.is]) provided reliable statistical inferences when no random effects exist, and we speculate that the same result will hold when only random intercept effects exist. Hence, we recommend the [VLM.sub.is] approach when there is either within-pair variation in the density of fish or in their length compositions.

A type of model we did not consider involves large-scale within-pair random variations in the densities of fish encountered by both vessels, and smaller-scale length-specific random variations in length compositions. When plotted like Figure 1, log density ratios would appear as a scatter of points about horizontal lines. The intercepts of the horizontal lines would depend on the large-scale random variation in the fish densities, and the variation in the scatter about these horizontal lines would depend on the smaller-scale length-specific random variations in length compositions. These types of effects can be modeled with hierarchical random effects (i.e., set, and length within set). This procedure would be fairly straightforward with a PQL approach but more difficult with marginal MLE. The reliability of estimates (relative efficiency and variance parameters) is uncertain. Even more complicated hierarchical random-effect models for both vessel and catch length-composition effects could be considered. This was beyond our scope but important to understand for reliable statistical inference. In reality, random effects in log density ratios may also have non-normal and skewed distributions, and it would be helpful to understand the robustness of [VLM.sub.is] to these types of model mis-specifications.

Another sensible simulation procedure is to specify the stock densities fished by both vessels from site to site, and generate random catches for both vessels. The stock densities could be specified by using a variety of spatial models, as long as the within-pair and between-pair variations in densities are reasonably consistent with what one might expect in practice. However, this would not be a conditional simulation because the total catches by both vessels (and for all sets) would vary from simulation to simulation. It would still be useful to examine if the conditional CIs are accurate in this more general setting. However, the results from our conditional simulations were very similar for each of the seven species scenarios we considered and we anticipate they would also be accurate in the more general setting.

Cadigan et al. (2006) used a random effect for [delta] [equivalent to] [delta](l) that was autocorrelated in length l, corr{[delta](l),[delta](l')}=[[gamma].sup.[absolute value of l - l]]. This is a strategy to model smooth functions (e.g., Brown and de Jong, 2001). We used a simpler approach based on the assumption that [delta](l) varied linearly with l. A GLMM in which [delta](l) is autocorrelated can be fitted by using PQL software, but is time consuming to simulate and therefore we decided to focus on Equation 7 which is much easier to estimate.

Another approach to estimate relative efficiency (i.e., Pelletier, 1998) is maximum likelihood based on the negative binomial (NB) distribution. We have pursued this approach; however, there are complications in estimating the NB over-dispersion parameter based on the joint likelihood of both trawl catches and this problem affects the accuracy of CIs. The conditional approach is also more complicated for the NB distribution. We will report on this elsewhere.

Fryer et al. (2003) showed how to use spline methods for smooth, but otherwise nonparametric, estimates of relative efficiency. This is a useful estimation approach, especially to check the adequacy of a parametric model. The simple logistic-linear model we considered may be sufficient to test whether there is a significant length-dependent vessel effect but the logistic-linear model may not be sufficiently flexible for reliable estimation of relative efficiency over all lengths.

Lewy et. al. (2004) advocated paired-trawls along the same trawl track-line to avoid complications due to spatial variations in stock densities. However, such trawling introduces a different complication which involves disturbance of the fish densities encountered by the second vessel because of the fishing activity of the first vessel.

Another potential advantage of GLMMs is less sensitivity to outliers. Figures 14 and 15 of Cadigan et al. (2) indicated that GLMM estimates of [beta] were less sensitive to outliers than GLIM estimates. This lack of sensitivity is a considerable advantage because identifying outliers is time consuming in practice when conversion factors are estimated for many species. In addition, standard errors may be too small when observations are incorrectly deemed to be outliers and removed from the analysis. Atlantic cod and thorny skate had mean total catches (for both vessels) that exceeded the 75th percentile (Table 2), indicating that there were a few large catches that may have undue influence on estimates. The GLMMs seem better suited for this type of data. It would be useful to test these methods through simulation to assess robustness to outliers. The randomization approach used by Benoit (1) is another appropriate and robust procedure (e.g., Cox and Hinkley, 1974, p. 180-181) to test for the statistical significance of vessel effects, and we recommend this approach in addition to the use of GLMMs. However, it does not provide robust estimates of vessel effects and the approach cannot replace a GLMM.

Acknowledgments

The authors are grateful for the expertise, assistance, and many discussions with S. Walsh and B. Brodie, of the Northwest Atlantic Fisheries Center, Fisheries and Oceans Canada (DFO). M. Koen-Alonso of DFO also provided comments on earlier draft.

Manuscript submitted 20 April 2009.

Manuscript accepted 14 September 2009.

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(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.] (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 Needler research vessels using a Campelen 1800 shrimp trawl in NAFO Subdivision 3Ps and Divisions 3LN. DFO Can. Sci.

Noel G. Cadigan (contact author) (1)

Jeff J. Dowden (2)

Email address for contact author: noel.cadigan@dfo-mpo.gc.ca

(1) Fisheries and Oceans Canada Northwest Atlantic Fisheries Center 80 East White Hills Road St. John's, NL, Canada AIC 5X1

(2) Research and Evaluation Department Newfoundland and Labrador Centre for Health Information 28 Pippy Place St. John's, NL, Canada AIB 3X4
Table 1
Definitions of variables and acronyms for models used to estimate
relative efficiency from comparative fishing data.

[R.sub.ij]          Random variable for catches obtained at the
                    i'th paired-tow station by survey protocol
                    j=c (control) or j=t (test)

[r.sub.ij]          Observation of [R.sub.ij]

[R.sub.i]           [R.sub.ic]+[R.sub.it]

[R.sub.ijk]         Catches at the i'th tow station and k'th
                    length class by survey protocol j

[R.sub.ik]          Total catch (from both vessels) at length
                    class [kappa] from set i,
                    [R.sub.ick]+[R.sub.itk]

n                   Total number of paired-tow stations

[n.sub.i]           Number of length classes caught in the i'th
                    pair of tows

n*                  Total number of sets and length classes, n *
                    = [[summation].sub.i][n.sub.i]

[[lambda].sub.ij]   Fish densities encountered at station i and
                    tow j

[S.sub.i]           log ([[lambda].sub.ic]/[[lambda].sub.it])

[q.sub.j]           Probability an encountered fish is captured,
                    j=c, t

[rho]               Relative efficiency, [rho]=[q.sub.c]/[q.sub.t]

[beta]              log([rho])

p                   Probability a captured fish was caught by the
                    control protocol

[D.sub.ij]          Tow duration at the i'th paired-tow station
                    by vessel j

[F.sub.ijl]         Subsampling fraction for length l fish

[Z.sub.il]          Logit offset,
                    [Z.sub.il]=log([D.sub.ic][F.sub.icl] /
                    [D.sub.it][F.sub.itl]

[PHI]               Binomial over-dispersion

[[sigma].sup.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.sub.i]         Vessel- and length-effects binomial model
                    with random intercept for each set; marginal
                    MLE

[VLP.sub.i]         Vessel- and length-effects binomial model
                    with random intercept for each set; PQLE

[VLM.sub.is]        Vessel- and length-effects binomial model
                    with random intercept and slope for each set;
                    marginal MLE

[VLP.sub.is]        Vessel- and length-effects binomial model
                    with random intercept and slope for each set;
                    PQLE

Table 2
Paired/tow fishing simulation parameters based on the species in
Cadigan et a1. (3) n is the total number of paired/sets, [n.sup.*]
is the total number of sets and length classes, R=[R.sub.t]/[R.sub.c],
is the total catch by the test (t) and control (c) vessels, and
[R.sub.t]/[R.sub.c], is the ratio of catches from each vessel.
The mean, median and coefficient of variation (CV) are for total
catch (R). R25 and R75 are the lower and upper 25th percentiles of
R. See Table 1 for definitions of notations.

Species                           n   [n.sup.*]   R

American plaice                  105   2035   11,494
(Hippoglossoides platessoides)
Atlantic cod                      91   1132     3926
(Gadus morhua)
Deepwater redfish                 63   1030   12,069
(Sebastes mentella)
Greenland halibut                 56    585     1359
(Reinhardtius hippoglossoides)
Thorny skate                      79    990     2394
(Raja radiata)
Witch flounder                    57    970     5046
(Glyptocephalus cynoglossus)
Yellowtail flounder               24    536     5795
(Limanda ferruginea)

Species                          [R.sub.t]/[R.sub.c]   mean    median

American plaice                         1.051          109.5     40.0
(Hippoglossoides platessoides)
Atlantic cod                            1.067           43.1      4.0
(Gadus morhua)
Deepwater redfish                       1.207          191.6     88.0
(Sebastes mentella)
Greenland halibut                       1.243           24.3     13.5
(Reinhardtius hippoglossoides)
Thorny skate                            1.124           30.3      9.0
(Raja radiata)
Witch flounder                          1.334           88.5     52.0
(Glyptocephalus cynoglossus)
Yellowtail flounder                     1.250          241.5    159.0
(Limanda ferruginea)

Species                          R25   R75   CV

American plaice                    9   169   126
(Hippoglossoides platessoides)
Atlantic cod                       2    31   217
(Gadus morhua)
Deepwater redfish                 14   379   104
(Sebastes mentella)
Greenland halibut                  4    36   120
(Reinhardtius hippoglossoides)
Thorny skate                       3    20   226
(Raja radiata)
Witch flounder                     6   151   108
(Glyptocephalus cynoglossus)
Yellowtail flounder                5   503   102
(Limanda ferruginea)

Table 3
Parameter estimates, standard errors (SE), and lower
and upper 95% confidence interval limits from various
models for paired-tow comparative fishing data. See Table
1 for definitions of model acronyms and parameters. The u
and l parameter subscripts indicate a vessel or fish-length
effect.

Model          Parameter                           Estimate     SE

VO             [[beta].sub.v]                        -0.153   0.069
               [PHI]                                  5.888      --
VP             [[beta].sub.v]                        -0.094   0.093
               [[sigma].sub.v.sup.2]                  0.301   0.088
VM             [[beta].sub.v]                        -0.097   0.093
               [[sigma].sub.v.sup.2]                  0.302   0.089
VLO            [[beta].sub.v]                        -0.166   0.037
               [[beta].sub.l]                        -0.053   0.052
               [PHI]                                  1.628      --
[VLP.sub.i]    [[beta].sub.v]                        -0.102   0.092
               [[beta].sub.l]                         0.058   0.052
               [[sigma].sub.v.sup.2]                  0.296   0.086
[VLM.sub.i]    [[beta].sub.v]                        -0.105   0.092
               [[beta].sub.l]                         0.058   0.052
               [[sigma].sub.v.sup.2]                  0.295   0.086
[VLP.sub.is]   [[beta].sub.v]                        -0.099   0.094
               [[beta].sub.l]                         0.059   0.080
               [[sigma].sub.[bar.[omega]].sup.2]      0.299   0.088
               [[sigma].sub.l.sup.2]                  0.119   0.061
[VLM.sub.is]   [[beta].sub.v]                        -0.103   0.095
               [[beta].sub.l]                         0.061   0.081
               [[sigma].sub.[bar.[omega]].sup.2]      0.303   0.091
               [[sigma].sub.l.sup.2]                  0.122   0.065

Model          Parameter                           Lower    Upper

VO             [[beta].sub.v]                      -0.290   -0.018
               [PHI]                                   --       --
VP             [[beta].sub.v]                      -0.279    0.091
               [[sigma].sub.v.sup.2]                   --       --
VM             [[beta].sub.v]                      -0.283    0.089
               [[sigma].sub.v.sup.2]                0.124    0.480
VLO            [[beta].sub.v]                      -0.239   -0.094
               [[beta].sub.l]                      -0.154    0.048
               [PHI]                                   --       --
[VLP.sub.i]    [[beta].sub.v]                      -0.286    0.082
               [[beta].sub.l]                      -0.045    0.160
               [[sigma].sub.v.sup.2]                   --       --
[VLM.sub.i]    [[beta].sub.v]                      -0.290    0.079
               [[beta].sub.l]                      -0.047    0.164
               [[sigma].sub.v.sup.2]                0.121    0.468
[VLP.sub.is]   [[beta].sub.v]                      -0.288    0.090
               [[beta].sub.l]                      -0.098    0.215
               [[sigma].sub.[bar.[omega]].sup.2]       --       --
               [[sigma].sub.l.sup.2]                   --       --
[VLM.sub.is]   [[beta].sub.v]                      -0.294    0.088
               [[beta].sub.l]                      -0.101    0.224
               [[sigma].sub.[bar.[omega]].sup.2]    0.121    0.484
               [[sigma].sub.l.sup.2]               -0.008    0.252

Table 4
The size of a vessel effect (i.e., change in relative efficiency,
[rho]-1, in %) that can be detected with power=0.8 or 0.95. Columns
are for values of 02 (i.e., the random effect variance) and rows are
for species simulation scenarios.

Species                          Power = 0.8 / [[sigma].sup.2]

                                   0.0   0.1   0.5   0.9

American plaice                      5    13    23    32
(Hippoglossoides platessoides)

Atlantic cod (Gadus morhua)         10    18    33    42
deepwater redfish

(Sebastes mentella)                  5    15    30    43

Greenland halibut                   16    23    39    54
(Reinhardtius hippoglossoides)

thorny skate (Raja radiata)         13    20    34    45

witch flounder (Glyptocephalus       8    17    34    49
cynoglossus)

yellowtail flounder (Limanda         8    25    62   101
ferruginea)

Species                          Power = 0.95 / [[sigma].sup.2]

                                   0.0   0.1   0.5   0.9

American plaice                      9    24    44    64
(Hippoglossoides platessoides)

Atlantic cod (Gadus morhua)         17    33    68    92
deepwater redfish

(Sebastes mentella)                  7    27    60    99

Greenland halibut                   28    44    86   134
(Reinhardtius hippoglossoides)

thorny skate (Raja radiata)         23    38    71   101

witch flounder (Glyptocephalus      13    31    72   113
cynoglossus)

yellowtail flounder (Limanda        13    50   164   372
ferruginea)
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Date:Jan 1, 2010
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