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ICES Journal of Marine Science: Journal du Conseil Advance Access originally published online on May 3, 2008
ICES Journal of Marine Science: Journal du Conseil 2008 65(6):873-881; doi:10.1093/icesjms/fsn072
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© 2008 International Council for the Exploration of the Sea. Published by Oxford Journals. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Calibration of bottom trawls for northern shrimp

Michael C. S. Kingsley1, Kai Wieland1,2, Bo Bergström1 and Michael Rosing1

1 Greenland Institute of Natural Resources, PO Box 570, DK-3900 Nuuk, Greenland
2 Danmarks Fiskeriundersøgelser, PO Box 104, DK-9850 Hirtshals, Denmark

Correspondence to M. C. S. Kingsley: tel: +299 36 1200/6; fax: +299 36 1212; e-mail: mcsk{at}natur.gl

Kingsley, M. C. S., Wieland, K., Bergström, B., and Rosing, M. 2008. Calibration of bottom trawls for northern shrimp. – ICES Journal of Marine Science, 65: 873–881.

The Skjervøy 3000 trawl used since 1988 in the West Greenland bottom-trawl survey has been replaced by a Cosmos 2000. To be able to compare old data on the northern shrimp (Pandalus borealis) with new data, calibration experiments were carried out by trawling twice consecutively along the same track, using either the same gear twice or the two different gears in one order or the other. Catch models were fitted to the shrimp data—both size-aggregated catch weights and size-specific counts—by likelihood and Bayesian methods. The catch in the second haul relative to that in the first depended not only on the gear used in the two hauls, but also on density, the second catches being a smaller proportion of first catches when densities were high, and often larger than the first catches at low-density stations. This density-dependence of the catch ratio was larger for small shrimp than for big ones. The Cosmos trawl was estimated to fish with ~87% of the catchability of the Skjervøy trawl after correction for its greater wingspread. Catchability ratio varied with the size of shrimp caught, but the differences were not statistically significant.

Keywords: Bayesian modelling, gear calibration, Pandalus borealis, trawl survey

Received 3 July 2007; accepted 30 March 2008; advance access publication 3 May 2008.


   Introduction
Top
 Introduction
Methods
 Results
 Discussion
 Conclusions
 Appendix 1
 Appendix 2
 Appendix 3
 References
 
Fisheries resource surveys are universally carried out to provide fishery-independent information for management purposes, as well as more generally for research. Just as universally, questions arise whether surveys executed with different gear types, or the same gear type but of different designs, or the same gear design but fished by different ships can be compared, and if so, how. Long series of survey results are prized, reflecting changes in stock size over long periods, and changes in vessels or gear that might compromise series homogeneity are a concern. This question arises with particular frequency for trawl surveys, partly because many aspects of trawling lend themselves to more or less exact measurement, partly because many different designs of trawl exist, intended to fish on different bottoms or for different species, or to be towed by ships of different size and power, and partly because demersal resources are a major preoccupation of many fisheries management agencies. The literature of trawl comparison and calibration is therefore extensive (Pelletier, 1998). Much of it deals with calibration of different ships (e.g. Fréchet, 2000), often in the context of trawl surveys executed in collaboration by different jurisdictions (Cotter, 2001; von Szalay and Brown, 2001; Oeberst and Grygiel, 2004), or where one ship replaces another to continue a survey series (Cadigan et al., 2006), even if the trawl used is the same or of a standard design. Less often, questions of calibration arise when the trawl long used in a survey series is replaced by one of different design, even if the same vessel continues to be used, and the continuity of the series becomes an issue. This article is concerned with such a case.

A bottom-trawl survey, carried out by the 722 grt trawler "Paamiut", has provided data for the monitoring and assessment of the West Greenland stock of the northern shrimp (Pandalus borealis) since 1988 (Carlsson et al., 2000). The survey extends from Kap Farvel in southern Greenland to 72°30'N, and consists of 180–200 stations ranging in depth from 150 to 600 m. The survey estimates indices of the total biomass of the stock and its size composition, and provides estimates of the distribution, with respect to depth and region, of the total stock and of different size or age classes. A Skjervøy 3000 trawl with steel-bobbin groundgear used since 1988 was recently replaced by a Cosmos 2000 with rubber-disk rock-hopper groundgear that was expected to fish better on rough seabed and to survey effectively on a wider range of bottoms. It was foreseen that this switch would cause a problem in using the entire series of survey data in assessments, so a calibration study was undertaken to provide information for converting the existing series of data obtained with the Skjervøy to make it comparable with that to be obtained with the Cosmos. The catchabilities of the two trawls were compared in a calibration study carried out in 2004 and 2005, and the results are presented here.


   Methods
Top
 Introduction
 Methods
Results
 Discussion
 Conclusions
 Appendix 1
 Appendix 2
 Appendix 3
 References
 
Calibration stations were selected as a part of the routine annual survey for northern shrimp and other demersal resources carried out on the West Greenland shelf south of 72°30'N, between 150 and 600 m deep. The survey is stratified, with a random semi-systematic placing of stations (Kingsley et al., 2004). The stations used in the calibration study were in the Sukkertoppen Deep and the Holsteinsborg Deep, and off the mouth of Disko Bay (Figure 1). The fishing conditions were the normal ones for the survey (Wieland and Bergström, 2005), fishing only between 07:00 and 17:00 local time. Both trawls had a 20-mm stretched-mesh liner in the codend. Injector International 7.5-m2 trawl doors installed for use with the Cosmos trawl were also used with the Skjervøy trawl during the calibration study, replacing the previously used Greenland Perfect doors.


Figure 1
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  		Figure 1. The central West Greenland coast, showing trawl survey stations used in an alternate-hauls catchability calibration study for Pandalus borealis between a Cosmos 2000 trawl and a Skjervøy 3000.

 
The study design followed Lewy et al. (2004), correcting for spatial variation in density by fishing twice consecutively over each trawl track. Sets were of four kinds: either one or the other gear was used twice, or the two different gears were used in one order or the other. Haul durations of 15 and 30 min were used in the survey and in the calibration trials, the two hauls at each station having the same duration. The interval between stopping the first haul and starting the second, which included retrieving the gear and emptying the codends, repositioning the ship back from the start of the targeted track, and sinking the trawl to the start of the second haul, varied between 37 and 115 min, with a mean of 59 min; changing gear was slightly quicker than re-using the same gear. The mean distance between the centres of the two tracks at a station was 295 m (4–1312 m). The two gears had different wingspreads and therefore swept different areas along a given length of track. The doorspread was measured by acoustic sensing and recorded at intervals during each haul; corresponding wingspreads were calculated from a simple geometric model of the trawl configuration that assumed that bridle and wing formed a single straight line (Wieland and Bergström, 2005). The mean wingspread in each haul was multiplied by the GPS-measured straight-line track length to estimate the area swept.

A sample, the "deck sample", of 2–8 kg depending on the size of the shrimp caught, or the entire catch if smaller, was taken from the codend on deck when the trawl was taken on board, before the catch was dropped into the holding tank. The whole catch was usually sorted and weighed, the shrimp weighed wet off the sorting belt. However, for some large catches only one codend was processed, the other being assumed to weigh the same as the one. The deck sample was sorted in the laboratory by species and sex. The wet weight of P. borealis in the deck sample was divided by the species catch to determine the "sampling fraction". The carapace lengths (CL) of all P. borealis in the deck sample were measured, to the nearest half millimetre, obliquely from the posterior margin of an eye socket to the centre of the posterior rim of the carapace.

The principle behind the analysis was that a "disturbance effect" could be calculated from the results of fishing twice with the same gear, and used to correct the difference in catch when different gears were used, to end up with a calibration factor (i.e. a catchability ratio) between the two gears. The disturbance effect was necessarily assumed to be a property of the first gear used on each track. The catchability ratio was estimated on a density basis (weight or number of shrimp caught for each unit of swept area) rather than as a simple catch ratio, because the calculations of both biomass and the numbers of shrimp in the routine analysis of the annual survey are based on a density estimate at each station, which is obtained by dividing the catch by the swept area.

A catchability ratio was estimated by total catch weight of all sizes of shrimp combined. Catchability ratios by number were also estimated for different sizes of shrimp. The catches were disaggregated by CL class based on the size composition of the deck sample. CL classes of 0.5 mm were aggregated into five groups: 0–10, 10.5–15, 15.5–20, 20.5–25, and >25 mm.

Analyses of catchability ratio
For the first analyses, second density was plotted against first density on log–log scales (Figure 2). Three outliers for which the catch ratios were between 4 and 40 were rejected, as were two other pairs of sets, one of which had two extremely small catches and the other one small catch and one zero catch. The resulting dataset comprised 59 pairs of sets, 11 in which the Cosmos was run twice, 17 in which it was followed by the Skjervøy, 13 in which the Skjervøy was run twice, and 18 in which it was followed by the Cosmos.


Figure 2
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  		Figure 2. Shrimp densities in calibration experiments with a Cosmos and a Skjervøy trawl on the West Greenland shrimp grounds.

 
Catch-model analysis
It is assumed in the analysis of Lewy et al. (2004) that the second catch at a station is proportional to the first, the constant of proportionality depending only on the gears used in the two hauls. However, the first plots of the data indicated that the ratio of the two catches depended also on the density at the station: at high-density stations the second catch was lower, relative to the first, than it was at low-density stations. Therefore, a more complex model than that used by Lewy et al. (2004) was fitted to the data. It was based on an allometric relationship between the two catches at a station, the allometric line being allowed to have a slope different from unity.

The fitted model predicted the first catch at station s from the gear catchability (Ej), the swept area (Ai,s), and the station density (Ds):



Formula 072M1

(1)

However, the second catch at the station was predicted using not only the catchability of the second gear, the swept area, and a disturbance effect attributable to the first gear used, but also an allometric relationship between the predicted catches that depended on the density at the station:


Formula 072M2

(2)
where Ci,s is the predicted catch in haul i (i = 1, 2) at station s, Ds the density at station s, Ej the catchability of gear j (j = 1, 2), Gi,s the gear used in haul i at station s, Ai,s the area swept by haul i at station s, Fj the disturbance factor for gear j, and T the exponent of the allometric density-dependent relationship between the first and second catches. Then, for all hauls, the observed catch was considered to be lognormally distributed in relation to the predicted catch, with homoscedastic error:


Formula 072M3

(3)
where {sigma}2 is the mean squared deviation of observed catches from the predictions. The assumption of homoscedastic error was checked and found to be valid. A log-likelihood, adjusted for small sample sizes (Appendix 1), was calculated as:


Formula 072M4

(4)
where n was the number of observations (for the size class, where relevant), i.e. twice the number of stations, and {nu} the number of degrees of freedom for error. For size-group count data, the catch counts were predicted as above, then multiplied by the sampling fraction to correct them to predicted deck-sample counts and fitted to the recorded deck-sample counts for the size group.

The equations above did not completely specify the model, because the catchabilities and the densities could have free, but inversely related, estimates. The product of the catchabilities was therefore constrained to be 1, and their ratio gave the calibration factor that was the objective parameter.

Lewy et al. (2004) assumed Poisson distributions for numbers caught, which implies a binomial distribution for catch ratios, and used a logit transformation of the binomial in the analysis. This is mathematically similar to the approach used here, viz. an analysis of log-transformed catch quantities.

The model was fitted by two methods: maximum likelihood (ML) using general-purpose optimizers (Appendix 2), and Bayesian fitting using MCMC on the WinBUGS platform (Appendix 3). For either fitting method, the parameters to be fitted were one density for each station, two catchabilities (with a constraint on their product), two disturbance factors, one density effect, and one variance of the error of prediction. This model was fitted to data on the total catch weights at each station. For size-group count data, all these parameters were fitted for each size group; common values for some parameters for some sets of size groups were also subsequently fitted on an exploratory basis. Reduced models were tested by likelihood-ratio tests and by an Akaike's Information Criterion (AIC).

When fitting by ML, the uncertainty of parameter estimation was estimated by likelihood support intervals. Significance tests between parameter estimates were carried out with likelihood-ratio tests, using the {chi}2 approximation to changes in log-likelihood consequent on setting parameters (e.g. for different size groups) to be equal.

The Bayesian model was tested by calculating values of the deviance information criterion (DIC) for models with different parametrizations and checking changes in pD and DIC (Spiegelhalter et al., 2002). In Bayesian modelling, pD, the difference between the (posterior) mean deviance and the (smaller) deviance at the (posterior) mean of the fitted variables, may be considered to approximate the number of "effective" variables (Spiegelhalter et al., 2002). The DIC, like other information criteria, penalizes models that achieve good fits by having a lot of explanatory variables by adding the number of variables, in this case the pD value, to the deviance. The resulting DIC is an overall measure of a model's compromise between parsimony and good fit; smaller is better.

Analysis by size group
The counts by 0.5-mm size class for each haul were aggregated into 5-mm size groups. Both hauls of a pair were rejected from the analysis for a given size group unless there were at least ten shrimp in the size group in the two deck samples taken together. The resulting datasets were much reduced for the size group <10.5 mm, and slightly reduced for the others (Table 1).


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  		Table 1. Dataset sizes for CL size groups of Pandalus borealis in a paired-fishing calibration trial in West Greenland after rejecting stations for which the paired deck samples combined did not contain at least ten shrimp in the size group.

 
The 17 stations retained for the <10.5-mm size group yielded on average ~29 of these small shrimp in each deck sample. Lowering the threshold of 10 shrimp per station would not have included many more stations; in general, either these small shrimp were present in some numbers or they were absent; deck samples at the rejected stations had an average of 0.8 shrimp of this size. The 10.5–15-mm size group was better represented, but again, in the sets excluded because of low counts in the deck samples, shrimp of this size group were very few. The largest shrimp (>25 mm CL) were also much fewer than those between 15 and 25 mm, but they were more generally present and more stations could be retained (Table 1).


   Results
Top
 Introduction
 Methods
 Results
Discussion
 Conclusions
 Appendix 1
 Appendix 2
 Appendix 3
 References
 
The Cosmos trawl had a wingspread on average ~20% larger than that of the Skjervøy. However, it was necessary to assume that the disturbance factor allowed for in predicting the second catch at a station depended only on the first gear type, otherwise it was not possible to fit predictive models of catches.

Log–log regressions and preliminary analysis: size-aggregated weight data
When first density (weight/area) was plotted against second density in log–log-space, the aggregate dataset had a linear appearance (Figure 2). When a single line was fitted to the data, a quadratic term was very small (significant at 95%) and was ignored. Separate straight lines were fitted to the four types of set; their slopes were not significantly different (p = 0.143.) Parallel lines were fitted, and their common slope, 0.884, was significantly different from unity (p < 0.001). This implied that the ratio of second density to first density decreased, overall, with increasing catch. Inspection showed that among the stations where the biggest catches were made, the second catches did not average much more than 60% of the first catches, whereas among the low-catch stations, they were 170–180% of first catches. These differences were quite consistent between the catches made at these density levels with the different gears. We concluded that there was a straight-line relationship in log-space between the first and second catches, but that its slope was not unity. However, because parallel lines could be fitted, it appeared that a usable model might have both disturbance factors depending on density in the same way and a catchability ratio independent of density. Accepting a model of density-dependent disturbance factors affected the estimates of catchability ratio.

Model fit
The likelihood and the Bayesian models were both easy to fit to both the aggregated-weight data and the size-group data. The likelihood model converged to a stable solution, independent of starting values, which gave well-controlled standard errors (s.e.) and predicted catches that were closely related to observed catches (Figures 3 and 4). The Bayesian model converged to a stable posterior distribution, with significant updating of prior distributions. Median estimates of parameters from the Bayesian model generally agreed with the ML estimates from the likelihood model, and the uncertainties of estimation from the two models were also generally in reasonable agreement (Tables 2–4).


Figure 3
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  		Figure 3. Prediction of deck-sample count observations for Pandalus borealis 20–25 mm CL by a catch model in alternate-haul calibration trials between a Cosmos 2000 trawl and a Skjervøy 3000 trawl.

 
The results for the under-10-mm size group showed exceptions to these generalities, because so few stations provided data for it, counts were low (Tables 1 and 3), and the fit of the model was inferior to the fit for the other groups (Figures 3 and 4). For this size group the results from the likelihood model differed more from those of the Bayesian model, but the uncertainties of the parameter estimates were so large that the differences were not statistically significant (Tables 3 and 4).


Figure 4
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  		Figure 4. Prediction of deck-sample count observations for Pandalus borealis <10.5 mm CL by a catch model in alternate-haul calibration trials between a Cosmos 2000 trawl and a Skjervøy 3000 trawl.

 
Likelihood model
For aggregated catch-weight data, the density effect on disturbance factors, i.e. the slope of a straight line in log-space relating first and second catches at a station, was estimated to be 0.891, with an s.e. of 0.023, significantly different from 1. This model therefore confirmed the results from the regression analyses that the disturbance effect (i.e. the reduction in second catch relative to the first) was greater at high densities, but that at low densities, there was a tendency for catches to be larger in the second haul at a station.

A single overall estimate of the catchability ratio, on a weight basis (Cosmos relative to Skjervøy), was 86.8%, with an interquartile range (i.q.r.) of 13.0% and an s.e. of 9.6% (Table 2). For the normal distribution, the i.q.r. is ~1.349x the standard deviation. This ML estimate of the overall catchability ratio was not significantly different from unity. A reduced model testing the hypothesis that the two trawls fished the same (i.e. with unit catchability ratio and the same disturbance effect) reduced the maximum log-likelihood by 0.820 ({chi}22 = 1.64, p > 0.05; Table 2).


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  		Table 2. Results of catchability calibrations, aggregated catch-weight data, Cosmos 2000 relative to Skjervøy 3000, for Pandalus borealis in West Greenland.

 


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  		Table 3. Catchability ratios between Cosmos 2000 and Skjervøy 3000 trawls for CL size groups of Pandalus borealis, count data, ML estimates from likelihood model.

 


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  		Table 4. Catchability ratios between Cosmos 2000 and Skjervøy 3000 trawls for CL size groups of Pandalus borealis, count data, median estimates from Bayesian model.

 
Results from the analysis of count data by size group were affected by the small quantities of data for the smallest size groups. Estimates of catchability ratio varied for the different size groups, but the differences were not statistically significant, and a common catchability ratio of 0.854 with an s.e. of 0.069 could be fitted simultaneously to the five sets of count data for the five size groups, with a {Delta}ln(l’) of 1.71 (p > 0.05), indicating an improvement in information when only one catchability ratio was estimated. There were, however, still differences between the values of the other parameters in the model for the different size groups, parameters such as the disturbance factors and the density effect. Fitting a single set of all parameters to all five sets of count data, instead of five different sets, changed log-likelihood by 18.5 ({chi}216 = 36.9, p < 0.01), and this model was also rejected by the information criterion.

The three largest size groups had catchability ratios that were more closely similar: a model with common values of the catchability ratio (0.879, s.e. 0.077) and the density effect (–0.084, s.e. 0.019) for the three largest size groups, saving 4 degrees of freedom, changed the log-likelihood by just 0.67. Fitting common disturbance factors as well for these three size groups reduced the log-likelihood by a further 3.42 ({chi}24 = 6.84, p > 0.05). A minimum model that assumed that catchabilities and disturbance factors were the same for the two gears and for these three size groups could not be rejected by a significance test, although it was narrowly rejected, in favour of the preceding model, by the information criterion. When the count data for these three size groups were combined into one super-group dataset of 54 stations and 52 176 deck-sampled individuals, the catchability ratio between the two trawls was still estimated at nearly the same value, 0.875, but the estimated s.e. increased to 0.113, so the difference in catchabilities was not significant, and a model in which the two trawls had the same disturbance factors and the same catchabilities for this super-group was preferred, based on the information criterion.

Bayesian model
Results from the Bayesian fitting of the model to the total catch-weight data were similar to those of the likelihood model, but not identical. The median estimate of the power of density entering into the disturbance effects was –0.124 with an i.q.r. of 0.032—precisely estimated, and clearly different from unity—whereas the disturbance multiplier was estimated at 1.19 for the Cosmos trawl and 1.41 for the Skjervøy trawl (Table 2). The median estimate of catchability ratio for size-aggregated catch-weight data fitted by Bayesian methods was estimated to be 0.865 with an i.q.r. of 0.130.

For the size-group analysis, larger deviations between the likelihood and the Bayesian models were apparent, but the pattern of differences between the catchability ratios for the different size groups was similar. Again, the smallest size group had a greater catchability ratio than the other size groups, the second had the smallest ratio, and the three largest size groups had similar ratios near 0.9. A simplified model with the same catchability ratio for all size groups was fitted and the DIC values compared. The full model with five different catchability ratios had 232 parameters, estimated by the Markov Chain Monte Carlo (MCMC) pD statistic at 238.6 "effective parameters". The model with one catchability ratio had 228 parameters, which the MCMC pD statistic estimated as 234.4 "effective parameters"’. The mean deviance of the simpler model was, however, less than that of the full model, 8821.8 vs. 8827.3, so the DIC measure—9056.2 vs. 9065.9—strongly preferred the simpler model with one catchability ratio.


   Discussion
Top
 Introduction
 Methods
 Results
 Discussion
Conclusions
 Appendix 1
 Appendix 2
 Appendix 3
 References
 
Our study was aimed at comparing two different trawls fished by the same ship under the same conditions. The doors had already been changed, and the effect of the door change on the performance of the Skjervøy trawl was not investigated. Changing doors at sea between hauls is not only difficult and dangerous, but also takes time, and although door changes and rigging can affect catch rates (Byrne and Forrester, 1987; Lauth et al., 1998), the change from Greenland Perfect to Injector International doors was not expected to have a large effect on the Skjervøy trawl, especially because catch rates in the study, as in the survey, were standardized on swept area.

In some large multinational surveys, variation in catch rates between ships is minimized by very detailed specification of the trawl to be used (ICES, 2005, 2007), but this does not remove all differences, and additional gear modification may be needed to standardize trawl geometry to an acceptable level (Fréchet, 2000). Alternatively, calibration experiments between ships may still be needed. Most trawl calibration studies use one form or another of parallel trawling, where the vessels to be compared (fishing the same or different gears) fish simultaneously side by side (e.g. von Szalay and Brown, 2001; Cadigan et al., 2006), but to investigate specifically the performance of different gear, the alternate-hauls method (Wileman et al., 1996) is among the methods available. Disturbance effects in alternate-hauls trials were recognized and investigated in the context of the international trawl survey carried out in the Baltic Sea by Grygiel (2004).

Fishing twice on the same track is intended to remove the spatial variability that confounds the results of parallel trawling or randomized block designs, adding more or less noise to the data. Standard analyses (Lewy et al., 2004; Cadigan et al., 2006) can remove station density from the calculations and instead use catch or catch-rate ratios. Alternate-haul designs that attempt to go further in eliminating spatial variability by fishing on the same track must include other variables that have to be estimated from the same dataset, namely the disturbance effects. However, in the model developed here, ratios of first to second catches depended on density, so station densities had to be explicitly estimated anyway, as well as the density effect itself. The uncertainties associated with the extra variables and the need to estimate them may have reduced the accuracy with which the catchability ratio could be determined, and may have undermined the effectiveness of the method.

The basic model for this method of estimating a catchability ratio assumes that the second catch on a trawl track is proportional to the first, with a constant of proportionality depending on the two gears that were used along with their catchabilities and disturbance factors, but not depending on the density of the target species at the station (Lewy et al., 2004). The data we collected showed beyond doubt that this model does not apply to northern shrimp under the conditions of this study. For aggregated catch-weight data, and for size-group count data, we found, by both the model-fitting methods, that the ratio between the first and second catches depends on the density of shrimp at the station. Second catches at stations with large catches in the first haul were much smaller, but at stations with small catches on the first tow, second catches tended to be bigger. The ratio between the first and second catches was strongly associated with aggregate density (i.e. weight data) over a density range spanning four orders of magnitude, the relationship being very close to a power law. Count data analysed by size group appeared to show, however, that this density effect was greater for the smallest shrimp than for larger ones. Commercial fishers sometimes say that shrimp are attracted to trawl tracks, although without any suggestion as to why, but our results tend to show that this might apply more to small shrimp than to larger ones.

Data for estimating catchability ratios for the smallest shrimp were sparse. Only about one-quarter of the stations were retained in the analysis for the size group <10 mm, and their deck samples averaged only 29 shrimp, so the catch-model parameters could not be estimated precisely for this size group. For the three largest size groups, more stations were retained and counts were larger. Therefore, although there were indications that the catchability ratio between the two gears was different for the smallest shrimp from what it was for the others, the data did not reveal a statistically significant difference, and for the size groups for which the catchability ratio was more precisely estimated, the differences between size-group catchability ratios were small anyway. Reasons for the ratio to differ in this way with size are not easy to understand. The Cosmos is a slightly wider trawl, and the headline is higher. If the vertical distribution of different sizes is different—if the smallest shrimp have a strong tendency to be found in the water column and not on the bottom, even by day—a higher trawl opening could result in their being caught in larger numbers in the Cosmos, relative to shrimp of other sizes, giving a higher catchability ratio. However, the two types of trawl also have different groundgear, and this may also affect the catchability ratio for the smallest shrimp in a different manner from how it does for the larger ones.

The parameters of the model showed some system in the values that fitted the data for the different size groups. The density effect was 0.46 for the smallest shrimp, decreasing somewhat consistently for larger ones. For the smallest shrimp, there was more of a tendency for the catches to be relatively larger in the second haul at stations where the density of shrimp of this size was small, whereas for the larger shrimp, the ratio of catches in the two hauls was less influenced by density. The disturbance factor for the Skjervøy was much the same for all size groups, and appeared to induce about a 26% increase in catch in the second haul (at unit density), but the disturbance factor for the Cosmos varied with size, large for the smallest shrimp and decreasing with increasing size.

The three largest size groups, shrimp >15 mm CL, are the mainstay of the commercial fishery, provide most of the weight of the survey catch, and are the stock component whose biomass index is used in the quantitative assessment model. Catchability ratios were similar for these three groups. A best estimate for the catchability ratio for these size groups was 0.879, with an s.e. of ~0.077, and the hypothesis that catchabilities were equal could not be rejected. Pooled count data for a super-group of these three groups gave a similar estimate for the catchability ratio, but with a much larger s.e.; it is not clear why.

A caution has been sounded that correcting past data with small calibration factors that are not accurately estimated could import error into assessment processes (Munro, 1998). Results from the present study, in which the estimates of differences in catchability were of similar magnitude to their estimated s.e., could fall into this category. The assessment of the West Greenland shrimp stock uses the series of biomass index estimates from the survey as input data to a surplus-production model of the stock dynamics, which is fitted by Bayesian methods (Hvingel and Kingsley, 2006). It might be appropriate to introduce the estimated calibration factors to the assessment process not as rigid correction factors to past data, but with their s.e. as parameters of prior distributions of the true correction factors that the assessment model could then update according to the information in the other input series.

Part of the reason to change the trawl was to extend the range of the survey to a wider range of seabeds, reducing problems of gear getting stuck or being damaged. Experience since the changeover has shown that these hopes have been realized, but such an extension of the survey range might introduce another difference between estimated survey indices obtained with the two different trawls, one that our study could not investigate. Bottoms fishable by only one gear are a small proportion of the total study area, and probably not significant for the total survey, but are locally concentrated, and if densities there differed much from the stratum mean, this effect might be significant for some strata.


   Conclusions
Top
 Introduction
 Methods
 Results
 Discussion
 Conclusions
Appendix 1
 Appendix 2
 Appendix 3
 References
 
In calibration experiments carried out by fishing consecutively in the same place, it cannot necessarily be assumed that the ratio of the second catch to the first depends only on the gears used. For northern shrimp, it appears to depend also on the density of shrimp at the station, second catches tending to be relatively lower when densities are higher, and vice versa. The catch ratio closely follows a power law of density, and the departure from proportionality, as well as the tendency for second catches to be higher at low-density stations, was more marked for catches of smaller shrimp.

For the northern shrimp, a Cosmos 2000 trawl with rubber-disc rock-hopper groundgear caught, based on aggregate weight for each unit of swept area, 86.8% (s.e. 9.8%) as much as a Skjervøy 3000 with bobbins. Within the available size classes, differences in catchability ratio between more narrowly defined size groups were insignificant. Shrimp <10.5 mm CL were caught in smaller numbers, leading to a paucity of data, and the catchability ratio (Cosmos:Skjervøy) for this size group was estimated by likelihood methods at 104%, with an s.e. of 46%. Shrimp from 10.5 to 15 mm CL had an estimated catchability ratio of 67.8%, with an s.e. of 15%.


   Appendix 1
Top
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix 1
Appendix 2
 Appendix 3
 References
 
likelihood-ratio tests
Likelihood-ratio tests are flexible and convenient, and can be shown to be equivalent to variance-ratio tests where both are appropriate, if appropriate critical levels are selected. An approximation that the logarithm of the likelihood ratio is distributed as 2 x {chi}2 is asymptotically valid. However, where a statistical model expresses observed values in the form of prediction plus error, maximum-likelihood methods are known to produce biased estimates of the error mean square, equivalent to dividing the error sum of squares by the number of observations rather than the number of degrees of freedom. If the number of parameters is of similar magnitude to the number of observations, this bias can cause likelihood-ratio tests to be excessively sensitive.

An AIC corrected for small-sample-size bias is:


Formula

recommended whenever n, the number of observations, is less than ~40 times the number, K, of parameters. Writing


Formula

so that the correction is shifted from the information criterion to the likelihood term and c is the required factor, and setting a proportional relationship CAIC = dAICc, then


Formula

conveniently solved by c = d = (n – (K+1))/n, and giving an adjustment for the log-likelihood that is similar to the bias factor in ML estimation of residual mean squares.

In our study, the number of observations was not large, and the number of parameters to be estimated was greater than half the number of observations. To counter the negative bias in the estimation of error variance and the associated sensitivity of likelihood-ratio tests, the sample-size adjustment c above was applied to the log-likelihoods. This adjustment did not alter the maximum-likelihood estimates of parameters, but increased the estimated s.e. and made tests less sensitive. Adjusted likelihoods resulted in s.e. that were similar to those obtained from fitting the same models by Bayesian methods, critical levels similar to those from parallel variance-ratio tests where the latter were available, and, of course, an information-criterion ranking of candidate models similar to that given by the small-sample-size-corrected information criterion applied to unadjusted likelihoods.


   Appendix 2
Top
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix 1
 Appendix 2
Appendix 3
 References
 
maximizing likelihoods
Because shrimp density affected the ratio of first to second catches, a density at each station had to be estimated explicitly; for size-group analyses, one for each size group. Maximum- likelihood estimation generated a moderately large, sparse non-linear mathematical programming problem. Two platforms were used to maximize likelihoods: Excel® Solver® (Fylstra et al., 1998) and ‘optim’ under R (R Development Core Team, 2007). The generalized reduced gradient algorithm (Lasdon et al., 1974, 1978), documented as "a robust version of the BFGS (Broyden, Fletcher, Goldfarb, and Shanno) algorithm", was used in Solver, and the BFGS algorithm, with no gradient function supplied, in ‘optim’. Where applied to the same models, they gave, naturally, the same solutions. Solver's limit of 200 adjustable variables was dealt with by iterating appropriately chosen overlapping subproblems to joint convergence under control of Visual Basic. The flexibility of Solver in applying constraints made it the natural choice for applying likelihood-ratio tests to reduced models. Marginal s.e. were estimated either by inverting the Hessian matrix available from ‘optim’ or, in Solver (which does not make a Hessian matrix available), by maximizing, then minimizing, the variable under investigation with a constraint that the log-likelihood should not decrease by more than {Delta} from its maximum value, with W as the width of the resulting interval calculating the s.e. as W/{surd}8{Delta}. Values of {Delta} between 1/2 and 1/128 were used to check on the shape of the likelihood surface. Error correlations, where needed, were taken from the inverse of the Hessian matrix from ‘optim’, or from the procedure for estimating s.e. in Solver.


   Appendix 3
Top
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix 1
 Appendix 2
 Appendix 3
References
 
BUGS coding for a Bayesian model for catchability ratio analysis
BUGS stands for Bayesian inference using Gibbs sampling (Lunn et al., 2000). The coding below is for a catch model for aggregated weight data; a model for size-specific count data was very similar. The data provided were a table of paired values for first and second hauls at a station. Values were trawl (1 for Cosmos, 2 for Skjervøy), swept area (SWA), and catch. Most priors were uniform in log-space. The prior for the density-effect power was uniform; this parameter was very strongly decided by the data, and the prior probably did not have much influence on the posterior. Estimates of station density were also strongly decided by the catch data, and the priors probably did not have large influence. The catchability ratio was given a prior that was uniform in log-space, so its reciprocal and both catchabilities also had prior distributions of this form.

In the following coding, a tilde, ~, has the meaning "is distributed as"; <-, "has the value"; "pow" is a power function; "dunif" is a uniform distribution with the given limits; "dlnorm" is a lognormal distribution with parameters mean and precision (reciprocal of variance) in log-space; "dgamma" is a gamma distribution with parameters mean-squared over variance and mean over variance.

for (j in 1:stations) {

l.exp.catch[1,j] <- log(SWA[1,j]*Dens[j]*Catchability [Trawl[1,j]])

 Catch[1,j]~dlnorm(l.exp.catch[1,j], preccatch)

l.exp.catch[2,j] <- log(SWA[2,j]*Dens[j]*Catchability[Trawl [2,j]]*Dfact[Trawl[1,j]]*

 pow(Dens[j],Dens.Effect))

Catch[2,j]~dlnorm(l.exp.catch[2,j], preccatch)}

#Prior distributions and constraints

for (j in 1:stations) {

 Logten.Dens[j]~dunif(–6,6)

 Dens[j] <- pow(10,Logten.Dens[j])}

Logten.Catch.Ratio~dunif(–0.7,0.7)

Catchability.Ratio <- pow(10, Logten.Catch.Ratio)

preccatch~dgamma(0.05,0.002)

Catchability[1] <- sqrt(Catchability.Ratio)

Catchability[2] <- sqrt(1/Catchability.Ratio)

Dens.Effect~dunif(–1,1)

for (Gear in 1:2) { Logten.Dfact[Gear]~dunif(–1,1)

 Dfact[Gear] <- pow(10,Logten.Dfact[Gear])}

BUGS code is not a prescription for consecutive calculations; instead it defines relationships that are all simultaneously true.


   Acknowledgements
 
We thank Per Kanneworff for the careful fishing that placed our second tracks on top of our first, and the officers, crew, and laboratory staff of the FV "Paamiut" for their help. We also thank Carsten Hvingel for advice and assistance with construction and debugging of the Bayesian model in the Bugs language.


   References
Top
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix 1
 Appendix 2
 Appendix 3
 References
 

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