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## Latin american journal of aquatic research

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*On-line version* ISSN 0718-560X

### Lat. Am. J. Aquat. Res. vol.43 no.1 Valparaíso Mar. 2015

#### http://dx.doi.org/10.3856/vol43-issue1-fulltext-13

*Research Article*

**Sensitivity analysis of catch-per-unit-effort of Atlantic bigeye tuna*** *(*Thunnus obesus*)**data series applied to production model**

**Análisis de sensibilidad de la captura por unidad de esfuerzo del atún patudo del Atlántico** **( Thunnus obesus) **

**con series de datos aplicados a un modelo de producción**

**Humber A. Andrade ^{1}**

^{1} Universidade Federal Rural de Pernambuco (UFRPE) Departamento Pesca e Aquicultura (DEPAq) Av. Dom Manoel de Medeiros s/n, Dois Irmãos, CEP 52171-030, Recife-PE, Brazil

Corresponding author: Humber A. Andrade (humber.andrade@gmail.com)

Corresponding editor: José A. Alvarez

**ABSTRACT.** Different datasets and a Bayesian production model were used to assess the status of the Atlantic bigeye tuna (*Thunnus obesus*) stock. Several datasets convey little information hence estimations of parameters are imprecise unless a very restrictive prior is used. Modes of posteriors calculated for composite datasets are in between modes of the posteriors calculated for separated datasets. Most of the calculations indicate that biomass has decreased until the beginning of 1990's when the stock was overfished. Catches decreased after 1999 but there is doubt if the stock was recovering in 2000's. The answer depends on the dataset and on the prior distribution.

**Keywords:** stock assessment, biomass, production model, Bayesian model, adaptive importance sampling.

**RESUMEN.** Diferentes conjuntos de datos y un modelo bayesiano de producción fueron utilizados para evaluar el stock del atún patudo del Atlántico (*Thunnus obesus*). Varios conjuntos de datos transmiten poca información por lo tanto las estimaciones de los parámetros son imprecisas, salvo que se utilice una distribución *a priori* muy restrictiva. Las modas de las *a posteriori* calculadas para conjuntos de datos compuestos están entre las modas de las *a posteriori* calculadas para conjuntos de datos separados. La mayoría de los cálculos indica que la biomasa ha disminuido hasta principios de los 90 cuando el stock fue sobreexplotado. Las capturas disminuyeron después de 1999, pero hay dudas sobre si el stock se estaba recuperando en los años 2000. La respuesta depende del conjunto de datos y de la distribución *a priori* utilizada.

**Palabras clave:** evaluación de stock, biomasa, modelo de producción, modelo bayesiano, muestreo adaptable por importancia.

**INTRODUCTION**

Bigeye tuna (*Thunnus obesus*) are distributed throughout the Atlantic Ocean between 50°N and 45°S, but not in the Mediterranean Sea (ICCAT, 2010). Evidences, such as lack of identified genetic heterogeneity and the time-area distribution of fish and movements of tagged fish, suggest an Atlantic-wide single stock, which is currently the hypothesis accepted by the International Commission for Conservation of Atlantic Tunas (ICCAT).

In the Atlantic Ocean catches of bigeye are high if compared to catches of other tuna species. Only catches of the smaller and less valuable skipjack (*Katsuwonus pelamis*) and yellowfin tuna (*Thunnus albacares*) are higher than those of bigeye. Most of the bigeye has been caught by longliners but the species is also caught by bait-boat and purse-seine vessels (Miyake *et al.,* 2004; ICCAT, 2010). Several countries fish bigeye in the Atlantic Ocean but catches of Japan, Taiwan, Ghana and Spain were the higher ones in the last decade (ICCAT, 2010).

In the last three stock assessment analyses for bigeye tuna carried out by ICCAT Working Group (ICCAT WG) both simple (*e.g.,* production models-Schaefer, 1954) and complex models (*e.g.*, fully integrated Stock Synthesis-(Method, 1990)) were used (ICCAT, 2005, 2008, 2011). ICCAT WG noticed there were considerable uncertainty concerning data and methods; consequently, estimations of benchmarks were very different. For example, maximum sustainable yield varied between 70,000 and 90,000 ton. Regarding methods complex models are more realistic, but simple models can be useful when data is limited (Ludwig & Walters, 1985). Bayesian and conventional versions of simple production models are often used. Discussions about the limitations and the usefulness of production models can be found in Prager (2002), Maunder (2003) and Prager (2003).

Several datasets from different countries concerning different gears and vessels are available for analysis of the bigeye stock. Relative abundance indexes calculated for different fleets are available in the last stock assessment meeting report (ICCAT, 2011). In some situations those multiple series of indexes are analyzed separately, but analyzing them together and composite indexes are also alternatives adopted by ICCAT WG. However Richards (1991) suggests that the analyses might be conducted separately for each dataset and the results should be presented separately to the decision makers. Following the same line of reasoning Schnute & Hilborn (1993) provided a composite model structure that allows the information conveyed by each separated dataset to arise in the results. Hence the objective here is to assess what are the results if we look at separated relative abundance datasets in the case of the Atlantic bigeye. Comparisons with the results gathered when analyzing composite indexes are warranted.

A Bayesian version of the production model was used because it allows previous knowledge about the fish stock to be considered in the analysis. For further comments on the use of Bayesian approach in fish stock assessment see Punt & Hilborn (1997), McAllister & Kirkwood (1998) and Meyer & Millar (1999a). Several methods can be used to calculate the posterior distributions in the Bayesian approach. Analytical solutions may be difficult to achieve hence Monte Carlo approaches are the alternative. Numerical methods like Markov Chain Monte Carlo (MCMC) and importance sampling methods are often used (Berger, 1985; Oh & Berger, 1992; West, 1993). Some authors have argued that MCMC is less efficient than importance sampling methods (Smith, 1991; Givens, 1993) though Meyer & Millar (1999a, 1999b) have showed that Gibbs sampler can be efficient even when there are many parameters. Nevertheless, MCMC may not converge when the posterior is multimodal (Newton & Raftery, 1994). Among the sampling importance methods the Sampling Importance Resampling (SIR) was the more popular in fishery stock assessments studies carried out in 1990's (*e.g.,* Raftery *et al.,* 1995; McAllister & Ianelli, 1997; McAllister & Kirkwood, 1998). Successful application of the importance sampling methods depends on the skill to build an *importance function* that is easy to sample from and similar to the posterior but with heavier tails (Van Dijk & Kloek, 1983; Oh & Berger, 1992). In this paper Adaptive Importance Sampling (AIS) was used because it is an iterative procedure in which a finite mixture of multivariate probability density functions is automatically updated until a suitable importance function is achieved (West, 1993). Applications of AIS in fisheries analyses can be found in Kinas (1996) and Andrade & Kinas (2007).

**MATERIALS AND METHODS**

**Catch and relative abundance indexes**

Catches of bigeye in the Atlantic are shown in Figure 1. There was an increasing trend since commercial fishery began in the 1950's. After the peak in 1994 catches have decreased until 2002 and since then, there was no clear time trend. Catch-per-unit-effort (CPUE) data from commercial fisheries are often used to calculate relative abundance indexes. Several procedures can be used to "standardize" CPUE in order to obtain those indexes (Maunder & Punt, 2004). Hereafter I assume to start that standardized CPUEs appearing in the 2010 bigeye assessment meeting report are useful as relative abundance indexes. Those indexes are shown in Table 1. Composite indexes were calculated using a generalized linear model with fleet as factor (ICCAT, 2011). All the indexes time series are shorter than the catch time series. However most of the indexes datasets cover a long period, though Morocco time series is an exception (S6 in Table 1). Notice also that ICCAT WG have excluded some data in order to calculate three of the composite time series (*i.e*., C4, C5 and C6). The motivations to discard the data and the details on the methods the ICCAT WG have used to calculate the relative abundance indexes are in ICCAT (2011) and ICCAT (2012).

**Figure 1.** Catches of Atlantic bigeye

tuna as reported in the ICCAT 2010

bigeye assessment meeting. Catch

for 2009 is the provisional estimation

used in the meeting.

**Table 1.** Relative abundance indexes available for analysis in the 2010 ICCAT bigeye

assessment meeting. Letters "S" and "C" in the column "Label" stand for separated and

composite series respectively. Gears are longline (LL) and baitboat (BB). Relative abundance

indexes were calculated using data from Brazil (BRA), Japan (JPN), Morocco (MOR), Portugal-

Azores (POR-AZO), China Taipei (TAI), Uruguay (URU) and United States (USA) and, all data

pooled (ALL).

All available relative abundance indexes are showed in Figure 2. Variability of the composite datasets is lower than of the separated datasets. In addition, some of the relative abundance indexes are contradictory. For example, Japanese time series (S2) show an increasing trend until mid 1970's and then a decreasing trend until the end of 2000's. Relative abundance indexes as calculated using data of Taiwan (S5) decreased fast in early 1970's but slowly until the end of 1980's followed by an increase until mid 1990's and a decreasing trend until the end of the time series. Other separated time series show erratic pattern or a decreasing trend. While some of the separated time series are contradictory all across the years, the composite time series are conflicting only before 1970 (Fig. 2). Similar decreasing trends from 1971 to 2000 appear for all the composite time series.

**Figure 2.** Relative abundance index (available from 2010 ICCAT bigeye assessment

meeting). Time series for separated fleets are in the left panel, while composite indexes

are in the right panel. Labels are: United States (S1), Japan (S2), Uruguay (S3),

Brazil (S4), Taiwan (S5), Morocco (S6), Portugal-Azores (S7), all data pooled (C1),

all data pooled but weighted by total catch by year and fleet (C2), all data pooled

by weighted by the number of 5° ? 5° squares (latitude x longitude) covered by the

fleet in the year (C3), all data pooled but only 1971 forward and weighted by total

catch by year and fleet (C4), all data pooled except 1970 back Japanese longline but

weighted by total catch by year (C5) and, all data pooled except 1970 back Chinese

Taipei longline but weighted by total catch by year (C6). All separated times series

were calculated for longline gears except the one of Portugal-Azores which is baitboat.

**Stock assessment model**

Catches and relative abundance indexes are the input data for simple production models. Indexes are often assumed to be proportional to abundance. I have assumed the relationship between biomass (B) and relative abundance indexes (*I*) in the *t ^{th}* year is:

Where *v* is a normally distributed random variable with variance *V*, accounting for observational error. Catchability coefficient *q* is assumed to be constant or, at least, to change at random over the years. That assumption holds if the standardization of CPUE was successful.

I have used the following version of traditional logistic production model (Graham, 1935; Schaefer, 1954):

Where *C _{t}* is the catch in the

*t*year,

^{th}*r*is the population growth rate,

*k*is the carrying capacity biomass. Observation error in

*C*was assumed to be negligible and the process error was not considered.

_{t}Aside the nuisance (*V*) there are three parameters of interest: *r*, *k* and *q*. Because it is mathematically more convenient to deal with parameters defined over the real line let *θ** =* {log *r*, log *k*, log *q*} be the three dimensional parameter vector of interest.

**Bayesian approach**

Bayesian inference of *θ* is obtained by the product of prior probability density distribution π(*θ*) and the likelihood *L*(*θ*|*x*) calculated based on the data *x*. The posterior density distribution for *θ* is:

In some cases there is not analytical solution for the integral in the denominator. A couple of numerical procedures can be used to obtain a sample of *θ* from a distribution (importance function) that is assumed to be similar to the true posterior *p*(*θ*|*x*). When using such numerical procedure it is necessary to verify if the sample of *θ* indeed was drawn from an importance function similar to the posterior distribution. Hereafter the question about how close are the posterior and the importance function is denominated "convergence". Details about the solutions and equations I have used are in Kinas (1996) and Andrade & Kinas (2007) but, some explanations are warranted in the following sections.

**Likelihood**

After taking the logarithm of equation 1 and for notational convenience, I define *Y _{t} =* log

*I*and

_{t}*μ*

*(*

_{t}*θ*)

*=*log(

*qB*). The probability model for

_{t}*Y*is:

_{t}Where *N*(*μ** _{t}* (

*θ*),

*V*) is an one-dimensional normal distribution with mean

*μ*

*(*

_{t}*θ*) and variance

*V*. Let the complete data set be

*Y*(

*τ*) = {

*Y*

_{1},...,

*Y*

*}, hence the joint likelihood is*

_{τ}*L*(

*θ*,

*V*|

*Y*(

*τ*))

*= p*(

*y*(

*τ*)|

*θ*

*,V*). If independent prior distributions are assumed for

*θ*and for

*V,*say

*π*(

*θ*,

*V*)

*=*π(

*θ*)π(

*V*), then the likelihood

*L*(

*θ*

*|x*) is obtained by marginalization with respect to

*V*:

Kinas (1996) showed that if the prior for *V* is an inverse-gamma density distribution, say *π*(*V*)* ~ IG*(*α**,ß*), the solution is:

That choice about prior distribution for *V* is convenient because the inverse-gamma is conjugated with the normal probability in equation 5. The resulting likelihood equation is simple in the sense that it only depends on the residuals with respect to *μ** _{t}* (

*θ*) and on the prior parameters for

*V.*

In order to calculate the likelihood (equation 6), biomass in each year is estimated by using the equation 2. Nevertheless, an initial value of biomass for the first year needs to be defined in advance. I have assumed the initial biomass is equal to the carrying capacity *k *(*i.e.,* *B*_{0} = *k*) because catch was probably very small before the first year to show up in the available time series (Fig. 1). Furthermore, when *B*_{0} = *k* is assumed the biases of estimations related to "maximum sustainable yield" (*MSY*) are not large whenever the observational error is used to fit the model (Punt, 1990).

**Prior**

The priors for *r* and *k* used in this paper were based mainly on priors the ICCAT WG have used in the bigeye assessment meetings (*e.g.,* ICCAT, 2005). In 2007 the ICCAT WG decided to use a uniform prior for *k* bounded at 1.5x10^{5} and 2.5x10^{6} (ICCAT, 2008). The informative prior for *r* was lognormal, with mean on the original scale equal to 0.6, and standard deviation on logarithm scale equal to 0.3 (*e.g., SD*(*log r*) = 0.3) in 2004 and 2007 stock assessments (ICCAT, 2005, 2008). In the 2010 assessment meeting the ICCAT WG decided to use again priors similar to those used in the two previous meetings (ICCAT, 2011).

The priors I have used are in line to those mentioned above. I have used a wide uniform prior *U*(1x10^{5}, 3.5x10^{6}) for *k* . Two prior distributions for *r* were used. One informative equal to that used in 2004 assessment meeting mentioned above and other less informative uniform *U*(0,2), and finally, the prior for *q* remains to be defined. In the past 2004 and 2007 assessment meetings *q* was estimated using an analytical shortcut (ICCAT, 2005, 2008) while the parameters of the prior distribution does not appear in the report of the 2010 assessment meeting (ICCAT, 2011). Hence I shall not use information of those past meetings. Because I did not find any other independent estimation of *q* for the Atlantic stock or even for other stocks of bigeye worldwide, I have decided to use a wide uniform prior for *q* on logarithm scale *U*(-30,-5). That prior is equivalent to the Jeffrey's non-informative prior (Millar, 2002). In summary I have used two sets of priors, one more informative (hereafter just denominated as "informative") and one less informative (hereafter denominated as "non-informative"). The main difference between them concerns the prior for *r* , which is lognormal in the informative and uniform in the non-informative prior set. Priors were always uniform on original scale for *k* and uniform on logarithm scale for* q*.

**Adaptive Importance Sampling (AIS)**

In the sampling importance scheme the density *p*(*θ*|*x*) is replaced by an importance function *g*(·) from which independent and identically distributed (*i.i.d.*) samples of *θ* can be easily drawn. For each vector drawn from the *g* (·) importance function, say *θ** _{i}* with

*i*= {1,...,

*n*}, a weight

*w*(·) can be calculated as the ratio between the kernel

*f*(·) =

*π*(·)

*L*(·) of equation 3 and the importance function:

The normalized weights are

Let *w ^{*}* be the vector with the values of

*w*

_{i}*with*

^{*}*i*= {1,...,

*n*}. A random sample from

*p*(

*θ|x*) is then approximated by resampling the sampled

*θ*s with probabilities given by the vector

*w*.

^{*}In the above approach a good importance function *g*(·) should have similar shape but heavier tails than *p*(*θ|x*) (Oh & Berger, 1992; West, 1992, 1993). In order to find a suitable importance function, West (1992, 1993) suggests starting with a first importance function (*e.g.,* a multidimensional student), to draw a sample of *θ* of size *n* and to calculate the corresponding weights *w ^{*}* just like explained above. The importance function is then updated and the procedure is repeated a couple of times until the importance function becomes close to the posterior.

In order to update the importance function for a *p*-bidimensional vector *θ* a Monte Carlo estimation of the first sample covariance matrix (*p* x *p*) is calculated as:

where the Monte Carlo mean vector is:

The importance function is then updated by calculating:

Where the right-most term is a *p*-dimensional student density with *a* degrees of freedom, mean *θ _{i}*, and variance

*h*where

^{2}xC*h*a smoothing parameter is denoted "bandwidth". I have used

*a*= 9 and the bandwidth suggested by Silverman (1986):

Where *b* = *p* + 4. The mixture model *g*(*θ*) approaches *p*(*θ**|x*) for increasing sample size *n* if the bandwidth decays at a suitable rate. West (1993) suggests that this can be achieved if a fairly small sample size is used in the first step and if the sample increases in an appropriate rate in the following updating steps. In this work the sample size in each cycle was calculated by:

where *S* is a counter for the looping; *d* is some positive constant. The initial sample size I have used was *n _{s=}*

_{1}= 4000 and the constant was

*d*= 2.

**Relative entropy and the final sample**

The goal when using the algorithm described in the previous sections is to update the importance function before drawing a final sample. The criterion I have used to assess if the updating procedure was successful is the Relative Entropy (*RE*) (West, 1993):

It is easy to show that *RE* is close to 1 if the importance function *g*(*θ*) is close to the posterior *p*(*θ|x*).

In the updating procedure I have selected *RE *≥ 0.95 as criterion to assume the algorithm has converged. Because there is no guarantee that the updating procedure will reach that entropy in a few updating steps I used a limit of ten cycles. A final sample of *θ* was drawn with approximate size equal to 0.15 x *n* without replacement with weights *w** ^{*}*. The choice 0.15 is close to 1 in 10 suggested by Smith & Gelfand (1992) while sampling without replacement is expected to work better in ill behaved cases with few large weights (Gelman

*et al.,*1995). In order to gather a final sample with 2000 vectors

*θ*the AIS sample might exceed 2000/0.15 13,334 values hence the calculations stop when both the entropy is higher than 0.95 and the minimum sample size (13,334) have been reached or, in the tenth cycle in case of failure.

**Time trend as predicted by the models**

In order to assess the status of the bigeye Atlantic stock in the last ten years of the time series the slope of regression of catch rate predictions against the years were calculated based on the posteriors. Basic statistics summaries were calculated for the 2000 (posterior sample size) values of slopes estimated for each model. The choice concerning the amount of years (10) to assess the regression slopes is subjective but it is enough to give some idea of the most recent catch rate (proxy of biomass) time trend as predicted by the models.

**RESULTS**

**Convergence and model fittings**

Overall entropy values close to or higher than 0.95 were reached in all model runs, hence I have assumed most of the models had converged. The exception was the calculation for Uruguay dataset (S3) when using the informative prior. In this case the entropy after ten cycles was 0.83. Hence those results might be considered carefully. In general high entropy values were obtained sooner when using informative prior than when using non-informative prior.

There are twenty six model fittings (2 sets of priors x (7 separated + 6 composite datasets)), but because there are some similarities only four fits are shown as example to not clutter (Fig. 3). Most of the models do not fits well to the values of the beginning of the time series, but fits well to the end of the time series. See for example the models fitted to C2 and C3 composite datasets with non-informative prior (Figs. 3a-3b).

**Figure 3.** Model fittings for: a) C3 composite dataset, b) C2

composite dataset, c) S5 Taiwan dataset and d) S7 Portugal-

Azores dataset. Non-informative priors were used in the

calculations. Dashed lines stand for 0.5% and 99.5% percentiles

while solid lines stand for the median.

Overall most of the model's predicted catch rates decrease until 2000 but calculations after that year are controversial. Only four examples are shown in Figure 3 to not clutter. Some of the model fittings suggest that there is not a clear time trend in the end of the series (Figs. 3b-3c), while other models suggest increasing (Fig. 3a) or decreasing time trends (Fig. 3d). Some of the datasets show high variability all across the years or during some decades. See for example the large range of the 99% credibility interval (dashed lines) calculated for Portugal-Azores dataset (S7) (Fig. 3d). In opposition the precision is high especially for the composite datasets (Figs. 3a-3b).

Notice that in spite of the similar data source considered to calculate the C2 and C3 composite datasets, fittings of the models are quite different in the end of time series. Hence the weights used to calculate composite datasets clearly affect the results. Remind that the C2 composite dataset was calculated by using total catch of each fleet as weight, and the number of 5° x 5° (latitude x longitude) squares covered by fleets, were the weights to calculate the C3 dataset (Table 1).

Models fitted to Brazil, Morocco and Taiwan longline datasets did not show any time trend. See for example the model fitted to catch rate of Taiwan longline fleet (Fig. 3c) that is not flexible enough to fit the peaks and plunges. Brazilian and Morocco model fittings also do not show clear time trends.

Most of the models are biased for the beginning of the time series in the sense that the central trend of the residuals depart from zero, but they are not strongly biased for years after 1995 (Fig. 4). Those models that are not strongly biased for the end of the time series are grossly classified as "well behaved" models, while those showing strong bias in the end of the time series are the "ill behaved" models. "Intermediate" models are those showing moderate biases in the end of time series. Well behaved models are those fitted to S1, S2, C3, C4, C5, C6 by using both informative and non-informative priors, and to C2 with informative prior. See Figure 4a for example. Intermediate models are those fitted to C1, S4 and S7 by using both informative and non-informative prior and to C2, S3 and S6 with noninformative prior. See Figs. 4b and 4d for example. Models fitted to S5 by using both priors and to S3 and S6 datasets with informative prior are the ill behaved models (Fig. 4c).

**Figure 4.** Residuals for models fitted to: a) C3 composite dataset, b) C2

composite dataset, c) S5 Taiwan dataset, and d) S7 Portugal-Azores dataset.

Non-informative priors were used in the calculations.

Statistical summary of the linear regression slopes of catch rate predictions (*proxy* for biomass) over the last 10 years of the time series, together with the empirical probability that the slope is positive, are shown in Table 2. Positive slopes indicate that biomass was recovering in 2000's. Most of the empirical distributions of the slopes are approximately symmetric as indicated by similar values of median and means. Values of coefficient of variation are high especially when the mean is close to zero. If we rely on the empirical probability that the slope is positive, most (four) of the models fitted to separated datasets with non-informative prior indicate that the probability that the stock was recovering in the 2000's is low. In opposition, if we rely on the calculations based on separated datasets and on informative priors, most of the models indicate that the probability that the stock was recovering in 2000's is moderate or high. Most of the models fitted to composite datasets with noninformative prior indicate that the probability that the stock was recovering in 2000's is also low. The opposite pattern arise when informative prior is used in calculations. In summary most of the calculations with the non-informative prior are pessimistic while the calculations with the restrictive informative prior are optimistic.

**Table 2.** Mean, median and coefficient of variation (CV) of simple linear regressions

slopes fitted to predictions of catch rate for the last ten years, and empirical probability

(Pr) that the slope is positive. Datasets: USA longline (S1), Japan longline (S2), Uruguay

longline (S3), Brazil longline (S4), Taiwan longline (S5), Morocco longline (S6), Portugal-

Azores bait-boat (S7), all data combined but not weighted (C1), all data combined

weighted by catch per year and per fleet (C2), all data combined weighted by area (C3),

all fleets but only 1971 forward weighted by catch (C4), all data combined weighted by

catch but 1970 back Japan longline discarded (C5) and all data combined weighted by

catch but 1970 Taiwan longline discarded.

**Posteriors**

Joint and marginal posteriors and priors for *r* and *k* are in Figure 5. The characteristic "banana type" shape of joint posteriors indicate that estimations of *r* and *k* for most of the datasets are strongly correlated (Figs. 5a, 5d, 5g, 5j). Exceptions are the results for S4 (Brazil), S5 (Taiwan) and S6 (Morocco) datasets (Figs. 5a, 5d). As a matter of fact, those three time series convey little information about the parameters of the surplus production model.

**Figure 5.** Posteriors of *r* (intrinsic growth rate) and *k* (carrying capacity). Datasets: USA

(S1), Japan (S2), Uruguay (S3), Brazil (S4), Taiwan (S5), Morocco (S6), Portugal-Azores

(S7), all data pooled (C1), all data pooled but weighted by total catch by year and fleet

(C2), all data pooled by weighted by the number of 5° x 5° squares (latitude x longitude)

covered by the fleet in the year (C3), all data pooled but only 1971 forward and weighted

by total catch by year and fleet (C4), all data pooled except 1970 back Japanese longline

but weighted by total catch by year (C5) and, all data pooled except 1970 back Chinese

Taipei longline but weighted by total catch by year (C6). Panels a, b, c, g, h and i are for

analyses with noninformative prior, while results gathered with informative prior are in the

other six panels. Contour lines are at 0.01 of the largest density.

Overall the effect of the informative prior on joint posteriors was to shift the kernel of contour plot to *r* values close to 0.5 and to *k* values lower than 1,500,000 ton when analyzing datasets that convey information (S4, S5 and S6). Hence it is evident that the restrictive informative prior for *r* is also very informative about *k* due to the high correlation between them.

Marginal posteriors calculated for S1, S2, S3, S7 and for most of the composite datasets give high densities to small values of *r* (<0.3) if the noninformative prior is used (Figs. 5b, 5h). If the informative prior is used the marginal posteriors shift to right (Figs. 5e, 5k) because that prior gives little weight to low values of *r* (<0.25). Posteriors of *r* for S4, S5 and S6 datasets are strongly affected by the informative prior. Because those three datasets do not convey information about *r* , if the informative prior is used, the posteriors are equal to the prior, but they are flat if the non-informative prior is used.

All separated datasets convey little information about *k*, hence the precisions of the marginal posteriors are low if the non-informative set of priors are used in the calculations (Fig. 5c). As a matter of fact S4, S5 and S6 convey no information about *k* and the posteriors are flat and truncated in the right tail by the prior (Figs. 5c, 5f). Posteriors of *k* as calculated for the datasets S1, S2, S3 and S7 are much more precise when informative priors are used (Fig. 5f). Densities are high for values close to 800,000 ton.

Composite datasets convey information about *k* but the calculated posteriors are contradictory. For example, if the non-informative priors are used the mode of the posterior for C3 dataset (all data weighted by area) is close to 600,000 ton, while the mode of the posterior for C4 (only 1971 forward data weighted by catch) is close to 3,000,000 ton (Fig. 5i). The modes of the more precise posteriors as calculated with informative prior are close to 1,000,000 ton for all the composite datasets (Fig. 5l).

Posteriors for *q* are not shown to not clutter. Nevertheless most of the comments above concerning estimations and effects of priors on the posteriors of *r* and *k* are also valid for posteriors of q. In summary, all marginal posteriors were unimodal and the precision of the calculated posteriors with informative prior are usually higher than those calculated with the noninformative prior.

**Maximum Sustainable Yield and ratios** *F/F _{MSY}* and

*B/B*

_{MSY}Posteriors of yield at *MSY *(*Y _{MSY}*) are in Figure 6. Precision of posteriors calculated for USA (S1), Japan (S2), Uruguay (S3) and Portugal-Azores (S7) were large, while posteriors calculated for the other three separated datasets (S4, S5 and S6) were flat (Figs. 6a-6b). All posterior samples were pooled to summarize the results. The "pooled posterior" for separated datasets has a mode close to 73,000 ton when using non-informative prior (Fig. 6a) and a mode close to 91,000 ton as calculated using informative prior (Fig. 6b).

**Figure 6.** "Maximum sustainable yield" as calculated using alternative datasets:

United States (S1), Japan (S2), Uruguay (S3), Brazil (S4), Taiwan (S5), Morocco

(S6), Portugal-Azores (S7), all data pooled (C1), all data pooled but weighted by

total catch by year and fleet (C2), all data pooled by weighted by the number of

5° x 5° squares (latitude x longitude) covered by the fleet in the year (C3), all

data pooled but only 1971 forward and weighted by total catch by year and fleet

(C4), all data pooled except 1970 back Japanese longline but weighted by total

catch by year (C5) and, all data pooled except 1970 back Chinese Taipei longline

but weighted by total catch by year (C6). Densities for pooled samples of posteriors

are also showed (thick dashed lines). Calculations with non-informative priors are

in panels A and C while results gathered with informative priors are in panels b and d.

Estimations of *Y _{MSY}* for composite datasets as calculated with non-informative priors are contradictory (Fig. 6c). The more optimistic scenario is the one calculated for C3 (

*Ymsy*mode close to 100,000 ton) while calculations for the C4 and C5 datasets are pessimistic (

*Y*mode close to 50,000 ton). The modes of the posteriors of

_{MSY}*Ymsy*for composite datasets calculated with informative prior have shifted to right if compared to those calculated with non-informative prior. Contour plots based on the ratio between fishing mortality (

*F*) in 2009 and at

*MSY*and on the ratio between biomass (

*B*) in 2009 and at

*MSY*are shown in Figure 7. Optimistic scenarios are indicated by

*F*

_{2009}/

*F*≤ 1 and

_{MSY}*B*

_{2009}/

*B*≥ 1 while pessimistic scenarios are indicated by

_{MSY}*F*

_{2009}/

*F*1 and

_{MSY}>*B*

_{2009}/

*B*1. Calculations based on S1, S2, S3 and S7 datasets point for a pessimistic scenario while those results calculated with the datasets that do not convey information about

_{MSY}<*r*and

*k*(S4, S5, S6) are very precise and optimistic. The effect of the informative prior is to shift the kernel of contour plots for S1, S2, S3 and S7 datasets to a less pessimistic scenario and, to shift the kernel of contour plots for S4, S5 and S6 datasets to a less optimistic scenario. Calculations with non-informative priors for composite datasets are more pessimistic than calculations with informative priors (Figs. 7c-7d). Kernels of contour plots calculated for composite datasets are in between those calculated for the more informative (S1, S2, S3, S7) and less informative (S4, S5, S6) separated datasets.

**Figure 7.** Contour plots as calculated with non-informative priors (a and c)

and with informative priors (b and d). Datasets: United States (S1), Japan

(S2), Uruguay (S3), Brazil (S4), Taiwan (S5), Morocco (S6), Portugal-Azores

(S7), all data pooled (C1), all data pooled but weighted by total catch by

year and fleet (C2), all data pooled by weighted by the number of 5°x5°

squares (latitude x longitude) covered by the fleet in the year (C3), all data

pooled but only 1971 forward and weighted by total catch by year and fleet

(C4), all data pooled except 1970 back Japanese longline but weighted by

total catch by year (C5) and, all data pooled except 1970 back Chinese

Taipei longline but weighted by total catch by year (C6). Contour lines are

at 0.1 of the largest density.

Posterior probabilities calculated for optimistic, intermediate and pessimistic scenarios are in Table 3. If we rely on the calculations for S1, S2, S3 and S7 datasets we would be almost sure that the stock was overexploited in 2009, but if we rely on calculations for S4, S5 and S6 datasets we would be almost sure that the stock was underexploited. The effect of the priors is also important. Calculations for separated datasets based on the informative prior are more optimistic. Calculations for some of the composite datasets are also very sensitive to priors. Most of the calculations with the non-informative prior indicate that the stock was overexploited in 2009. However, only three models indicate overexploitation if the informative prior is used. Calculations for C2 and C6 datasets are the more sensitive to prior. If the non-informative priors are used, the results indicate that the probability of overexploitation is high, but the opposite arise when using informative priors (Table 3).

**Table 3.** Probabilities based on ratios between fishing mortality (*F*) and biomass

(B) in the last year (2009) with respect to *F* and *B* at *MSY. pa =* pr(*F*_{2009}/*F _{MSY }>*1

*B*

_{2009}/

*B*1);

_{MSY}<*pb =*Pr(

*F*

_{2009}/

*F*

_{MSY}*>*1

*B*

_{2009}/

*B*≥1);

_{MSY}*pc*= Pr(

*F*

_{2009}/

*F*≤ 1

_{MSY}

*B*

_{2009}/

*B*<1);

_{MSY}*Pd =*Pr(

*F*

_{2009}/

*F*≤1

_{MSY}

*B*

_{2009}/

*B*≥1). Datasets: USA longline (S1),

_{MSY}Japan longline (S2), Uruguay longline (S3), Brazil longline (S4), Taiwan longline (S5),

Morocco longline (S6), Portugal-Azores longline (S7), all fleets combined - no weight

(C1), all fleets combined - weighted by catch (C2), all fleets combined - weighted by

number of 5°x5° squares covered by fleet in the year (C3), all fleets but only 1971

forward - weighted by catch (C4), all fleets except 1970 back Japan - weighted by

catch (C5), all fleets except 1970 back Taiwan - weighted by catch (C6).

In summary the assessment results were very sensitive to priors and CPUE series used. Model estimations were less robust when non informative priors were used; in the sense that calculations based in different datasets are contradictory. Overall, the assessment results were optimistic when informative priors and composite CPUE series were used.

**DISCUSSION**

Surplus production model is not the only way to calculate *r*. Some of the other methods to calculate *r* are based on population dynamics parameters (*e.g.,* growth and, age and length at maturity) like, for example, solving classical Euler-Lotka equation (Fisher, 1930) for *r.* There is an example on how to calculate *r* by that method in the site of FLR-project (http://flr-project.org) that is a collection of packages to conduct quantitative fisheries science. Coincidentally the example is about Indic Ocean bigeye stock. After running the code one gets *r* 0.66. As a matter of fact, the informative prior I and ICCAT WG (*e.g.,* ICCAT, 2005, 2008, and 2011) have used gives high weight to values close to 0.66. However, the posteriors give more weight to low values of *r* , the information conveyed by most of the datasets contradicts that informative prior. In the past ICCAT stock assessment meetings (*e.g.,* ICCAT, 2005) the restrictive prior for *r* was considered not ideal but necessary because the data were not informative enough to get meaningful estimations of the production model's parameters. Hence the conclusion is that the commercial CPUE dataset are not very useful to run production models for bigeye.

The shortcomings of using CPUE from commercial fisheries in assessment of fish populations are well-documented (*e.g.*, Beverton & Holt, 1957; Hilborn & Walters, 1992; Swain & Sinclair, 1994; Harley *et al.,* 2001). In the case of the Atlantic bigeye there are at least three questions: *i*) the fishing scenario was the "one way trip" (monotonous increasing trend of fishing effort and of catch along with monotonous decreasing trend of CPUE) which result in poor relative abundance indexes (see Hilborn & Walters, 1992). In the bigeye case the monotonous pattern appeared during most of the analyzed period (1950 through mid 1990's); *ii*) the relationship between CPUE and abundance is not linear and violates the usual assumption; and *iii*) the available relative abundance indexes are not very useful because data and/or the model used to standardize the CPUE are not the ideal. The solution of the question *i* depends of the future development of fishery that is related to complex political, ecological and socioeconomic issues. Questions *ii* and *iii* could be further studied and maybe overcome if the quality and quantity of the data improve and if fishery independent data are collect. However, scientific programs for highly migratory species are very expensive and difficult to implement (Bishop, 2006).

The restrictive informative prior used to overcome the poor relative abundance indexes has a strong effect on the calculations and conclusions about the status of the stock especially in the end of the time series. Let *Y _{MSY}* as benchmark to illustrate. If annual catches of bigeye are compared to posterior of

*Y*benchmark the conclusion is that the stock was overfished from the beginning of 1990's until the beginning of 2000's if one relies in the results gathered with informative prior. However, results calculated with non-informative prior indicate that the stock was already overfished in the end of 1980's and that it has been overfished since them. The less pessimistic scenario calculated with the informative prior arise because it constrains

_{MSY}*r*to high values, while the majority of the datasets point for a much less productive stock.

In spite of the recommendation to use separated indexes (Richards, 1991) or to use models that allow the information conveyed by separated indexes to show up in the results (Schnute & Hilborn, 1993), ICCAT WG have been often adopted composite indexes. The motivations are: a) "to produce a single series for input in an assessment model either to minimize model convergence problems arising from conflicting indexes"; and b) "to integrate and summarize the information provided by multiple indexes into a single trend ... to compare trends of abundance for the overall population ..." (ICCAT, 2012). The objective in this paper was not to confront those conflicting points of view in an attempt to identify which is better (separated *x* composite datasets). The objective was to assess what are the results and conclusions when using the different dataset types when analyzing Atlantic bigeye. Sensitivity analyses concerning the effect of discarding suspected data when calculating composite time series were carried out in the last two stock assessments (ICCAT, 2008, 2011). In the last assessment of Atlantic bigeye stock, Japanese and Taiwanese catch rate data of the very beginning of the time series were considered suspected because of the increasing and sharp decreasing time trends respectively. Increasing trends are not expected when the fishery starts. Also quick decreasing trend of relative abundance indexes are not expected because the catches were not high in the beginning of the fishery. Therefore three of the six composite time series calculated by ICCAT WG do not include those suspect data. On the other hand, Brazil and Morocco datasets were all considered in the calculations of the composite datasets in spite of the meaningless results that arise when those data are analyzed separated. As a matter of fact, Brazilian, Moroccan and even the Taiwanese datasets do not convey any information about the parameters of the production model. A key question is why to consider such datasets when calculating composite indexes?

In a report published recently there are protocols for the inclusion or use of CPUE series in assessment models (ICCAT, 2012). The protocols are based on several characteristics of the data used in the standardization calculations, on the diagnostic of the model used to standardize CPUE, on biological plausibility of the CPUE time series and on many other criteria. However, to assess the quality of the information, it is an alternative to check if the parameter calculations based on the datasets are meaningful. If this criterion is adopted, Brazilian, Moroccan and Taiwanese datasets should not be considered in the calculations of the composite indexes. If the results of the stock assessment gathered with the different composite datasets point for the same stock status, the discussion about weights and procedures used to calculate composite datasets are of secondary importance. However this is not the case for bigeye. Posteriors were more affected by the choice about the procedure used to calculate the composite datasets (*e.g.*, weighting by catch or by area) than by the choice about discarding suspected data of the beginning of the time series. While the choice of the weights is of major importance it is usually subjective. Alternative approaches to calculate composite indexes are in ICCAT (2012) but there is not guidance to assess how suitable are a given weighting criterion. Uncertainty that arise in the posteriors when analyzing the original separated dataset is not properly depicted in the calculations for the composite datasets. One might keep in mind that because the composite datasets are "averages", the high precision of the calculations is artificial. Therefore the results of stock assessment based on composite datasets should be carefully considered whenever they are used as guidelines for management decisions and recommendations.

If one rely on *Y _{MSY}* benchmark, most of the results indicate that probably the Atlantic bigeye stock have been overfished during 1990's. Also, the calculations of probability that

*F*

_{2009}/

*F*1 and that

_{MSY}>*B*

_{2009}/

*B*1 indicate that bigeye stock was still overfished in the end of 2000's. Nevertheless, there is doubt about if the stock has begun to recover since the beginning of 2000's. If we rely on those separated datasets that convey some information (S1, S2, S3 and S7) or, in the results obtained for composite datasets with non-informative priors, the answer is no, probably the stock was not recovering in the 2000's. If we rely on the results gathered with composite datasets and informative prior, the opposite answer arises. In summary the diagnostic about the time trend of the abundance of Atlantic bigeye depends on the dataset and on the prior considered.

_{MSY}<Finally it is important to recognize that the analysis showed in this paper is just one among several possible approaches. The model used here is a simple single species one, and the usefulness of such models as tool for stock assessment and management is debatable (Hollowed *et al.,* 2000; Prager, 2002, 2003; Maunder, 2003; Walters *et al.,* 2005). Therefore, this paper might not be the only document considered when assessing the Atlantic bigeye stock. Instead the paper was wrote to show how informative are the available datasets to run production models and how important are the issues regarding the alternatives composite *versus* separated datasets and non-informative *versus* informative priors for Atlantic bigeye tuna.

**CONCLUSIONS**

Relative abundance indexes for bigeye as calculated using commercial data are not very informative about parameters of surplus production models; hence scientific programs to obtain fishery independent relative abundance indexes for Atlantic bigeye are encouraged. Most of the results of the production models indicate that bigeye stock was overexploited in 1990's. However the conclusion about the possibility that the stock was recovering in 2000's depends on the dataset and on the prior.

**AKNONWLEDGMENTS**

I am grateful to two anonymous reviewers for the valuable comments and constructive criticisms.

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*Received: 28 June 2013; Accepted: 14 October 2014*