ANALYSIS OF AGGREGATED FISHING DATA USING DELTA GENERALIZED LINEAR MODELS: WHITE MARLIN (TETRAPTURUS ALBIDUS) CAUGHT BY THE BRAZILIAN LONG-LINE FLEET

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1 SCRS/2005/105 Col. Vol. Sci. Pap. ICCAT, 59(1): (2006) ANALYSIS OF AGGREGATED FISHING DATA USING DELTA GENERALIZED LINEAR MODELS: WHITE MARLIN (TETRAPTURUS ALBIDUS) CAUGHT BY THE BRAZILIAN LONG-LINE FLEET Humber A. Andrade 1 SUMMARY Commercial catch and effort fishing data are often modeled using generalized linear models to standardize the catch rates in an attempt to estimate relative abundance of tuna and tuna-like populations. When several fishing operations result in catches equal to zero, the use of a delta model is one alternative. In the delta model framework the positive catches and the proportion of positive catches are analyzed separated and then the results of both models are used to obtain the estimation of relative abundances. The binomial density distribution is suitable for modeling the proportion of positive catches. The application of this approach is straightforward when fishing sets are detailed in reports, then each fishing operation can considered as a try in a Bernoulli experiment. However, detailed fishing data sets are not available in several historical statistics reports. In the Brazilian longline fishing statistics the effort has been aggregated per year, month, kind of fleet and square (e.g. 5º lat. x 5º long.). In each line of the aggregated data set there is the total number of hooks and the catch of the species gathered in all fishing sets carried out per year, month, fleet and square. Hence, the estimations of the number of tries and the number of successes are not straightforward. In this paper I tried out three approaches to estimate the proportion of positive fishing operations in order to standardize commercial catch rate using generalized linear models. I used data on white marlin (Tetrapturus albidus) caught by the Brazilian national and leased fleets as case of study. The standardized catch rates were estimated by back-transforming the coefficients of factor year using the inverse of the link function. The results indicated that the standardized estimations were not strongly affected by the three methods used to approximate the number of tries and success from the aggregated fishing data sets. RÉSUMÉ Les données commerciales de prise et d effort de pêche sont souvent modélisées à l aide de modèles linéaires généralisés visant à standardiser les taux de capture afin d essayer d estimer l abondance relative des populations de thonidés et d espèces apparentées. Lorsque plusieurs opérations de pêche entraînent des prises égales à zéro, l utilisation de modèles delta représente une alternative. Dans le cadre du modèle delta, les captures positives et la proportion des captures positives sont analysées séparément et les résultats des deux modèles sont ensuite utilisés pour obtenir l estimation des abondances relatives. La distribution de la densité binomiale est appropriée pour modéliser la proportion des captures positives. L application de cette approche est simple lorsque les opérations de pêche sont détaillées dans les rapports, chaque opération de pêche pouvant alors être considérée comme un «essai» dans l expérimentation de Bernoulli. Toutefois, des jeux de données de pêche détaillées ne sont pas disponibles dans plusieurs rapports statistiques historiques. Dans les statistiques de la pêche palangrière brésilienne, l effort a été agrégé par année, mois, type de flottille et carré (p. ex. 5º lat. x 5º long.). Sur chaque ligne du jeu de données «agrégées», figure le nombre total d hameçons et la prise des espèces capturées dans toutes les opérations de pêche réalisées par an, mois, flottille et carré. C est pourquoi il n est pas simple d estimer le nombre «d essais» et le nombre de «succès». Dans le présent document, j ai essayé trois approches visant à estimer la proportion des opérations de pêche positive dans le but de standardiser le taux de capture commercial à l aide de modèles linéaires généralisés. J ai eu recours à des données sur le makaire blanc (Tetrapturus albidus) capturé par les flottilles brésiliennes nationales et en location en tant qu étude type. Les taux de capture standardisés ont été estimés en 1 UNIVALI/CTTMar Rua Uruguai, , Itajaí (SC), Brazil; humber.andrade@univali.br. 335

2 rétrotransformant les coefficients du facteur «année» en utilisant l inverse de la fonction lien. Les résultats ont indiqué que les estimations standardisées n ont pas été fortement affectées par les trois méthodes utilisées pour se rapprocher du nombre «d essais» et de «succès» à partir des jeux de données de pêche agrégées. RESUMEN Los datos de captura comercial y esfuerzo pesquero se modelan a menudo utilizando modelos lineales generalizados para estandarizar las tasas de captura en un intento de estimar la abundancia relativa de las poblaciones de túnidos y especies afines. Cuando varias operaciones de pesca tienen como resultado unas capturas iguales a cero, una alternativa es la utilización de modelos delta. En el marco de un modelo delta, las capturas positivas y la proporción de capturas positivas se analizan de modo independiente, y después se utilizan los resultados de ambos modelos para obtener la estimación de las abundancias relativas. La distribución de densidad binomial es apropiada para modelar la proporción de capturas positivas. La aplicación de este enfoque es sencilla cuando se detallan los lances de pesca en los informes, entonces cada operación pesquera puede considerarse un intento en un experimento de Bernoulli. Sin embargo, en muchos informes estadísticos históricos no se dispone de conjuntos de datos pesqueros detallados. En las estadísticas pesqueras de la pesquería de palangre de Brasil, el esfuerzo se ha agregado por año, mes, tipo de flota y cuadrícula (por ejemplo 5º latitud x 5º longitud). En cada línea del conjunto de datos agregados se incluye el número total de anzuelos y la captura de las especies agrupadas en todos los lances de pesca realizados por año, mes, flota y cuadrícula. De aquí que las estimaciones del número de intentos y del número de éxitos no sean sencillas. En este documento he intentado tres enfoques para estimar la proporción de operaciones positivas de pesca con el fin de estandarizar la tasa de captura comercial utilizando modelos lineales generalizados. Utilicé los datos de aguja blanca (Tretapturus albidus) capturada por las flotas brasileñas nacional y fletada como caso de estudio. Las tasas de captura estandarizadas se estimaron retro-transformando el coeficiente de factor año utilizando la inversa de la función vínculo. Los resultados indicaban que las estimaciones estandarizadas no se veían muy afectadas por los tres métodos utilizados para acercar el número de intentos y éxitos a partir de los conjuntos de datos de pesca agregados. KEYWORDS Catch/effort; Catchability; Fishing effort; Longlining; By-catch 336

3 1. Introduction Generalized linear models (GLM) are often used to standardize catch rate in order to estimate relative abundance estimations, which are necessary for stock assessment analysis (e.g. Gavaris, 1980; Hilborn and Walters, 1992). In the tuna long-line fisheries, catches of some species (e.g. white marlin-tetrapturus albidus) are often zero. The usual approach is to model the proportion of positive catches and the catch rate of the positive catches separately. The standardized catch rate index is the product of the standardized estimations gathered with the two models (Lo et al., 1992; Vignaux, 1994). The binomial distribution is the alternative to model the proportion of positive catches. The lognormal density distribution is often used to model the positive catches. The models built using this approach are usually called delta-lognormal models (e.g. Punt et al., 2000; Maunder and Punt, 2004). If each fishing set (long-line set and retrieval) is reported and if the number of hooks and the soaking time of each fishing operation is the same, the estimation of the proportion of positive fishing sets is easy. Each fishing operation could be considered a try in a Bernoully experiment. Positive catches are successful tries. The binomial model can be used in the analysis of the proportion of successes. However, in several fishing data sets there is not detailed information on each fishing operation. For example, the effort and catch data is aggregated in the task 2 ICCAT data set. In this data set there are information on total number of hooks and the total catch of all fishing operations carried out per year, month, area and fleet. Therefore the estimation of the proportion of positive fishing sets is not straightforward. There is not a detailed and comprehensive data set about the Brazilian long-line fishery on the west of the South Atlantic Ocean. Thought it is not the ideal, to analyze the task 2 aggregated data set is the alternative. The main obstacle is to properly estimate the number of tries and successful tries using the aggregated data. In this paper I used Brazilian catches of white marlin as case of study. I evaluated three alternatives to estimate the proportion of positive fishing sets. The standardized catch rates as calculated using the three alternatives were compared. 2. Data and analysis 2.1 Data set and the model for positive fishing sets The data set from the Brazilian long-line fleet (national plus leased boats) that operated from 1991 to The data set used is the same that was described in the paper of Andrade (2006) (this volume). Hence, here I just highlight some issues that were not detailed there. Catch (tones) and effort (number of hooks) are aggregated and summed per year, month, flag and area (e.g. 5º latitude x 5º longitude) in the data set. The catch rate I considered is C (1) U = 1000 f where C is the catch (t) and f is the nominal effort (number of hooks). The soaking time in the Brazilian fishing sets has been approximately the same for all fleets and it has not changed across the years. Usually the long-line is set in the end of the day and it is retrieved in the morning of the next day. Hence soak time is approximately constant and it was not taken into account in the equation 1. Although the number of hooks in the long-lines usually ranges from 500 to 2500, they are usually close to 1000 hooks. Hence I assumed that U is the nominal catch rate in each fishing operation with 1000 hooks. The vector of U values larger than zero is the response variable in the generalized linear model built to analyze the positive fishing sets. The year, month, area and kind of fleet factors are the explanatory variables included in the model. These factors are those defined in the paper of Andrade (2006). There are two levels in the factor area : south of 15º S and north of 15º N (Figure 1). The levels of the factor kind of fleet are fleet 1 (Brazilian national plus leased boats from Honduras, Spain and USA) and fleet 2 (leased boats from Saint Vincent and Taiwan). 337

4 Andrade (2006) showed that the swordfish is one of the main fishing targets of the fleet 1, while albacore and bigeye are the main targets of the fleet 2. The catch rate has been assumed to have a lognormal distribution in most of the GLM analyses (e.g. Gavaris, 1980; Kimura, 1981; Hilborn and Walters, 1992). I also assumed the lognormal density distribution here. The notation I adopted is similar to that showed in Andrade (2006). The response variable in the model for positive observations is (2) y log( U ) The expected value of y in the s th fishing set ( μ s ) is (3) μ = θ + s 0 p m j j= 1 k = 1 θ x jk ( i) jk + p 1 p m m j j' j= 1 j' = j+ 1k = 1 k' = 1 γ ' ' ( j, k )(, j, k ) () i () i x x + ε jk j' k' s where the explanatory components are θ 0 - expected value of log(u) when all factors are taken at their baseline θ jk - effect of level k (k = 2,..., m j ) of factor j on μ s with respect to the baseline θ j1 0 γ ( j, k )(, j', k' ) - interaction effect between θ jk and θ j'k' with respect to the baseline γ ( j,1)(, j', k' ) 0 k' = 1,..., m j' γ ( j, k )(, j',1) 0 k = 1,..., m j () i 1, if level k of factor jis present in observation i x jk = 0, otherwise 2 ε ~ independent identically distributed (i.i.d) random variables with Normal distribution ( 0, σ ) s N. In order to select the factors that significantly affect the catch rate I used the Akaike Information Criterion - AIC 2 (Akaike, 1974) and the deviance analysis ( χ -test - α = 0.05) (McCullagh and Nelder, 1989; Dobson, 2002). The residuals were analyzed in order to verify if the fit of models were acceptable. 2.2 Model for the probability of a positive observation In the available data set it was reported aggregated the total number of hooks used and the total catch gathered in several fishing sets. The explanation of the three approaches I used to estimate the number of fishing sets (or tries) (n) and the number of successful fishing sets (k) for each line of the data set follows bellow. Procedure A Regardless of the number of hooks (i.e. effort) reported, each line of the data set was considered as a try. If the white marlin catch was positive, the try was considered a successful one (k = 1). Procedure B - All long-lines were assumed to have close to 1000 hooks, hence the total number of fishing sets (or tries) is: (4) n = max[ 1, round( f /1000)] It was also assumed that all the white marlin weight reported in each line was caught in one fishing set, hence the number of positive fishing sets is one (i.e. k = 1). Procedure C The number of fishing sets were calculated as described above in the procedure B. The average weight of white marlin caught per successful fishing set ( w ) was calculated by: i p n 1 w i i= 1, n= 1 =, (5) = w = 338

5 where i = {1,..., p} is an index of the p lines of the data base where only one fishing set was reported (n = 1). is the weight of white marlin caught reported in the i th line of the data set. The number of positive fishing sets is then estimated by: w i (6) k 0, if C = 0 = max 1, round( [ C / w) ], if C > 0 The three approaches described above were used to estimate the proportion of positive fishing sets ( π = k / n ). Usually the binomial density distribution and the logit link function are used to model proportion data (McCullagh and Nelder, 1989; Dobson, 2002). Hence the response variable of interest is π (7) z log 1 π and the model for the expected value of z in the d th crossing level of the factors year, month, kind of fleet and area ( ω d ) is similar to that showed for μ s in equation (3). However for consistency the notation for the 2 error term is ε d instead of ε s. The AIC and the χ -test for the deviance (α = 0.05) were used to select the significant factors. 2.3 Standardized catch rate calculations In order to obtain the standardized catch rates the first step was to back-transform the coefficients estimated for the factor year by using the inverse of the link functions. The product of the back-transformed values gathered with the lognormal and the binomial models were used as estimations of standardized catch rates (e.g. Searle, 1980; Vignaux, 1994; Stefánsson, 1996). This approach is valid when the coefficients estimated for the levels of year factor are not biased due to the effects of catchability variations (Hilborn and Walters, 1992). 3 Results and discussions The number of reports before the end of 1990 s is very small. Data on fleet 1 level of kind of fleet was available only after 1999 (Figure 2), hence the design matrix is unbalanced for year and kind of effort crossing levels. The balance among crossing levels of the other factors are reasonable. Positive catch rates were generally large in 1991 and 1997 (Figure 3 A). The large variability of the positive catch rates in these two years are also of concern. The catch rates increased from August to October but they were generally large in the end of the years (Figure 3 B). There are no large differences between the catch rates of the two levels of kind of fleet and area factors (Figures 3 C and D). The differences among the proportions of positive fishing sets as calculated using the procedures A, B and C were large for factors year, month and kind of fleet (Figures 4, 5, and 6). The proportion of positive observations estimated for 1991 and 1992 were large as calculated using the procedures A and C, while large proportions were found from 1999 to 2004 when the procedure B was applied. The proportions of positive observations were large in the last months of the years as calculated using the procedure C, but the distribution of the positive proportions was uniform across the years as calculated using procedures A and B. Finally, if the procedure B is used, the proportion of positive fishing sets becomes remarkably small for kind of fleet 2 that fishes mainly albacore and bigeye. All the main factors were included in the selected models (Table 1). Despite the main factors month, area and kind of fleet were sometimes not significant, they were kept in the models because there were part of significant interactions. Notice also that the factors month and kind of fleet were not significant in the model for the positive fishing operations (i.e. positive catch rates) but they were usually significant in the models for the proportion of positive catch rates. The interactions between year and kind of effort were excluded because there were not balanced in the design matrix (see Figure 2). All other interactions comprising year were not strongly relevant, hence they were excluded. 339

6 The coefficient of determination was 0.28 for the lognormal, and 0.53, 0.36 and 0.39 for the binomial models using the procedures A, B and C, respectively. In all the models the final residual deviances were larger than the residual degrees of freedom. The deviance of the selected binomial model as calculated using the procedure C was particularly large. Normal plots of standardized residuals are in Figure 7. Notice that the residuals distribution of the model C is of concern. The estimations of the model C are probably biased. All coefficients estimated for the year factor were negative. The excerption was the estimation for 1997 in the lognormal model, but it is not significantly different of zero (α = t-test) (Table 2). Most of significant coefficients estimated for month factor were positive. The positive effects appeared in the middle of the year, mainly around September and October. The effects of the south area were negative in the binomial models A and B, but positive in the lognormal and in the binomial model C. Finally, notice that the effect of level 2 of the kind of effort factor is positive in the models A and C, but negative in the binomial model B. The standardized indexes and the means of nominal catch rates are shown in Figure 8. All the estimated values were divided by the estimation of the first year to show up in the data set. Hence the values showed in the Figure 8 are scaled with respect to the effect of the 1991 year. Time trends of the standardized catch rates were similar, though the mean nominal catch rate gathered when used the procedure B was different from the others. Estimations of nominal catch rates as calculated using procedure B were larger from 1999 to 2004 than the estimations gathered using procedures A and B. In three time series of standardized indexes appeared a sharp and probably unrealistic catch rate decline from 1991 to In the middle of 1990s the standardized indexes increased peaking in 1997, but after 1999 there was a slight decreasing trend. The large variances of the catch rates calculated for the positive fishing data set in 1991 and 1997 are of concern (Figure 3). Estimations of the proportions of positive fishing operation of 1991 and 1992 gathered using the procedures A and C, and of 2004 gathered using the procedure B, were are also too large (Figures 4, 5 and 6). The issues cited above indicate that the catch or/and the effort data reported for 1991, 1992, 1997 and 2004 are suspicious. Notice that the scaled standardized catch rates estimations were strongly affected by the data of 1991, 1992 and 1997 (Figure 8). If the estimations gathered for 1991, 1992 and 1997 are not taken into account the standardized catch rate estimations indicate that there were not large oscillations of the relatively abundance from 1993 to On the other hand, if data from 1991, 1992 and 1997 are taken into account, the diagnosis would be that a sharp decline in the relative abundance may have occurred in the beginning of 1990 s. Therefore, those suspicious data should be critically examined if the intention is to use the standardized catch rates estimations as relative abundance indexes. The three time trends of the standardized catch rates as calculated by back-transforming the estimations of the coefficients for factor year were similar. The results of the standardization analyses were not affected by the procedures of calculating the proportion of positive fishing operations. Therefore, the approaches used to transform aggregated data using methods to re-estimate lost details did not worth. The more complex approach C, probably results in more realistic estimation of the proportion of positive sets, it did not result in improvements in the estimation of the standardized catch rates. If the intention is to standardize catch rates, among the three approaches I tested the simplest is also the best alternative (i.e. procedure A) of estimating the proportion of positive fishing sets when analyzing aggregated data set. References AKAIKE, H A new look at the statistical identification model. IEEE Transactions on Automatic Control, 19: ANDRADE, H. A Standardized CPUE for blue marlin (Makaira nigricans) caught in the west of South Atlantic. Col. Vol. Sci. Pap. ICCAT, 59(1): DOBSON, A. J An introduction to generalized linear models. 2 nd Edition. Chapman & Hall/CRC. 225 pp. GAVARIS, S Use of a multiplicative model to estimate catch rate and effort from commercial data. Can. J. Fish. Aquat. Sci., 37:

7 HILBORN, R and C. Walters Quantitative Fisheries Stock Assessment Choice, Dynamics and Uncertainty. London, Chapman and Hall. 570p. KIMURA, D. K Standardized measures of relative abundance based on modeling log (CPUE) and their application to Pacific Ocean perch (Sebastes alutus). J. Cons. Int. Explor. Mer, 39: LO, N. C., L. D. Jacobson and J. L. Squire Indices of relative abundance from fish spotter data based on delta-lognormal models. Can. J. Fish. Aquat. Sci. 49: MAUNDER, M. N. and A. E. Punt Standardizing catch and effort data: a review of recent approaches. Fish. Res. 70: McCULLAGH, P and J. A. NELDER Generalized Linear Models. London, Chapman & Hall. 513p. PUNT, A. E, T. I. Walker, B. L. Taylor and F. Pribac Standardization of catch and effort data in a spatially-structured shark fishery. Fish. Res. 45: SEARLE, S. R., F. M. Speed and G. A. Milliken Population marginal means in the linear model: An alternative to least squares means. The American Statistician 34(4): STEFÁNSSON, G Analysis of groundfish survey abundance data: combining GLM and delta approaches. ICES J. Mar. Sci. 53: VIGNAUX, M Catch per unit effort (CPUE) analysis of west coast South Island Cook Strait spawning hoki fisheries, NZ Fisheries Association Research Document No. 94/

8 Table 1. Analysis of deviance table for the lognormal and binomial models as calculated using the procedure A, B and C to estimate the proportion of positive fishing sets. LOGNORMAL MODEL Df Deviance Resid. Df Resid. Dev P(> Chi ) NULL Year Month E-08 Kind of fleet Area Month:Kind of fleet Month:Area BINOMIAL MODEL A Df Deviance Resid. Df Resid. Dev P(> Chi ) NULL Year Month Kind of fleet E-08 Area E-06 Month:Kind of fleet E-11 BINOMIAL MODEL B Df Deviance Resid. Df Resid. Dev P(> Chi ) NULL Year Month Kind of fleet Area Month:Kind of fleet Kind of fleet:area E-06 BINOMIAL MODEL C Df Deviance Resid. Df Resid. Dev P(> Chi ) NULL Year Month Kind of fleet E-10 Area E-08 Month:Kind of fleet Month:Area Kind of fleet:area E

9 Table 2. Coefficients estimated for the main factors using lognormal and binomial density distributions to model the positive catch rates and the proportion of positive fishing sets. Letters A, B and C indicate the results gathered using the three alternatives to estimate the proportion of positive fishing sets. All coefficients for year are in the table, but only the significant estimations for factors month, kind of effort and area are shown bellow. Lognormal model Binomial model A Binomial model B Binomial model C Estimate Std. Error Pr(> t ) Estimate Std. Error Pr(> z ) Estimate Std. Error Pr(> z ) Estimate Std. Error Pr(> z ) (Intercept) (Intercept) (Intercept) E-17 (Intercept) year year year year year E-08 year E-06 year E-07 year year E-05 year E-09 year E-05 year year year E-06 year E-05 year year year E-09 year E-05 year year year E-10 year year year E-05 year E-08 year E-06 year year year E-12 year E-05 year year year E-14 year E-09 year year E-05 year E-13 year E-09 year year E-05 year year E-11 year year E-05 year year E-09 year year E-08 year E-10 year E-05 year month month month month E-06 month month month month E-07 month month month month E-06 area kindofeffort kindofeffort E-05 month E-07 area area E-05 month month E-10 month E-15 month month month E-14 kindofeffort E-08 area

10 Figure 1. Schematic map of the fishing grounds used by Brazilian (national plus leased) long-line fleet. North (A) and South (B) areas. Figure 2. Histogram of frequency of reports of the main factors ( year, month, kind of fleet and area) included in the models. 344

11 Figure 3. Catch rate (t/1000 hooks) of the white marlin (Tetrapturus albidus) caught by Brazilian fleet. Year, month, kind of fleet and area are the main factors considered in the models. The levels of kind of effort are: (1) Fishing sets of the Brazilian national and leased boats from Honduras, Spain and USA, and (2) Fishing sets of the Brazilian leased boats of Saint Vincent and Taiwan. The levels of area are: (1) north and (2) south fishing areas. Figure 4. Proportion of positive fishing sets for the main factors as calculated using the procedure A. Figure 5. Proportion of positive fishing sets for the main factors as calculated using the procedure B. 345

12 Figure 6. Proportion of positive fishing sets for the main factors as calculated using the procedure C. Figure 7. Normal plot of standardized residuals of the binomial models fitted to proportion of positive fishing sets. Three approaches (A, B and C) were used to calculate the proportion of positive fishing sets. Figure 8. Standardized catch rates of the white marlin (Tetrapturus albidus) caught by the Brazilian fleets (national plus leased boats) as estimated using delta-lognormal generalized linear models. Three alternatives methods (A, B and C) were used to estimate the proportion of positive fishing sets. 346