SCRS/2009/154 Collect. Vol. Sci. Pap. ICCAT, 65(4): 1469-1485 (2010)

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AN EVALUATION OF U.S. ALBACORE TUNA LENGTH-WEIGHT DATA AND DEVELOPMENT OF A REGIONAL RELATIONSHIP

Kristin L. Erickson, Todd Gedamke, Stephen Turner,

Matthew Maiello, and Kenneth Keene1

SUMMARY During the 2009 ICCAT albacore tuna (Thunnus alalunga) stock assessment meeting U.S. catch-at-size data were questioned when a marked increase in catch at length was observed after 2003. In response, staff at the U.S. Southeast Fisheries Science Center initiated a review of the database and methodology used in the catch at size calculations. Length-weight parameters used for ICCAT were found to over-estimate our observed length-weight data. Data from the U.S. pelagic observer program were utilized to generate a length-weight relationship for the western Atlantic. Over 4000 records were used to estimate length-weight parameters. The results differ from the accepted ICCAT north Atlantic relationship and suggest that regional parameters may be more appropriate.

RÉSUMÉ

Au cours de la réunion d’évaluation du stock de germon (Thunnus alalunga), tenue par l’ICCAT en 2009, les données de prise par taille des Etats-Unis ont été contestées lorsqu’une augmentation accusée de la prise par taille a été observée après 2003. En réponse, le personnel du Centre des sciences halieutiques du Sud-Est des Etats-Unis a procédé à la révision de la base de données et de la méthodologie utilisée dans les calculs de la prise par taille. Les paramètres longueur-poids utilisés par l’ICCAT se sont avérés surestimer nos données de longueur-poids observées. Les données du programme d’observateurs pélagiques des Etats-Unis ont été utilisées pour créer une relation longueur-poids pour l’Atlantique Ouest. Plus de 4.000 registres ont été utilisés afin d’estimer les paramètres longueur-poids. Les résultats diffèrent de la relation acceptée de l’ICCAT pour l’Atlantique Nord et suggèrent que des paramètres régionaux pourraient être plus appropriés.

RESUMEN

Durante la reunión ICCAT de evaluación de atún blanco (Thunnus alalunga) de 2009, se cuestionaron los datos de captura por talla estadounidenses al observarse un incremento en la captura por talla a partir de 2003. Respondiendo a este cuestionamiento, el personal del U.S. Southeast Fisheries Science Center inició una revisión de la base de datos y de la metodología utilizada en los cálculos de la captura por talla. Se halló que los parámetros talla-peso utilizados por ICCAT sobreestimaban nuestros datos talla-peso observados. Los datos del programa estadounidenses de observadores de la pesquería pelágica se utilizaron para generar una relación talla-peso para el Atlántico occidental. Se utilizaron más de 4.000 registros para estimar los parámetros talla-peso. Los resultados difieren de la relación de ICCAT para el Atlántico norte aceptada y sugerían que podría ser más apropiado utilizar parámetros regionales.

KEYWORDS

Length-weight, albacore tuna, Thunnus alalunga, Atlantic, catch-at-size

1 National Marine Fisheries Service, Southeast Fisheries Science Center, National Oceanic and Atmospheric Administration, 75 Virginia Beach Dr., Miami, FL 33149. Kristin.Erickson@noaa.gov; Todd.Gedamke@noaa.gov; Steve.Turner@noaa.gov; Matthew.Maiello@noaa.gov; Kenneth.Keene@noaa.gov.

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1. Introduction A valid length-weight relationship is a critical component to most stock assessments. At the 2009 ICCAT (The International Commission for the Conservation of Atlantic Tunas) albacore tuna (Thunnus alalunga) assessment the U.S. predicted length data were questioned when unexplained changes in predicted length frequencies were observed starting in 2004. As a result staff at the U.S. Southeast Fisheries Science Center (SEFSC) initiated a review of the methodology used to produce the U.S. catch-at-size (CAS) data. An initial evaluation of western Atlantic length-weight data also suggested that length-weight parameters may differ between the eastern and western Atlantic. Currently, in ICCAT assessments, separate relationships are used only for the south Atlantic, north Atlantic, and Mediterranean stocks. For the north Atlantic, the accepted length-weight parameters were derived by Santiago (1993) from fish of the Bay of Biscay, adjacent waters and the Azores. We utilize fourteen years of empirical data from the U.S. pelagic observer program to develop a length-weight relationship for the western Atlantic. In this report we compare an albacore tuna length-weight relationship from fish caught in the western Atlantic to those derived in other locations. 2. Materials and methods Data were obtained by trained scientific observers onboard U.S. pelagic longline vessels. Between 1992 and 2006, length and round weight measurements were collected from 4,012 Atlantic albacore tuna (Figure 1) from the western Atlantic. Round weight was converted to whole weight using a conversion factor of 1.25. A comparison of the ICCAT Santiago (1993) relationship to the one used in the SEFSC catch at size system (hereafter referred to as SEFSC-Oracle) and available raw data was conducted. Problems in the SEFSC-Oracle predicted values were noted and both the parameters being used and the algorithms were carefully evaluated. Estimating weight from length The length-weight relationship was initially investigated using the nonlinear length-weight model: W=aLb (1) where L corresponds to lower jaw–fork length, W corresponds to whole weight, and a and b are parameters estimated from the data. While a clear relationship between length and weight was evident, a number of data points appeared to be statistical outliers (Figure 1). Least trimmed squares (LTS) and iteratively re-weighted least squares (IRLS) approaches were used to minimize the affect of true outliers. The first step in both procedures was to fit equation (1) to all available data using PROC NLIN (SAS system 9.2, 2008) and scale each observed residual (RO by the residuals mean ( R ) the residual standard deviation (Rσ) as (Rousseeuw 1984):

OS

R RR

R

(2)

For the LTS procedure data points with absolute scaled residuals (RS) values greater than 2.5 were removed from the analysis (Rousseeuw and Leroy 1987; Restrepo and Powers 1997). For the IRLS procedure weights were assigned using the Beaton and Tukey (1974) bi-weight function:

(3)

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where wi is the weight for each data point and K is a scaling factor. The scaling factor or tuning constant (K, set at 4.685) and the cutoff for the LTS (set at 2.5) are generally picked to produce 95% efficiency when the errors are normally distributed (Hamilton 1992). Equation (1) was then refitted to the trimmed data for the LTS procedure and the weighted data for the IRLS approach to obtain the final parameter estimates a and b. Estimating length from weight In the United States, weight data is generally more readily available than length data so we also evaluated the inverse of equation (1) to estimate length from weight:

1

W bLa

(4)

Equation (4) could be used in the construction of catch-at-size matrices. In our preliminary analyses of equation (4) we encountered some convergence problems and sensitivity to parameter starting values during the final parameter estimates for the IRLS method. A fine scale grid search was conducted over a reasonable range of starting values for both parameters a and b to insure that the global solution was identified. We also investigated a commonly used re-parameterized version of equation (4):

L W (5) Note that we generally found the notation in the literature for the parameters (typically a and b) to be ambiguous and have modified it to insure that α and β in equation (5) are not confused with the a and b in equation (1) and (4). For comparison the two terms are equivalent as:

11

ba

and,

1

b

(6)

Following the same LTS and IRLS procedures described in fitting equation (1) to our data, we estimate parameters for equations (4) and (5). As a diagnostic of model fit a visual inspection of residuals was conducted. In addition, the Hougaard measure for skewness was calculated for the LTS and IRLS methods (Hougaard 1982). The Hougaard measure reflects the degree of nonlinearity in the model and reflects the reliability of standard errors and confidence intervals. For the interpretation of the Hougaard measure we use the rules provided by Ratkowsky (1990):

- Absolute values < 0.1 indicate very close to linear behavior and very reliable standard error and confidence interval estimates.

- Absolute values between 0.1 and 0.25 indicate reasonably close-to-linear behavior. - Absolute values > 0.25 indicate very apparent nonlinear behavior and uninformative standard error and confidence interval inferences. Non-linearity was suggested for equation (1) so a log-transformed approach was utilized in an attempt to better describe uncertainty for comparison to the Santiago (1993) model. For the trimmed data in the LTS approach the natural log of both the weight and length data were used and model parameters were calculated using PROC REG in SAS (SAS system 9.2, 2008): ln(W)=ln(a) + b*ln(L) (7) The slope was equivalent to b in equation (1) and the calculated intercept was equivalent to ln(a). Confidence intervals and standard error were calculated within the PROC REG procedure.

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We conducted our analyses and present our results in pounds (lbs.) as this is the convention in the United States for data collection. For use in the international arena, we also converted the data to metric units, re-estimated the parameters, and present final results in terms of kilograms. 3. Results Our initial comparison between the western Atlantic raw data, the ICCAT Santiago (1993) relationship and the SEFSC-Oracle relationships indicated some problems in the SEFSC-Oracle estimation procedures and a difference between the Santiago (1993) relationship and the observed data (Figure 2). Length predictions from the algorithms and length-weight parameters used in the SEFSC-Oracle calculations did not accurately reflect western Atlantic sizes. In the United States data are generally collected in terms of weight in pounds and must be converted to length in centimeters. The accepted ICCAT length-weight parameters for north Atlantic albacore (Santiago 1993) are presented in terms of converting length in centimeters to weight in kilograms and are therefore not comparable at first glance. However, even when converted to comparable units, the SEFSC-Oracle a and b parameters did not correspond to those of Santiago (1993) and accepted by ICCAT. Estimating weight from length In the first fit (base estimate) of equation (1) to all of the data, parameter a was estimated to be 0.000323 and parameter b was estimated to be 2.5781 (Table 1). Given the resulting residuals, the LTS approach identified 74 of 4012 data points to be statistical outliers while the IRLS method assigned a weight of 0 (see equation 3) to 84 of the 4012 data points. The final parameter estimates from both approaches were very similar with a estimated to be 0.000170 and 0.000158 and b estimated to be 2.7159 and 2.7319 from the LTS and IRLS methods respectively. The residuals showed no discernable pattern (Figure 3). Final results for the LTS, IRLS, and log transformed estimation methods were very similar and the fitted lines were graphically identical (Table 1; Figure 4). The Hougaard measure for skewness for parameter a was approximately 0.3 for all analyses while values for b were less than 0.05 in all cases. While parameter estimates were unbiased, the non-linear behavior indicated by values for parameter a suggested that standard error and confidence intervals were likely to be overestimated. The 95% confidence intervals produced by the log transformation (Equation 7) showed no significant improvement over those generated by the LTS or IRLS approaches (Figure 4; Table 1). Estimating length from weight The initial base estimates from equation (4) fit to all of the data, estimated parameter a as 1.08 x 10-6 and parameter b as 3.8078 (Table 2). The LTS approach identified 84 of 4012 data points to be statistical outliers while the IRLS method assigned a weight of 0 (see equation 3) to 35 of the 4012 data points. Parameter estimation for equation (4) proved extremely sensitive to starting values. A fine scale grid search was utilized. The results of the final parameter estimates for both methods were again very similar with a estimated to be 4.96 x 10-6 and 5.72 x 10-6 and b estimated to be 3.4781 and 3.4478 from the LTS and IRLS methods respectively. The residuals showed no discernable patterns (Figure 5). The Hougaard measure of skewness values for parameter a were once again greater than 0.3 indicating non-linear behavior and reflecting the possibility of biased standard error and confidence intervals (Table 2). The base estimate of equation (5) to all of the data estimated parameter α as 36.9182 and parameter β as 0.2626 (Table 2). The LTS approach identified 84 of 4012 data points as statistical outliers while the IRLS method assigned a weight of 0 (see equation 3) to 27 of the 4012 data points. The resulting final parameter estimates from both approaches were again very similar with α estimated to be from 33.4995 and 33.2901 and β estimated to be 0.2875 and 0.2891 from the LTS and IRLS methods respectively. No discernable pattern was detected during the inspection of the residuals (Figure 6). The final results for the LTS, IRLS methods were very similar and the fitted lines are almost graphically identical (Table 1; Figure 6). The Hougaard values for parameters α and β in both the LTS and IRLS analyses were less than 0.03 reflecting close to linear behavior and standard error and confidence interval estimates are unbiased. The log-transformed approach was deemed unnecessary for equation (5) based on the Hougaard results.

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4. Discussion This study was initiated to explore the apparent inconsistencies in the United States albacore tuna size frequency and mean weight data observed during the 2009 ICCAT albacore tuna assessment. We have determined that the algorithms and length-weight parameters used in the SEFSC-Oracle calculations did not accurately predict length from weight. We have limited our investigation to the identification of the problem and have not recalculated the mean weights or size frequencies in this report. We have recommended revising the SEFSC-Oracle database and algorithms using conventional parameters and calculations so that literature values are comparable and potential errors are more easily avoidable in the future. We used over 4,000 observations collected over 14 years of western Atlantic albacore data, developed a regional length-weight relationship. The data exhibited a clear relationship and provided an opportunity to evaluate the applicability of current ICCAT parameters to the western Atlantic region. We chose our parameter estimation methodologies carefully to minimize the effect of outliers. The data were not overly noisy; however, there were a few obvious instances of data recording or data-entry errors (Figure 1). Thus, two high breakdown robust regression techniques for nonlinear models were selected: LTS and IRLS. Both methods were chosen due to their statistical efficiency and ability to retain the highest breakdown value (i.e., maintain robustness) while mitigating the influence of outliers in an otherwise normally-distributed data set (Rousseeuw and Leroy 1987; Restrepo and Powers 1997; Rousseau and Van Driessen 1999). For both methods standard values were utilized for the tuning constant (4.685; IRLS method) and cutoff value (2.5; LTS method) and results proved to be insensitive to minor changes in these values. This indicated a strong informational content of the data and an overall successful application of the approaches to remove the effect of outliers. Our confidence in the estimated parameters was also bolstered by the consistency among the final results from all three approaches used (LTS, IRLS, and log-transformation) and a lack of discernable patterns in residuals (Tables 1, 2; Figure 3, 5, 6). None of the methods clearly outperformed the others. The only difficulties we encountered were in the fitting of equation (4) and calculation of length to weight confidence intervals. Specifically, the optimization of equation (4) was sensitive to starting values and we recommend the equivalent weight to length equation (5) which did not suffer from the same problems. As indicated by the Hougaard measure of skewness the estimated confidence intervals for equations 1 and 4 were unreliable. While a technique such as bootstrapping would be a suitable approach to estimate an appropriate measure of uncertainty, we did not feel this was a necessary exercise at this point. The quantity of data, the lack of patterns in the residuals, and the valid confidence intervals for equation (5) indicated very little uncertainty in the relationship between length and weight. The length-weight relationship we derived for western Atlantic albacore tuna differs from the one currently used by ICCAT for the entire north Atlantic stock (Santiago 1993). The Santiago (1993) relationship, which was estimated using only eastern Atlantic fish is similar at smaller sizes but it predicts heavier fish at sizes above 90-100 cm (Figure 4; ~16 lb. difference at 130 cm). This is illustrated by a clear pattern in the residuals (Figure 8). A difference in the relationship is also apparent when predicting length from weight (Figure 7). It should be noted that our parameters were estimated by fitting models to both raw length and weight data. It should also be noted that for comparison to our weight to length relationships, we needed to invert the Santiago (1993) length to weight parameters. As a result the parameters are not directly comparable as the assumptions regarding error structure are different depending on which is deemed to be the independent variable. While an error in variables approach would be optimal, we would recommend that, at a minimum, the relationships be described and parameters presented to predict values in both directions. When comparing our estimates for western Atlantic fish to relationships derived from the eastern Atlantic and Mediterranean, there appears to be regional difference that should be recognized. All four studies from the eastern Atlantic predict heavier fish at larger sizes than the U.S. pelagic fleet (Figure 9). In fact, at the largest sizes our relationship is closer to that of Mediterranean fish (Megalofonou 2000) which have been designated as a separate stock. Our work does not suggest to separate the North Atlantic albacore stock into eastern and western Atlantic stocks, but given the robustness of our estimates and observed regional differences in the length-weight relationships we recommend the use our LTS parameters for western Atlantic fish:

L=42.0486 · W 0.2875 and W= 7.70 x 10-5 · L2.7159 for weights in kilograms, and L=33.4995 · W 0.2875 and W= 1.70 x 10-4 · L2.7159 for weights in pounds.

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Acknowledgements We would like to thank Guillermo Diaz for his guidance during this study and his constructive comments in the preparation of this report. We would also like to thank Loyd Darby and Adriana Serra for providing critical database and computer support. References Beardsley, G.L. 1971, Contribution to the population dynamics of Atlantic albacore with comments on potential yields. Fishery Bulletin. U.S. 69(4): 845-857. Beaton, A.E. and Turkey, J.W. 1974, The Fitting of Power Series, Meaning polynomials, Illustrated on Band- Spectroscopic Data. Technometrics. 16: 147-185. Hamilton, A.J.S. 1992, Astrophys J. 385, L5. Hougaard, P. 1982, Parameterizations of non-linear models. J.R. Statistic. Soc. B. 44: 244-252. Lee, L.K. and Yeh, S.Y. 1993, Studies on the age and growth of South Atlantic albacore (Thunnus alalunga) specimens collected from Taiwanese longliners. Collect. Vol. Sci. Pap. ICCAT, 40(2): 354-360. Megalofonou, P. 2000, Age and growth of Mediterranean albacore. J Fish Biol 50: 700-715. Mejuto, J. and Gonzales-Garces, A. 1985, Relation talla-peso de atun blanco juventile del Atlantico norte.

Collect. Vol. Sci. Pap. ICCAT, 23(2): 278-281. Penney, A.J. 1994 Morphometric relationships, annual catches and catch-at-size for South African caught South

Atlantic albacore (Thunnus alalunga). Collect. Vol. Sci. Pap. ICCAT, 42(1): 371-382. Ratkowsky, D. 1990, Handbook of nonlinear regression models. Marcel Dekker: New York and Basel, pp.241.

Restrepo, V. and Powers. 1997. Application of high-breakdown robust regression to tunas stock assessment models. Fishery Bulletin. 95: 149-160.

Rousseeuw, P.J. 1984, Least Median of Squares Regression. J Am Stat Assoc. 79: 871-880. Rousseeuw, P.J. and Leroy, A.M. 1987, Robust Regression and Outlier Detection, Wiley-Interscience, New York. Series in Applied Probability and Statistics. pp.329 Rousseeuw, P.J. and Van Driessen, K. 1999, A fast Algorithm for the Minimum Covariance Determinant Estimator. Technometrics. 41: 212-223. Santiago, J. 1993, A new Length-weight relationship for the north Atlantic albacore. Collect. Vol. Sci. Pap. ICCAT, 40(2): 316-319. SAS Institute Inc. SAS® 9.2 Enhanced Logging Facilities, Cary, NC: SAS Institute Inc., 2008.

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Table 1. Results of LTS and IRLS analyses and parameter estimates for equation (1) calculated with all weights in pounds and lengths in cm (* = not calculated). Sample Parameter Standard 95% 95% Hougaard

Equation Method Parameter size Estimate Error LowerCL UpperCL tValue Probt Skewness

W = aL^b Base estimate

a

4012

3.23 x 10-4

4.52 x 10-5

2.3 x 10-4

4.1 x 10-4

7.15

1.00 x 10-12

*

LTS a 3938 1.70 x 10-4 1.77 x 10-5 1.4 x 10-4 2.0 x 10-4 9.63 1.01 x 10-21 0.309

IRLS a 3991 1.58 x 10-4 1.58 x 10-5 1.3 x 10-4 1.9 x 10-4 9.97 3.83 x 10-23 0.298

Log ln(a) 3938 -8.95 0.10 -9.15 -8.76 -90.43 0 *

Base estimate

b

4012

2.58

0.03

2.52

2.64

85.87

0

*

LTS b 3938 2.72 0.02 2.67 2.76 121.83 0 0.003

IRLS b 3991 2.73 0.02 2.69 2.77 126.82 0 0.003

Log b 3938 2.77 0.02 2.73 2.82 129.67 0 *

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Table 2. Results of LTS and IRLS analyses and parameter estimates for equation (4) and (5) calculated with all weights in pounds and lengths in cm (* = not calculated).

Table 3. Results of LTS analyses and parameter estimates for equation (1) and (5) calculated with all weights in pounds and lengths in cm.

Sample Parameter Standard 95% 95% Hougaard

Equation Method Parameter size Estimate Error LowerCL UpperCL tValue Probt Skewness

L=(W/a)^(1/b) Base estimate

a

4012

1.08 x 10-6

2.05 x 10-7

6.75 x 10-7

1.48 x 10-6

5.25

1.62 x 10-7

*

LTS a 3928 4.96 x 10-6 6.15 x 10-7 3.76 x 10-6 6.17 x 10-6 8.07 9.62 x 10-16 0.327 IRLS a 3978 5.71 x 10-6 6.54 x 10-7 4.43 x 10-6 7.00 x 10-6 8.74 3.45 x 10-18 0.302 L= W^β Base

estimate α

4012

36.92

0.41

36.11

37.72

89.99

0

*

LTS α 3928 33.50 0.29 32.93 34.07 115.38 0 0.025 IRLS α 3985 33.29 0.28 32.75 33.83 119.97 0 0.024 L=(W/a)^(1/b) Base

estimate b

4012

3.81

0.04

3.73

3.89

92.65

0

*

LTS b 3928 3.48 0.03 3.43 3.53 130.09 0 0.045 IRLS b 3978 3.45 0.02 3.40 3.50 139.71 0 0.042 L= W^β Base

estimate β

4012 0.26

0.003

0.26

0.27

92.65

0

*

LTS β 3928 0.29 0.002 0.28 0.29 130.09 0 0.001 IRLS β

3985 0.29 0.002 0.28 0.29 136.03 0 0.001

Equation Method Parameter Sample size

Parameter Estimate

Standard Error

95% LowerCL

95% UpperCL tValue Probt

Hougaard Skewness

W=aL^b LTS a 3938 7.70 x 10-5 8.02 x 10-6 6.10 x 10-5 9.30 x 10-5 9.63 0 0.309

LTS b 3938 2.72 0.022 2.62 2.76 121.83 0 0.003

L= W^β LTS α 3928 42.05 0.291 41.48 42.62 144.43 0 0.019

LTS β 3928 0.29 0.002 0.28 0.29 130.09 0 0.001

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Figure 1. Raw length-weight data for western Atlantic albacore tuna collected by U.S. pelagic observer program in the western Atlantic Ocean, 1992-2006 (n= 4012).

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Figure 2. Western Atlantic weight-length observation (black dots) and corresponding relationships from Santiago (1993), SEFSC-Oracle calculation, and the current base case weight-length analysis. In order to compare our results from equation (5) to Santiago (1993), the parameters were inverted by the equivalences in (6).

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Figure 3. Residuals from current study using equation (1) and the LTS method. Data points with residuals greater than 2.5 were identified as outliers and are represented by an open circle.

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Figure 4. Comparison between the current study (LTS and IRLS methods were graphically identical), Santiago (1993) and 95% confidence intervals of the LTS method and the log transformation. Note that 95% confidence intervals are likely too wide due to slight non-linearity (Hougaard measure of 0.309) of parameter a. Data ranged from 60 to 129 cm and 8 to 90 lbs once outliers were removed.

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Figure 5. Residuals from current study using equation (4) and the LTS method. Data points with residuals greater than 2.5 were identified as outliers and are represented by an open circle.

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Figure 6. Residuals from current study using equation (5) and the LTS method. Data points with residuals greater than 2.5 were identified as outliers and are represented by an open circle.

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Figure 7. Albacore tuna (Thunnus alalunga) weight-length observations from western Atlantic compared to length-weight relationships from the current study, Santiago (1993), and the previous SEFSC-Oracle relationship. Note 95% confidence intervals for equation (5) show very linear behavior according to the Hougaard measure (Table 2) and are deemed to be reliable.

40

60

80

100

120

140

0 20 40 60 80 100 120

Len

gth

(cm

)

Round Weight (lbs)

Predicted (LTS & IRLS)

Santiago Predicted

SEFSC-Oracle

95% Confidence Intervals (LTS & IRLS)

Observed Outliers

Observed

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Figure 8. Residuals from Santiago (1993) relationship applied to length-weight western Atlantic observations. Data points with residuals greater than 2.5 were identified as outliers in the LTS and are represented by an open circle.

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Figure 9. Comparison of length-weight relationships presented by several authors for albacore tuna from the Mediterranean Sea and Atlantic Ocean. Note that outliers identified during the LTS procedure were not included.