Empirical length and weight conversion equations for blue marlin,
white marlin, and sailfish from the North Atlantic Ocean
Michael H. Prager,
Eric D. Prince,
and
Dennis W. Lee
Running Head: Length and weight conversions for billfishes
1 ABMItACr
Because commercially caught billfishes are usually dressed at sea, and because measurements of size usually are riot obtained before processing, there is a need for conversion
among different measures of size (length or weight) for assessment and management purposes.
Here, we present empirical equations for converting among measures of size for blue marlin,
Makaira nigricang white marlin, Tetraptunis albidus-, and sailfish, Isdophorus platypterus,
from the North Atlantic Ocean. A series of length-length equations allows conversion from
any of six length measures to lower jaw-fbrk length; and a series of length-weight and
weight-length equations allow conversion between lower jaw-fork length and round weight.
Th estimate the equations, we used large data sets that have recently become available. We
incorporated bias corrections into the length-weight and weight-length equations and used
robust regressions for the length conversions. Equations for each species are given by sex and
also for combined sexes, so that conversions can be made whether the sex of the specimen is
known or not
2 INTRODUCMN
About 90% of the world's landings of billfishes (Isfiophoridae) are taken as incidental catch by offshore longline fisheries targeting tuna and swordfish (King, 1989). Billfishes caught in this manner are usually dressed at sea, with heads, fins, and viscera removed and carcasses frozen for off-loading at transhipment ports months later (Prince and
Brown, 1991). Although only one or two methods of dressing the target species of tuna and swordfish are normally practiced in the Atlantic, billfishes are dressed in more than a dozen ways (Prince and Miyake, 1989). Because length measurements on billfish from the longline fisheries are usually obtained after the fish have been dressed, there is a need for empirical equations for conversion among different measures of whole and dressed length (Table 1;
Prince and Brown, 1991). Because such fish am rarely weighed, there is also a need for equations to pern-tit conversion between length and weight.
Empirical equations for converting among size measures of these three species in
North Atlantic waters have been presented by several previous authors. De Sylva and Davis
(1963) gave length-weight relationships for postspawning male and female white marlin collected off the mid-Atlantic coast of the United States. Jolley (1974) presented similar equations for Atlantic sailfish, based on observation of 412 specimens. Lenarz and Nakamura
(1974) estimated a variety of conversion equations for the three species considered here. Rivas
(1974) provided illustrations of length-weight curves for blue marlin, fitted by eye to data on
58 males and 104 females. Baglin (1979) presented weight-to-length estimation equations for white marlin, by sex, in the North Atlantic and the Gulf of Mexico. Prince and Lee (1989) gave empirical conversion equations for estimating lower jaw-fork length (LJFL) from four other length measures on Atlantic billfishes: eye orbit-fork length (EOFL), pectoral-second
3 dorsal length (PDL), dorsal-fork length (DFL), and pectoral-fork length (PFL). Lee and Prince
(1990) presented empirical equations for converting LJFL to total length (TL) and TL to LJFL, but several of their equations contained typographical errors. Lee and Prince (1990) also gave conversions between TL and round weight and between LJFL and round weight. The preceding authors gave results for each sex separately, but did not develop equations for use on specimens of unknown sex. However, the sex of fish taken in these offshore fisheries is rarely known, unless sex data are collected by scientific observers (Prince and Brown, 1991).
Thus it is often necessary to estimate weight fmm. the length of fish of unknown sex.
Prager et al. (1992) reanalyzed the data of Lee and Prince (1990) to arrive at slightly revised equations for the length-weight and weight-length conversions. The revisions comprised correction of the equations for bias and development of equations intended for use when the sex of a fish is riot known.
The present paper has two objectives. The first is to present revised versions of many of the existing conversion equations. these new equations are based on large data sets that have recently become available. We also include a new equation for conversion of pectoral-anus length (PFL) to LJFL. The second objective is to present a comprehensive set of conversion equations in a single readily accessible reference. In addition, our use of robust
regression, a statistical technique relatively new to fishery science, may be of methodological
interesL
DATA AND MoDas
Data were obtained from the files of the National Marine Fisheries Service, Southeast
Fisheries Science Center, Mianti, Florida. These data include observations made during
billfish-survey field sampling by NMFS personnel (Father et al., 1992) and data obtained
4 through the ICCAT Enhanced Research Program for Billfish, particularly in the western
Atlantic Ocean (Carter 1992). From these data, equations were developed to estimate LTFL from any of six other measures of length (Fable 1) and to convert between LTFL and round
weight Before die data were used for estimation, they were visually screened for gross
outliers, which were removed.
Conversions between Length and Weight.-Equations for conversion between length and
weight used the allometric (power-equation) form. In fitting allometric equations to data, we
used logarithmic transformations: Instead of fitting directly a length weight relationship of the
form
w = a-I', (1)
(where w is the weight of a fish, I is its length, and a and b are estimated constants), we took
the natural logarithm of each side of the equation so that it became
IrKw) = ln(a) + b -In(l). (2)
Use of this transformation allows estimating the constants a and b by ordinary least squares. In
addition, the transformation is theoretically consistent with the observation that the coefficient
of variation of length at weight (or of weight at length) tends to be constant
The logarithmic transformation, however, introduces a bias into the resulting equation
when it is used for prediction. IU correction for this bias was reported in the ecological
literature by Baskerville (1972) and VVhiftaker and Marks (1975); a minor correction was given
5 by Sprugel (1983). The correction is not complex. If a prediction from equation (2) is denoted yOLS and the correction factor is denoted CF, then the corrected prediction is
(3)
The correction factor as given by Sprugel (1983) is
SSE J (4) CF = exp ( 2(n-2) ,
where SSE is the sum of squared errors from the OLS regression [equation (2)] and n is the sample size.
We fit allometric equations for the most common measure of billfish length, lower jaw-fork length (LJFL), computed the correction factors, and incorporated them directly into the equations as presented here. Because sample sizes were large, the correction factors were within I % of unity. All allometric conversion equations were estimated with the REG procedure of the Statistical Analysis System (SAS, 1988).
Conversions among Measures of Length.-Conversions among length measures (Table 1) can generally be accomplished with simple linear regression models. To increase robustness to other possible outliers or data errors, the equations presented here were fit with a robust- regression method, least-absolute-values (LAY) regression. In the LAY technique, the quantity minimized is not the sum of the squares of the residuals, as in ordinary-least-squares (OLS) regression, but instead is the sum of the absolute values of the residuals (Krasker, 1988; Berk,
1990). When the data contain no outliers, estimates from LAY regression are nearly identical to those from OLS regression. However, the OLS parameter estimates can be influenced quite
6 strongly by relatively few outliers, especially if they occur near the extremes of the observations on the predictor variable. In contrast, the parameter estimates from LAV regression are much less influenced by outliers; hence the term "robust regressiom" Equations derived from LAV are applied in the same way as those derived from OLS.
Estimation of LAV regressions can be accomplished with most commercially available nonlinear estimation software; we used the NONLIN procedure of Systat (Wilkinson, 1990).
We were unable to obtain correct results from version 6.04 of the NLIN procedure of SAS for
Personal Computers (SAS 1988), and we have concluded that this program contains an error that makes its use with non-OLS loss functions unreliable.
RESULTS
For predicting round weight from lower-jaw fork length, we use the notation
0 (5)
For predicting lower-jaw fork length from round weight, the corresponding notation is
C-Wd. (6)
The bias-corrected coefficients a, b, c, and d of equations (5) and (6) are tabulated in Table 2.
Conversions among length measures are given in Table 3, Table 4, and Table 5.
7 Discussm
Statistical Commems^The equations in Tables 3 through 5 are predictive equations; i.e., when estimating them, the effors in the estimated quantity (LJFL) were minimized. This procedure is optimal when the equations am to be used for prediction or estimation, but the pre4lictive equations should not be inverted and used to estimate other length measures from
LJFL. If necessary, equations for estimating LJFL from other measures could be derived from the same data, but these would not be equivalent to the inverses of the equations given.
The bias con-ection used in the allometric equations is simple to compute and apply. In the present case, the models fit well and the sample sizes were large, so that the correction factors were within 1% of unity. However, corrections encountered in other cases can be much more significant, For example, SaUa et al. (1988, p. 149) found that a correction factor of about 1.1 was necessary in an age-fecundity relationship for yellowtail flounder. Because it seems logical to correct for a known bias, we recommend that this correction be incorporated routinely into equations for conversion between length and weight.
Robust regression is a useful statistical technique with a large literature. It was motivated by the fact that OLS can be quite inefficient when the effors in the data have heavier tails than under a normal distribution (Krasker 1988), and this may be the case for many fisheries data, especially when quality control is not perfect. Data sets should always be screened for obvious outliers, but in many practical cases questionable points will remain after this is done. Then the investigator is faced with a difficult choice-whether to discard a few potentially influential (in the statistical sense) observations because they might be wrong. The use of robust regression is not a replacement for quality control nor a panacea for all
8 problems, yet it provides a practical solution to the type of difficulties mentioned, as it allows questionable data to be retained while reducing their influence on the final estimates.
Biological Comments.-Iower jaw-fork length is regarded as the most reliable measure of length for the Istiophoridae (Rivas 1956), but because of the nature of the fisheries, other length measures are usually collected instead. We hope that the models given here will allow most size and size-frequency data to be converted into a standard unit of measurement (LJFL).
which will facilitate data collection and study. The large number of conversions presented here
should suffice for Most conversions of billfish length and weight categories, particularly those
obtained from offshore longline fisheries.
Most of the length data used in this study were taken by placing measuring tapes over
the curve of the body (curved body measurements). Many of the length measurements
available from the offshore longline fleets are straight measurements (i.e., taken by fixing a
ruler to the deck and not including the body curvature), so the equations presented here am
slightly biased when applied to data from those fleets. Because billfishes are elongated species,
the differences between straight and curved body measurements (and hence the biases) are
probably quite small. However, it would be desirable to reduce this known source of error.
ACKNOWLEDGMENTS
We thank R. Carter and L. Muccio for help with data preparation. Earlier versions of
this paper were distributed as Working Documents of the international Commission for the
Conservation of Atlantic Minas. Contribution NUA-92/93-26 of the Southeast Fisheries
Science Center, NUFS.
9 LNERATURE CUED
Baglin. R. E. 1979. Sex composition, length-weight relationship, and reproduction of the white
marlin, Terrapturus albidus, in the western North Atlantic Ocean. Fish. Bun. U.S. 76: 919-926.
Baskerville, 0. L. 1972. Use of logaritlunic regression in the estimation of plain biomass. Can.
J. For. Res. 2: 49-53.
Berk, R. A. 1990. A primer on robust regression. Chapter 7 (pp. 292-324) in J. Fox and J.S.
Long (eds.), modem methods of data analysis. Sage Publications, Newbury Park,
Cahlbinia. 442 pp.
Carter, R. L. 1992. A summary of shore-based and at-sea sampling in the western Atlantic
Ocean 1987-1992: ICCAT Enhanced Research Program for Billfish. Int. Comm.
Conserv. AtL Tunas (Madrid), Working Document SCRS/92/59.
De Sylva, D. P, and W. P. Davis. 1963. White marlin, TetWturus albidus, in the middle
Atlantic bight, with observations on the hydrography of the fishing grounds. CDpeia
1963: 81-99.
Farber, M. I., J. A. Browder, and J. P. Contillo. 1992. Standardization of recreational fishing
success for marlin in the western North Atlantic Ocean, 1973-1991, using generalized
linear model techniques. Int. Comm. Crinserv. Ad. Tunas (Madrid), Working
Document SCRS/92/62.
Jolley, J. W., Jr. 1974. On the biology of Florida east coast Atlantic sailfish, (Istiophorus
plarypterus). Pages 121-125 in R. S. Shomura and R Williams (eds.), Proceedings of
the International BMfish Symposium, Kailua-Kona, Hawaii, 9-12 August 1982. Part 2:
10 Review and contributed papers. U.S. Dept. Commer., NOAA Tech. Rep. NMFS
SSRF-675.
King, D. M. 1989. Economic trends affecting commercial billfish fisheries. Pages 89-102 in R.
H. Stroud (ed.), Planning the firture of billfishes, research and management in the 90s
and beyond. Nat. Coll. Mar. Cons.. Mar. Rec. Fish. 13. Savannah, Ga.
Krasker, W. S. 1988. Robust regression. Pages 166-169 in S. Kotz and N.L. Johnson (eds.).
Encyclopedia of statistical sciences, vol 8. Jolm. Wiley & Sons, New York. 870 pp.
Lee, D. W., and E. D. Prince. 1990. Further development of length and weight regression
parameters for Atlantic blue marlin, white marlin, and sailfish. Int. Comm. Conserv.
Atl. Tunas (Madrid), Coll. Vol. Sci. Pap. 32(l): 418-425.
Lenarz, W. H., and E. L. Nakamura. 1974. Analysis of length and weight data on three species
of billfish from the western Atlantic Ocean. In R. S. Shomura and F. Williams (eds.),
Proceedings of die International Billfish Symposium, Kaflua-Kona, Hawaii, 9-12
August 1982. Part 2: Review and contributed papers, p. 121-125. U.S. Dept. Commer.,
NOAA Tech. Rep. NUFS SSRF-675.
Prager, M. H., D. W. Lee, and E. D. Prince. 1992. Bias-coffected length-weight relationships
for Atlantic blue marlin, white marlin, and sailfish. Int. Comm. Conserv. Ad. Tunas
(Madrid), Coll. 'vbl. Sci. Pap. 34(3): 656-658.
Prince, E. D., and B. E. Brown. 1991. Coordination of the ICCAT Enhanced Research
Program for Billfish. Am. Fish. Soc. Symp. 12: 13-18.
Prince, E. D., and D. W. Lee. 1989. Development of length regressions for Atlantic
Istiophoridae. Int. Comm. Conserv. All. Tunas (Madrid), Cc)U.'Vbl. Sci. Pap. 30(2):
364-374.
11 Prince, E. D., and P. M. Miyake. 1989. Methods of dressing Atlantic billfishes (Istiophoridae)
by ICCAT reporting countries. Int. Comm. Conserv. AtL TV= (Madrid), Coll. Nbl.
Sci. Pap. 30(2): 375-381.
Rivas, L. R. 1956. Definitions and methods of measuring and counting in the billfishes
(Istiophoridae, )Uphiidae). Bull. Mar. Sci. Gulf Carib. 6: 18-27.
Rivas, L. R. 1974. Synopsis of biological data on blue marlin, Makaira ind(cia (Cuvier), 1831.
Pages 17-27 in R. S. Shomura. and F. Williams (eds.), Proceedings of the International Billfish Symposium, Kailua-Kona, Hawaii, 9-12 August 1982. Part 3: Species
Synopses. U.S. Dept. Commer., NOAA Tech. Rep. NNIFS SSRF-675.
Saila, S. B., C. W. Recksiek, and M. H. Prager. 1988. Basic fishery biology programs: a compendium of microcomputer programs and manual of operation. Elsevier Science
Publishers, Amsterdam. 230 pp.
SAS (SAS Institute, Inc.). 1988. SAS/STAT user's guide, release 6.03 edition. SAS Institute,
Inc., Cary, N.C. 1028 pp.
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209-210.
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H. Uith and R. H. Whittaker (eds.). Primary productivity of the biosphere.
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ADDRMS: Southeast Fisheries Science Center, National Marine Fisheries Service, 75 Virginia
Beach Drive, Miami, Florida, 33149.
12 Table 1. Length measures used in this study, with mnemonic codes used in text and tables.
For description of length measures, see Carter (1992).
Mnemonic code Length measure
LJFL Lower jaw-fork length
PAL Pectoral-arms length
PFL Pectoral-fork length
PDL Pectoral-second dorsal length
TL Total length
EOFL Eye orbit-fork length
DFL Dorsal-fork length
13 Table 2. Coefficients of bias-corrected equations for predicting round weight (kg) from lower jaw-fork length (cm) using the equation a -1 1 or predicting lower jaw-fork length from round weight using the equation c -w '. Bias-correction factors have been incorporated into parameters a and c so that no further correction is necessary. For explanation, see text
Sample Weight Length Coefficients
size range, range, a b C d Sex (n) kg clu
Blue Marlin
9 3267 0.06-540.9 23.0-378.5 1.9034 x 10-4 3.2842 61.731 0.28180
d' 1978 0.06-178.0 23.0-277.0 2.4682 x 10' 3.2243 61.961 0.28137
9 cr 5245 0.06-540.9 23.0-378.5 1.1955 x 10-6 3.3663 62.010 0.28065
White marlin
9 3149 2.7-67.1 91.4-205.0 3.9045 x 10-6 3.0694 78.423 0.23191
d' 1719 3.6-41.3 96.0-195.5 1.9556 x 10-5 2.7487 76.847 0.23548
9 d' 4868 2.7-67.1 91.4-2.5.0 5.2068 x le 3.0120 76.460 0.23888
14 91
TOUZ'O Z96'ZL 6EWE 01 x 698Z'l ^,170Z-I*Lz L,z5-to*o L81Z lp t
op 161 SZ'O L06*OL USIT 9-01 x ZZ69-1 0*881-1*LZ IN-Wo L06
09tl9z*o V19,17t C89VE 9-01 x 1"1.1 5'tOV--J'LZ Llg-WO 08zl t
qsgl!gs Table 3. Blue marlin from the North Atlantic Ocean. Coefficients of robust-regression
equations for predicting lower jaw-fork length ~ (em) from another measure of length A.I
(em) using the equation Ao = (X + ~A.l. "Approx. length range" refers to the predictor variable in the equation. A question mark in column 2 indicates that the sex of some specimens was not known.
Predictor Sample Approx. length Intercept Slope
variable A.I Sex(es) size (n) range (em) (X ~
123 34-120 32.063 2.504
PAL d' 249 35-90 86.874 1.660
$ d'? 453 30-120 72.161 1.920
243 80-270 10.388 1.246
PFL d' 387 100-220 16.183 1.197
732 65-280 9.486 1.248
140 85-190 23.533 1.681
PDL d' 276 66-150 49.441 1.354
$ d'? 482 60-190 12.626 1.738
69 250-490 -1.331 0.773
TL 153 200-330 26.458 0.663
16 9 e ? 258 30-500 8.886 0.738
.9 113 130-300 10.737 1.090
EOFL cr 104 135-210 13.682 1.068
e ? 250 120-300 10.127 1.092 ......
9 115 125-280 10.134 1.200
DFL d' 125 115-200 8.985 1.197
9 d? 271 100-280 8.303 1.206
17 Table 4. White marlin from the north Atlantic. Coefficients of robust-regression equations for predicting lower jaw-fork length 4 (cm) from another measure of length X, (cm) using the equation X0 = a + OX,. "Approx. length range" refers to the predictor variable in the equation.
A question mark in column 2 indicates that the sex of some specimens was not known.
Predictor Sample Approx- length Intercept Slope variable X, Sex(es) size (n) range (cm) a 0
9 105 40-66 81.559 1.545
PAL d' 123 40-85 80.254 1.538
V d' ? 272 35-85 88.904 1.375 ......
9 188 92-145 13.667 1.241
PFL e 172 80-180 31.623 1.083
cr ? 424 80-180 25.291 1.139 ......
9 127 72-115 40.686 1.366
PDL d' 121 68-110 66.208 1.060
9 d? 294 65-115 42.753 1.333 ......
9 51 190-245 32.069 0.615
TL (F 65 130-235 9.944 0.710
V d' ? 127 130-280 15.986 0.685
18 65 128-165 18.372 1.037
EOFL 30 115-160 6.897 1.112
102 115-165 13.742 1.068
75 115-150 35.785 1.000
DFL 47 105-150 17.482 1.131
129 105-150 21.965 1.102
19 Table 5. Sailfish from the north Atlantic. Coefficients of robust-regression equations for predicting lower jaw-fork length ~ (em) from another measure of length Al (em) using the equation Ao = a + ~Al' "Approx. length range" refers to the predictor variable in the equation.
A question made in column 2 indicates that the sex of some specimens was not known.
Predictor Sample Approx. length Intercept Slope
variable Al Sex(es) size (n) range (em) a ~
652 30-90 120.170 0.798
PAL 455 35-80 111.175 0.907
~ d'? 1553 30-100 107.196 0.999 ...... •......
~ 728 75-175 36.766 1.025
PFL d' 484 90-150 34.211 1.043
~ d'? 1810 75-180 29.441 1.083
113 55-120 44.570 1.268
PDL 42 75-110 19.074 1.526
330 55-120 38.322 1.332
83 120-260 32.188 0.623
TL 52 110-245 21.961 0.657
142 40-270 18.171 0.686
20 .9 58 85-175 18.235 1.015
EOFL e 27 105-155 21.707 0.987
e ? 251 85-175 11.240 1.076 ......
V. 59 75-165 39.104 0.951
DFL e 21 110-145 1.555 1.221
.9 e ? 252 75-165 38.438 0.958
21