# Biodiversity of Fishes Length-Weight Relationships Rainer Froese (27.11.2014)

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Biodiversity of Fishes Length-Weight Relationships Rainer Froese (27.11.2014)

How to Measure Size in Fishes Length as a proxy for weight Total length (TL; tip of snout to caudal fin end ) Standard length (SL; tip of snout to caudal peduncle ) Fork length (FL; tip of snout to mid of forked caudal fin ) Other length measurements ( e.g. width in rays ) Length as proxy for size overestimates weight in eels, underestimates in puffers and boxfishes

Relationship Between Weight and Length W = a * L b with weight W in grams and length L in cm For parameter estimation use linear regression of data transformed to base 10 logarithms log W = log a + b * log L Plot data to detect and exclude outliers, and to check for growth stanzas

LWR Plot I Weight-length data for Cod taken in 1903 by steam trawlers from Moray Firth and Aberdeen Bay. Data were lumped by 0.5 cm length class and thus one point may represent 1-12 specimens.

LWR Plot II Double-logarithmic plot of the data in LWR Plot I. The overall regression line is W = 0.00622 * L 3.11, with n = 468, r 2 = 0.999, 95% CL of a = 0.00608 – 0.00637, 95% CL of b = 3.101 – 3.114. Note that the mid- length of length classes was used such as 10.25 cm for the length class 10 - 10.49 cm and the number of specimens per length class (1 - 12) was used a frequency variable in the linear regression. log W = -2.206 + 3.11 * log L W = 0.0062 L 3.11 a = 0.00608 – 0.00637 b = 3.10 – 3.11 n = 468, r 2 = 0.999

Growth Stanzas Double-logarithmic plot of weight vs. length for Clupea harengus, based on data in Fulton (1904), showing two growth stanzas and an inflection point at about 8 cm. For the first growth stanza: n = 5 (92), r 2 = 0.998, 95% CL of a = 0.00125 – 0.00134, 95% CL of b = 3.66 – 3.72. For the second growth stanza: n = 46(400), r 2 = 0.999, 95% CL of a = 0.00301 – 0.00312, 95% CL of b = 3.28 – 3.29.

How to Report LWR W = 0.0121 * L 3.03 r 2 = 0.994 n = 54 sex = mixed Length range = 30 -72 cm TL 95% CL a = 0.0101 – 0.0148 95% CL b = 2.99 – 3.09 Species: Gadus morhua Linnaeus, 1758 Locality: Kiel Bight, Germany Gear: Bottom trawl with 6 cm mesh size. Sampling duration: Mid April to mid May, 2005 Remarks: Beginning of spawning season.

Fulton’s Condition Factor K = 100 * W / L 3 Used to compare ‘fatness’ or condition of specimens (of the same species) of similar size, e.g. to detect differences between sexes, seasons or localities. Example: Condition of a specimen of 10 grams weight and 10 cm length 1 = 100 * 10 / 1000

Relationship between a and K If b is reasonably close to 3 (2.95 – 3.05) then K = 100 * a and K applies to the whole length range from which LWR was derived

Condition as a Function of Size and Season Log-log plot of condition vs. length of Comber Serranus cabrilla taken in spring, summer, fall and winter, respectively, in the Aegean Sea. The dotted line shows the condition factors associated with geometric mean a and mean b across all available LWRs for this species ( Froese 2006) b = 2.96 b = 2.91 b = 2.77 b = 2.60

Understanding b Frequency distribution of mean exponent b based on 3,929 records for 1,773 species, with median = 3.025, 95% CL = 3.011 – 3.036, 5th percentile = 2.65 and 95th percentile = 3.39, minimum = 1.96, maximum = 3.94; the normal distribution line is overlaid (Froese 2006).

b as Function of Size Range Absolute residuals of b=3.0 plotted over the length range used for establishing the weight-length relationship. The length range is expressed as fraction of the maximum length known for the species. A robust regression analysis of absolute residuals vs. fraction of maximum length resulted in n = 2,800, r 2 = 0.0065, slope = -0.0505, 95% CL -0.0735 – -0.0274 (Froese 2006).

Mean b as a Function of Studies Absolute residuals of mean b from b = 3.0, plotted over the respective number of weight-length estimates contributing to mean b, for 1,773 species. The two outliers with about 10 weight-length estimates belong to species with truly allometric growth (Froese 2006).

Understanding b b = 3 Isometric growth and small specimens have same condition as large specimens. Default. b << 3 Negative allometric growth or small specimens in better condition than large ones. b >> 3 Positive allometric growth or large specimens in better condition than small ones.

Understanding a If b ~ 3 then a is a form-factor with a = 0.001 -> eel-like (eel) a = 0.01-> fusiform (cod, tunas) a = 0.1-> spherical (puffers, boxfish)

Understanding a Frequency distribution of mean log a based on 3,929 records for 1,773 species, with median a = 0.01184, 95% CL = 0.0111 – 0.0123, 5th percentile = 0.00143, 95th percentile = 0.0451, minimum = 0.0001, and maximum = 0.273 (Froese 2006).

Interdependence of a and b Any increase in slope b will decrease intercept a, and vice-versa.

log a vs. b Plot Plot of log a over b for 25 length-weight relationships of Oncorhynchus gilae. The black dot was identified as outlier by robust regression analysis (robust weight = 0.000).

Oncorhynchus gilae in FishBase Variability discussed in Froese (2006)

Multi-species comparison Scatter plot of mean log a (TL) over mean b for 1,232 species with body shape information. Areas of negative allometric, isometric and positive allometric change in body weight relative to body length are indicated. The regression line is based on robust regression analysis for fusiform Species (Froese 2006).

More Information Froese, R., 2006. Cube law, condition factor, and weight- length relationships: history, meta-analysis and recommendations. Journal of Applied Ichthyology 22(4):241-253 (372 Thomson Reuters citations and 606 Google Scholar citations in November 2014) Froese, R., J.T. Thorson and R.B. Reyes, Jr. 2013. A Bayesian approach for estimating length-weight relationships in fishes. Journal of Applied Ichthyology, doi: 10.1111/jai.12299

Exercise In FishBase, find species with at least 5 LWR studies Discuss the variability of a and b (do log a over b plot in Excel) Select a study that describes LWR well and justify your selection