Length-Weight Relationships Rainer Froese (PopDyn SS )

2 How to Measure Size in Fishes Length as a proxy for weight Total length (TL) Standard length (SL) Fork length (FL) Other length measurements Length as proxy for size overestimates weight in eels, underestimates in puffers and boxfishes

3 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

4 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.

5 LWR Plot II Double-logarithmic plot of the data in LWR Plot I. The overall regression line is W = * L 3.108, with n = 468, r 2 = , 95% CL of a = – , 95% CL of b = – Note that the mid- length of length classes was used such as cm for the length class cm and the number of specimens per length class (1 - 12) was used a frequency variable in the linear regression.

6 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 = , 95% CL of a = – , 95% CL of b = 3.66 – For the second growth stanza: n = 46(400), r 2 = , 95% CL of a = – , 95% CL of b = 3.28 – 3.29.

7 How to Report LWR W = * L 3.03 r 2 = n = 54 sex = mixed Length range = cm TL 95% CL a = – % 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.

8 Fulton’s Condition Factor K = 100 * W / L 3 Used to compare ‘fatness’ or condition of specimens 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

9 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. b = 2.96 b = 2.91 b = 2.77 b = 2.60

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

11 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 = , slope = , 95% CL –

12 Mean b as a Function of Studies Absolute residuals of mean b per species 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.

13 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.

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

15 Understanding a Frequency distribution of mean log a based on 3,929 records for 1,773 species, with median a = , 95% CL = – , 5th percentile = , 95th percentile = , minimum = , and maximum =

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

17 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 (see text) by robust regression analysis (robust weight = 0.000).

18 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.

19 More Information Froese, R., Cube law, condition factor, and weight- length relationships: history, meta-analysis and recommendations. Journal of Applied Ichthyology 22(4): Thank You