Transcript Biodiversity of Fishes Length

Length-Weight Relationships
Rainer Froese
(PopDyn SS 3.6.2008)
How to Measure Size in Fishes
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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
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Relationship Between
Weight and Length
W = a * Lb
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
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LWR Plot I
14000
12000
Weight (g)
10000
8000
6000
4000
2000
0
0
20
40
60
80
100
120
Length (cm )
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.
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LWR Plot II
100000
10000
Weight (g)
1000
W = 0.00622 * L3.108
100
10
1
0.1
1
10
100
1000
Length (cm )
Double-logarithmic plot of the data in LWR Plot I. The overall regression line is W = 0.00622 * L3.108, with
n = 468, r2 = 0.9995, 95% CL of a = 0.00608 – 0.00637, 95% CL of b = 3.101 – 3.114. Note that the midlength 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.
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Growth Stanzas
1000
100
Weight (g)
W = 0.00307 * L3.28
10
1
W = 0.00130 * L3.69
0.1
0.01
1
10
100
Length (cm)
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), r2 = 0.9984,
95% CL of a = 0.00125 – 0.00134, 95% CL of b = 3.66 – 3.72. For the second growth stanza: n = 46(400),
r2 = 0.9996, 95% CL of a = 0.00301 – 0.00312, 95% CL of b = 3.28 – 3.29.
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How to Report LWR
W = 0.0121 * L3.03
r2 = 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.
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Fulton’s Condition Factor
K = 100 * W / L3
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
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Condition as a Function of Size and Season
0.3
Condition factor (log K)
0.2
0.1
Fall
b = 2.96
Winter
b = 2.91
Summer
b = 2.77
Spring
b = 2.60
0.0
-0.1
-0.2
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Length (log; cm)
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
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with geometric mean a and mean b across all available LWRs for this species.
Understanding b
Frequency (n)
300
200
100
0
2.0
2.5
3.0
3.5
4.0
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,
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minimum = 1.96, maximum = 3.94; the normal distribution line is overlaid.
b as Function of Size Range
Residuals of b
1.5
1.0
0.5
0.0
0.0
0.5
1.0
Fraction of maximum length
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
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in n = 2,800, r = 0.0065, slope = -0.0505, 95% CL -0.0735 – -0.0274.
Mean b as a Function of Studies
1.2
Residuals of b
1.0
0.8
0.6
0.4
0.2
0.0
1
10
100
Number of L-W estimates
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
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10 weight-length estimates belong to species with truly allometric growth.
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.
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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)
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Understanding a
Frequency (n)
300
200
100
0
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
log 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.
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Interdependence of a and b
100000
10000
Weight (g)
1000
W = 0.00622 * L3.108
100
10
1
0.1
1
10
100
1000
Length (cm )
Any increase in slope b will decrease intercept a, and vice-versa.
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log a vs. b Plot
-1.0
log a
-1.5
-2.0
-2.5
-3.0
2.5
2.7
2.9
3.1
3.3
b
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).
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Multi-species comparison
-0.5
o short & deep
+ fusiform
- elongated
— eel-like
-1.0
-1.5
log a
-2.0
-2.5
-3.0
-3.5
-4.0
negative allometric
isometric
positive allometric
-4.5
1.5
2.0
2.5
3.0
3.5
4.0
4.5
b
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
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body length are indicated. The regression line is based on robust regression analysis for
fusiform Species.
More Information
Froese, R., 2006. Cube law, condition factor, and weightlength relationships: history, meta-analysis and
recommendations. Journal of Applied Ichthyology
22(4):241-253
Thank You
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