LengthWeight Relationships

1
Length-Weight Relationships

• Rainer Froese
• (PopDyn SS 3.6.2008)

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 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

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 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 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.
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), 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.
7
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.

8
Fultons 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

9
Condition as a Function of Size and Season
b 2.96
b 2.91
b 2.77
b 2.60
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.
10
Understanding b
Frequency distribution of mean exponent b based
on 3,929 records for 1,773 species, with median
3.025, 95 CL 3.011 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 b3.0 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, r2 0.0065, slope
-0.0505, 95 CL -0.0735 -0.0274.
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 ltlt 3
• Negative allometric growth or small specimens in
  better condition than large ones.
• b gtgt 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 0.001 -gt eel-like (eel)
• a 0.01 -gt fusiform (cod, tunas)
• a 0.1 -gt spherical (puffers, boxfish)

15
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.
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., 2006. Cube law, condition factor, and
  weight-length relationships history,
  meta-analysis and recommendations. Journal of
  Applied Ichthyology 22(4)241-253
• Thank You