Allometric exponents support a 3/4 power scaling law
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Allometric exponents support a 3/4 power scaling
law
Catherine C. FarrellNicholas J. Gotelli
Department of BiologyUniversity of VermontBurlington, VT 05405
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Gotelli lab, May 2005
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Allometric Scaling
• What is the relationship metabolic rate (Y) and body mass (M)?
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Allometric Scaling
• What is the relationship metabolic rate (Y) and body mass (M)?
• Mass units: grams, kilograms
• Metabolic units: calories, joules, O2 consumption, CO2 production
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Allometric Scaling
• What is the relationship metabolic rate (Y) and body mass (M)?
• Usually follows a power function:
• Y = CMb
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Allometric Scaling
• What is the relationship metabolic rate (Y) and body mass (M)?
• Usually follows a power function:
• Y = CMb
• C = constant
• b = allometric scaling coefficient
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Allometric Scaling: Background• Allometric scaling equations relate basal
metabolic rate (Y) and body mass (M) by an allometric exponent (b)
0
2
4
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12
0 20 40 60 80 100 120
M
Y
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
Log (M)
Log (Y)
Y = YoMb Log Y = Log Yo + b log M
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Allometric Scaling: Background• Allometric scaling equations relate basal
metabolic rate (Y) and body mass (M) by an allometric exponent (b)
0
2
4
6
8
10
12
0 20 40 60 80 100 120
M
Y
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
Log (M)
Log (Y)
Y = YoMb Log Y = Log Yo + b log M
b is the slope of the log-log plot!
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Allometric Scaling
• What is the expected value of b?
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
Log (M)
Log (Y)
??
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Hollywood Studies Allometry
Godzilla (1954)
A scaled-up dinosaur
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Hollywood Studies Allometry
The Incredible Shrinking Man (1953)
A scaled-down human
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Miss Allometry
Raquel Welch
Movies spanning > 15 orders of magnitude of body mass!
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1 Million B.C. (1970)
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Fantastic Voyage (1964)
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Alien (1979) Antz (1998)
Hollywood (Finally) Learns Some Biology
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Hollywood’s Allometric Hypothesis:
0
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1.5
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2.5
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3.5
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4.5
0 0.5 1 1.5 2 2.5
Log (M)
Log (Y)
b = 1.0
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Surface/Volume Hypothesis
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
Log (M)
Log (Y)
b = 2/3
Surface area length2 Volume length3
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Surface/Volume Hypothesis
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
Log (M)
Log (Y)
b = 2/3
Surface area length2 Volume length3
Microsoft Design Flaw!
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New allometric theory of the 1990s
• Theoretical models of universal quarter-power scaling relationships– Predict b = 3/4– Efficient space-filling energy transport
(West et al. 1997)– Fractal dimensions (West et al. 1999)– Metabolic Theory of Ecology (Brown
2004)
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Theoretical Predictions
• b = 3/4– Maximize internal exchange efficiency– Space-filling fractal distribution networks (West et al.
1997, 1999)
• b = 2/3– Exterior exchange geometric constraints– Surface area (length2): volume (length3)
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Research QuestionsMeta-analysis of published exponents
1. Is the calculated allometric exponent (b) correlated with features of the sample?
2. Mean and confidence interval for published values?
3. Likelihood that b = 3/4 vs. 2/3?
4. Why are estimates often < 3/4?
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Allometric exponent
Species in sample
Taxon Source
0.71 391 mammals (Heusner 1991)
0.713 321 mammals (McNab 1988)
0.69 487 mammals (Lovegrove 2000)
0.737 626 mammals (Savage et al. 2004)
0.74 10 mixed (Kleiber 1932)
0.76 228 mammals (West et al. 2002)
0.724 35 passerine birds
(Lasiewski and Dawson 1967)
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Research Questions
1. Is the calculated allometric exponent (b) correlated with features of the sample?
2. Calculate mean & confidence interval for published values?
3. Likelihood that b = 3/4 vs. 2/3
4. Why are estimates often < 3/4?
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Question 1
• Can variation in published allometric exponents be attributed to variation in– sample size– average body size– range of body sizes measured
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Allometric exponent as a function of number of species in sample
Other
P = 0.6491
0 100 300 500 7000.60
0.65
0.70
0.75
0.80
0.85
0.90
Number of species in sample
Mammals
Allo
met
ric E
xpon
ent
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Allometric exponent as a function of midpoint of mass
P = 0.5781
Weighted by sample size P = 0.565
0 500 1000 1500 20000.60
0.65
0.70
0.75
0.80
0.85
0.90
Midpoint of mass
Mammals
Other
Allo
met
ric E
xpon
ent
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Allometric exponent as a function of log(difference in mass)
P = 0.5792
Weighted by sample size: P = .649
Mammals
Other
0 1 2 3 4 5 60.60
0.65
0.70
0.75
0.80
0.85
0.90
Log(difference in mass)
Allo
met
ric E
xpon
ent
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Non-independence in Published Allometric Exponents
• phylogenetic non-independence – species within a study exhibit varying levels of
phylogenetic relatednessBokma 2004, White and Seymour 2003
• data on the same species are sometimes used in multiple studies
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Independent Contrast Analysis
• Paired studies analyzing related taxa (Harvey and Pagel 1991)
– e.g., marsupials and other mammals
• Each study was included in only one pair• No correlation (P > 0.05) between difference in the
allometric exponent and– difference in sample size,
– midpoint of mass
– range of mass
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Question 1: Conclusions
• Allometric exponent was not correlated with– sample size– midpoint of mass– range of body size
• Reported values not statistical artifacts
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Research Questions
1. Is the calculated allometric exponent (b) correlated with features of the sample?
2. Calculate mean & confidence interval for published values?
3. Likelihood that b = 3/4 vs. 2/3
4. Why are estimates often < 3/4?
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Question 2: What is the best estimate of the allometric
exponent?
Mammals Birds Reptiles
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Allo
met
ric E
xpon
ent
Mammals Birds Reptiles
0.60
0.65
0.70
0.75
0.80
0.85
0.90
b = 3/4
b = 2/3
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Allo
met
ric E
xpon
ent
Mammals Birds Reptiles
0.60
0.65
0.70
0.75
0.80
0.85
0.90
b = 2/3
b = 3/4
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Allo
met
ric E
xpon
ent
Mammals Birds Reptiles
0.60
0.65
0.70
0.75
0.80
0.85
0.90
b = 2/3
b = 3/4
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Question 2: Conclusions
Reptiles
Variation is due to small sample sizes and variability in experimental conditions
Mammals and Birds
Results suggest the true exponent is between 2/3 and 3/4
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Research Questions
1. Is the calculated allometric exponent (b) correlated with features of the sample?
2. Calculate mean & confidence interval for published values?
3. Likelihood that b = 3/4 vs. 2/3?
4. Why are estimates often < 3/4?
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Question 3: Likelihood Ratio
b = 3/4 : b = 2/3
All species 16 074
Mammals 105
Birds 7.08
Reptiles 2.20
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Research Questions
1. Is the calculated allometric exponent (b) correlated with features of the sample?
2. Calculate mean & confidence interval for published values?
3. Likelihood that b = 3/4 vs. 2/3?
4. Why are estimates often < 3/4?
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Allo
met
ric E
xpon
ent
Mammals Birds Reptiles
0.60
0.65
0.70
0.75
0.80
0.85
0.90
b = 3/4
b = 2/3
Question 4: estimates often < 3/4?
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Linear Regression
• Most published exponents based on linear regression • Assumption: x variable is measured without error • Measurement error in x may bias slope estimates
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Measurement Error
• Limits measurement of true species mean mass
• Includes seasonal variation
• Systematic variation
• “Classic” measurement errors
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Simulation: Motivatione.g. y = 2xtrue
0 20 40 60 80 100
0
50
100
150
200
True measurement
Slope = 2.0
Slope = 1.8
0 20 40 60 80 100 120
0
50
100
150
200
Error in measurement
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Simulation: Assumptions
Assumed modelYi = mi 0.75
Add variation in measurement of mass
Yi = (mi + Xi)b
Simulate error in measurement
Xi = KmiZ
Z ~ N(0,1)
Y = met. Rate
m = mass
X = error term (can be positive or negative)
b = exponent
K = % measurement error
Z = a random number
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Circles: mean of 100 trialsTriangles: estimated parametric confidence intervals
Allo
met
ric E
xpon
ent
0.05 0.10 0.15 0.200.70
0.71
0.72
0.73
0.74
0.75
0.76
Proportion Measurement Error
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Question 4: Conclusions
• Biases slope estimates down
• Never biases slope estimates up
• Parsimonious explanation for discrepancy between observed and predicted allometric exponents for homeotherms.
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Slope Estimates Revisited
• Other methods than least-squares can be used to fit slopes to regression data
• “Model II Regression” does not assume that error is only in the y variable
• Equivalent to fitting principal components
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Ordinary Least-Squares Regression
• Most published exponents based on OLS • Assumption: x variable is measured without error • Fitted slope minimizes vertical residual deviations
from line
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Reduced Major Axis Regression
• Minimizes perpendicular distance of points to line • Does not assume all error is contained in y variable • “Splits the difference” between x and y errors
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Reduced Major Axis Regression
• Slope of Major Axis Regression is always > slope of OLS Regressions
• Major Axis Regression slope = b / r2
increasing b
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Re-analysis of Data• Adjusted slope for n = 5 mammal data sets
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Conclusions
• Measured allometric exponents not correlated with features of sample
• Published exponents cluster tightly for homeotherms – values slightly lower than the
predicted b = 3/4.
• Published exponents highly variable for poikilotherm studies
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Conclusions
• Body mass measurement error always biases least-squares slope estimates downward
• Observed allometric exponents closer to 3/4 than 2/3
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Acknowledgements
Gordon Research Conference Committee
Metabolic Basis of Ecology
Bates College
July 4-9, 2004