Allometric exponents support a 3/4 power scaling law

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Allometric exponents support a 3/4 power scaling
law

• Catherine C. Farrell
• Nicholas J. Gotelli
• Department of Biology
• University of Vermont
• Burlington, 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)

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)

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?

??
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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 gt 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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Hollywood (Finally) Learns Some Biology

• Alien (1979) Antz (1998)

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Hollywoods Allometric Hypothesis
b 1.0
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Surface/Volume Hypothesis
b 2/3
Surface area ? length2
Volume ? length3
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Surface/Volume Hypothesis
Microsoft Design Flaw!
b 2/3
Surface area ? length2
Volume ? length3
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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

• Is the calculated allometric exponent (b)
  correlated with features of the sample?
• Mean and confidence interval for published
  values?
• Likelihood that b 3/4 vs. 2/3?
• Why are estimates often lt 3/4?

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Research Questions

• Is the calculated allometric exponent (b)
  correlated with features of the sample?
• Calculate mean confidence interval for
  published values?
• Likelihood that b 3/4 vs. 2/3
• Why are estimates often lt 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
P 0.6491
Mammals
Other
Allometric Exponent
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Allometric exponent as a function of midpoint of
mass
P 0.5781 Weighted by sample size P
0.565
Mammals
Other
Allometric Exponent

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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
Allometric Exponent

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Non-independence in Published Allometric
Exponents

• phylogenetic non-independence
• species within a study exhibit varying levels of
  phylogenetic relatedness
• Bokma 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 gt 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

• Is the calculated allometric exponent (b)
  correlated with features of the sample?
• Calculate mean confidence interval for
  published values?
• Likelihood that b 3/4 vs. 2/3
• Why are estimates often lt 3/4?

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Question 2 What is the best estimate of the
allometric exponent?

• Mammals Birds Reptiles

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b 3/4
Allometric Exponent
b 2/3
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b 3/4
Allometric Exponent
b 2/3
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b 3/4
Allometric Exponent
b 2/3
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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

• Is the calculated allometric exponent (b)
  correlated with features of the sample?
• Calculate mean confidence interval for
  published values?
• Likelihood that b 3/4 vs. 2/3?
• Why are estimates often lt 3/4?

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Question 3 Likelihood Ratio
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Research Questions

• Is the calculated allometric exponent (b)
  correlated with features of the sample?
• Calculate mean confidence interval for
  published values?
• Likelihood that b 3/4 vs. 2/3?
• Why are estimates often lt 3/4?

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Question 4 estimates often lt 3/4?
Allometric Exponent
b 3/4
b 2/3
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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
Slope 2.0
Slope 1.8
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Simulation Assumptions

• Assumed model
• Yi 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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Allometric Exponent
Circles mean of 100 trials Triangles estimated
parametric confidence intervals
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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 gt 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