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Allometric exponents support a 3/4 power scaling law

Allometric exponents support a 3/4 power scaling law. Catherine C. Farrell Nicholas J. Gotelli Department of Biology University of Vermont Burlington, VT 05405. Gotelli lab, May 2005. Allometric Scaling. What is the relationship metabolic rate (Y) and body mass (M)?. Allometric Scaling.

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Allometric exponents support a 3/4 power scaling law

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  1. Allometric exponents support a 3/4 power scaling law Catherine C. Farrell Nicholas J. Gotelli Department of Biology University of Vermont Burlington, VT 05405

  2. Gotelli lab, May 2005

  3. Allometric Scaling • What is the relationship metabolic rate (Y) and body mass (M)?

  4. 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

  5. Allometric Scaling • What is the relationship metabolic rate (Y) and body mass (M)? • Usually follows a power function: • Y = CMb

  6. 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

  7. 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

  8. 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!

  9. Allometric Scaling • What is the expected value of b? ??

  10. Hollywood Studies Allometry Godzilla (1954) A scaled-up dinosaur

  11. Hollywood Studies Allometry The Incredible Shrinking Man (1953) A scaled-down human

  12. Miss Allometry Raquel Welch Movies spanning > 15 orders of magnitude of body mass!

  13. 1 Million B.C. (1970)

  14. Fantastic Voyage (1964)

  15. Hollywood (Finally) Learns Some Biology Alien (1979) Antz (1998)

  16. Hollywood’s Allometric Hypothesis: b = 1.0

  17. Surface/Volume Hypothesis b = 2/3 Surface area  length2 Volume  length3

  18. Surface/Volume Hypothesis Microsoft Design Flaw! b = 2/3 Surface area  length2 Volume  length3

  19. 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)

  20. 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)

  21. 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 < 3/4?

  22. 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 < 3/4?

  23. Question 1 • Can variation in published allometric exponents be attributed to variation in • sample size • average body size • range of body sizes measured

  24. Allometric exponent as a function of number of species in sample P = 0.6491 Mammals Other Allometric Exponent

  25. Allometric exponent as a function of midpoint of mass P = 0.5781 Weighted by sample size P = 0.565 Mammals Other Allometric Exponent

  26. Allometric exponent as a function of log(difference in mass) P = 0.5792 Weighted by sample size: P = .649 Mammals Other Allometric Exponent

  27. 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

  28. 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

  29. Question 1: Conclusions • Allometric exponent was not correlated with • sample size • midpoint of mass • range of body size • Reported values not statistical artifacts

  30. 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 < 3/4?

  31. Question 2: What is the best estimate of the allometric exponent? Mammals Birds Reptiles

  32. b = 3/4 Allometric Exponent b = 2/3

  33. b = 3/4 Allometric Exponent b = 2/3

  34. b = 3/4 Allometric Exponent b = 2/3

  35. Question 2: Conclusions Mammals and Birds Results suggest the true exponent is between 2/3 and 3/4 Reptiles Variation is due to small sample sizes and variability in experimental conditions

  36. 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 < 3/4?

  37. Question 3: Likelihood Ratio

  38. 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 < 3/4?

  39. Question 4: estimates often < 3/4? Allometric Exponent b = 3/4 b = 2/3

  40. Linear Regression • Most published exponents based on linear regression • Assumption: x variable is measured without error • Measurement error in x may bias slope estimates

  41. Measurement Error • Limits measurement of true species mean mass • Includes seasonal variation • Systematic variation • “Classic” measurement errors

  42. Simulation: Motivatione.g. y = 2xtrue Slope = 2.0 Slope = 1.8

  43. Assumed model Yi = mi0.75 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 Simulation: Assumptions

  44. Allometric Exponent Circles: mean of 100 trials Triangles: estimated parametric confidence intervals

  45. Question 4: Conclusions • Biases slope estimates down • Never biases slope estimates up • Parsimonious explanation for discrepancy between observed and predicted allometric exponents for homeotherms.

  46. 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

  47. 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

  48. 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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