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First published online October 31, 2008
Journal of Experimental Biology 211, 3581-3587 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.023317
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Traditional allometric analysis fails to provide a valid predictive model for mammalian metabolic rates

Gary C. Packard1,* and Geoffrey F. Birchard2

1 Department of Biology, Colorado State University, Fort Collins, CO 80523, USA
2 Department of Environmental Science and Policy, George Mason University, Fairfax, VA 22030, USA


Figure 1
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  		Fig. 1. Values for 51 samples exclusive of the elephant were taken from Appendix 2 in the study by Savage and colleagues (Savage et al., 2004Go). (A) A straight line was fitted to log-transformed data by ordinary least squares. (B) Residuals from the analysis in A are plotted against predicted values for logBMR. The parabolic pattern to the display indicates that a straight line is not an adequate function for describing values in the bivariate plot and calls into question the utility of the transformation. Departures from linearity were noted by Dodds and colleagues (Dodds et al., 2001Go) and Kozlowski and Konarzewski (Kozlowski and Konarzewski, 2005Go) in similar analyses, but both groups of investigators nonetheless continued to work with values in the logarithmic domain. (C) A two-parameter power function was fitted to arithmetic data by non-linear regression. (D) Residuals from the non-linear regression are plotted against body mass on a log scale to illustrate how variance increases with body size.

 

Figure 2
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  		Fig. 2. Validation of alternative regression models derived from analyses summarized in Fig. 1. (A) Equations fitted to data by non-linear regression (solid black line) and by back-transformation from logarithms (dashed red line) are displayed together in a double logarithmic plot. (B) Equations fitted to data by non-linear regression (solid black line) and by back-transformation from logarithms (dashed red line) are displayed together in an arithmetic plot. (C) Arithmetic values for BMR are displayed against values for mass on a log scale. The upward trajectory for the lines is a consequence of the semi-log plot. The inset expands that part of the scale for animals weighing between 100 and 10,000 g.

 

Figure 3
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  		Fig. 3. (A) A straight line was fitted by ordinary least squares to log-transformed data for mammals weighing less than 260 g. (B) A straight line was fitted by ordinary least squares to log-transformed data for mammals weighing more than 260 g. (C) A two-parameter power function was fitted by non-linear regression to values for metabolic rate and body mass for mammals weighing less than 260 g. (D) A two-parameter power function was fitted by non-linear regression to values for metabolic rate and body mass for mammals weighing more than 260 g. (E) Equations fitted to data for small mammals by non-linear regression (solid black line) and back-transformation (dashed red line) are displayed against the backdrop of values in the original scale. (F) Equations fitted to data for large mammals by non-linear regression (solid black line) and back-transformation (dashed red line) are displayed against the backdrop of values in the original scale.

 

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