Mammals, mass, and one suspiciously movable number
The Exponent That Bends
Fit one real mammalian metabolism dataset two ways. A moving mass window makes the fitted exponent drift from about two-thirds toward three-quarters, while a quadratic fit exposes the curvature that a single power law leaves behind.
Each point is a published species-average row from the mammalian BMR compilation used by Savage and colleagues in 2004. Both axes are logarithmic. Drag the three-decade window from shrews toward cattle.
Published species-average rows loadingmass in grams · BMR in watts · OLS on log10 values
inside fitoutside fitwindow OLSfull quadratic
The width is a choice copied from the moving-window analysis in Kolokotrones et al. Figure 2. It is not a biological boundary.
Fitted exponent
…
calculating
Rows in fit
…
same source, selected only by mass
Window result: calculating from the embedded measurements.
Start withB = aMp
Take log10log B = log a + p log M
Fit a straight lineslope = p
A power law becomes a straight line after both axes are logged. Its exponent is simply the slope. That is why the number above is not supplied by a theory or selected from a menu. The browser calculates ordinary least squares from only the green points.
At the small preset, the slope sits near 2/3. Move the same-width window right to the larger preset and it sits near 3/4. No measurement changed. Only the part of the mass range offered to a straight line changed.
This does not make either fraction foolish
Two-thirds has a geometric attraction: surface area grows as length squared while volume and mass grow as length cubed. Three-quarters has a network argument and a long empirical history. The plot makes a narrower claim. If the log-log relationship bends, forcing one straight slope onto different portions must return different exponents.
Layer 2: answer the noise objection
The bend has a coefficient.
A drifting window could still be dismissed as sampling noise. So fit the whole dataset twice: once with a straight line, once with the next simplest curve.
Both models refit in this browsersame … rows · one extra coefficient
linearquadratic
Show on the full-range plot
Curvature b2
calculating
positive means convex on log-log axes
Classical 95% CI
calculating
two-sided t test
Residual sum of squares
calculating
linear → quadratic, log10 units
AIC improvement
calculating
larger is more support for the bend
Curvature test: calculating from the embedded measurements.
The fitted model is log10 B = b0 + b1 log10 M + b2(log10 M)². The browser estimates all three coefficients. The confidence interval for b2 stays above zero, the residual error falls, and AIC rewards the curved model even after charging it for the extra term.
The gain in explained variance looks small because body mass already explains most of the four-order-of-magnitude rise in BMR. A small fraction of total variance can still be a systematic shape in what remains. On these data the quadratic removes … of the straight model's residual sum of squares.
Descriptive curve, not a law of infinity
The quadratic is a local description of the observed mammalian range. Its slope is b1 + 2b2 log10 M, so extrapolating far enough makes that slope grow without bound. Kolokotrones and colleagues explicitly warned that sparse data among the largest animals makes such extrapolation unsafe.
The derivation
West, Brown, and Enquist, 1997
Inside a model of resource delivery, three structural assumptions and an optimization criterion lead to quarter-power scaling.
A hierarchical branching supply network
A network that fills the organism's volume
Terminal units that do not scale with body size
Transport energy minimized by network geometry
within the model → B ∝ M3/4
This is a real derivation from stated assumptions. It is not a license to relabel every empirical deviation as error.
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The separate criticisms
Dodds, Rothman, and Weitz, 2001
Their reanalysis found little evidence in the datasets they studied for rejecting 2/3 in favor of 3/4. They argued that available derivations of 3/4 depended on assumptions that did not compel rejection of the geometric null.
This critique predates the 2010 curvature result.
It also noticed a possible increase in slope at larger mammalian masses.
Criticizing assumptions is not the same as disproving every network model.
Does the 2010 paper refute WBE?
No. The pure 3/4 power law is rejected as a full-range description by these data. The original WBE model has its own theoretical criticisms, which are a separate matter. Kolokotrones and colleagues showed that a finite-size extension of the original model, with positive coefficients under its original assumptions, bends the opposite way from the data. They also proposed modifications that can produce convex curvature, and one modified version fit nearly as well as the quadratic. That is a constraint on the theory, not a funeral.
Why ordinary species-by-species OLS is not the final statistical word
Species share evolutionary history, so 626 rows are not 626 wholly independent draws from nature. Closely related mammals can resemble one another in both mass and physiology. The paper used phylogenetic generalized least squares on its McNab and Sieg datasets. The quadratic term remained significant, with a smaller coefficient in some fits. This page cannot rerun that analysis because the Savage table embedded here has no phylogenetic tree or temperature field. The limitation stays visible rather than being silently ignored.
1997
West, Brown, Enquist
A space-filling distribution network derives quarter-power scaling under explicit assumptions.
2001
Dodds, Rothman, Weitz
Older datasets give little reason to reject two-thirds, and theory alone does not settle the exponent.
2003
White and Seymour
After controls for temperature, digestive state, and phylogeny, their mammalian analysis supports two-thirds.
2004
Savage and colleagues
A broad compilation argues that quarter powers predominate. Its mammalian species averages power this page.
2005
Glazier
A wide review emphasizes taxonomic, physiological, environmental, and ontogenetic variation beyond one universal exponent.
2010
Kolokotrones and colleagues
The log-log relation is convex. Fitted exponents depend on which masses a dataset emphasizes.
The check
Nothing in this table is typed in.
live calculation
result
cross-check
species-average rows parsed
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expected 626
small window slope, 100 to 103 g
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near 2/3
larger window slope, 102 to 105 g
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near 3/4
full linear slope
…
paper: 0.7115
full quadratic b2
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paper: 0.0320
quadratic b2 p-value
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classical OLS
AIC, linear → quadratic
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same points
Independent spine
The browser parses the embedded Savage species averages, takes logarithms, solves both normal-equation systems with pivoted Gauss-Jordan elimination, estimates covariance, evaluates a Student t tail, and computes full-likelihood Gaussian AIC, counting the fitted residual variance as a parameter.
The dependency-free verifier repeats those steps from the page's embedded data:
Measured and compiledMass and BMR values come from the Savage et al. compilation, which reconciled multiple earlier sources and averaged repeated species measurements. These are published observations, not values measured for this page.
Duplicate label retainedThe table has … fitted rows but … distinct species-name strings. Perameles gunnii appears twice, and the preserved source marks the second as a repeat. Both rows are retained because that exactly reproduces the published 626-row regression.
Free choiceThe moving window is three log10 units wide. That follows the 2010 paper's display. The small and larger presets are chosen because this dataset makes the two historic fractions visible there.
Model assumptionThe page's confidence interval is classical OLS. It treats logged residuals as independent with constant variance. The paper used robust variance estimates and further models.
PhylogenySpecies are evolutionarily related. The page does not have a tree and cannot perform PGLS. The paper did, and curvature remained significant with changed coefficients.
TemperatureThis embedded Savage table has mass and BMR, not body temperature. The 2010 analysis found curvature in separate datasets with and without a temperature term.
Open mechanismA significant quadratic describes shape. It does not identify the physiological cause, prove one scaling theory, or refute every network account.
No universal extrapolationThe quadratic is not claimed beyond the observed … range. Its unbounded eventual slope is a warning against extrapolation.
No verdict of wrongThree-quarters is not branded wrong, and Kleiber's law is not declared refuted. The verified result is that one exponent does not describe this full range.
Sources and apparatus
Tom Kolokotrones, Van M. Savage, Eric J. Deeds, and Walter Fontana, “Curvature in metabolic scaling,” Nature 464, 753-756 (2010), with its Supplementary Information and Supplementary Data workbook. This is the source of the curvature claim, cross-check coefficients, moving-window design, temperature analysis, sensitivity analysis, and PGLS caveat.
Van M. Savage, James F. Gillooly, William H. Woodruff, Geoffrey B. West, Andrew P. Allen, Brian J. Enquist, and James H. Brown, “The predominance of quarter-power scaling in biology,” Functional Ecology 18, 257-282 (2004). The … mammalian species-average rows embedded here are the paper's Appendix 1 dataset, as preserved in the CRAN asbio dataset and in the 2010 supplementary workbook.