Does Metabolism Really Scale to the Three Quarter Power

Ninety Years, One Sixth of a Decimal Place

The quantity under dispute is 0.0833. That is the whole of it. One camp says an animal's metabolic rate goes as its mass to the power 0.75, the other says 0.6667, and the distance between those two positions is one twelfth. Careers have been spent there.

We came to this expecting to referee. What we found is that the two camps are arguing about a number that our data decline to hold still. Here is the shape of it. Fit 422 mammals by the standard method and the exponent is 0.7091, which is not three quarters and is not two thirds, and which sits 0.00077 from the exact midpoint of the two positions, about one tenth of one standard error away from splitting the difference. We did not arrange that. We cannot explain it either, and we suspect it is a coincidence, but it made the club laugh.

Change the estimator and the answer moves. Change which animals you include and it moves further. Ask for the slope at 10 g rather than at 100 kg and it moves further still, by more than three times the disputed gap. None of this is a failure of the data. It is what the data have been saying for decades to anyone who fitted them more than one way.

So this article is a referee's report on an argument, and the finding is that both parties have been partly right about a quantity that was never a constant. Birds really do come out near two thirds, and carnivores and primates really do come out near three quarters. Rodents, all 155 of them, come out at 0.6666, which is two thirds to four decimal places and is the single funniest number in the study.

The Compilation, the Fields We Took, and the Ones We Ignored

The club has no respirometer. We have never put an animal in a chamber and measured its oxygen consumption, and nothing in this article was observed by us. What we have is arithmetic performed on other people's published measurements, which is what a meta-analysis is, and the only honest way to describe it.

The data are AnAge, the Animal Ageing and Longevity Database, Build 15, released 3 July 2023, part of the Human Ageing Genomic Resources [1]. We downloaded https://genomics.senescence.info/species/dataset.zip on 14 September 2026, cached the archive under analysis/data/, and the script prints the SHA-256 of both the archive and the extracted table at run time so anybody can confirm they have the same bytes we had. The download succeeded on the first attempt. Had it failed we would have said so here rather than reaching for a substitute.

AnAge Build 15 holds 4,645 entries. We kept every entry carrying both a positive body mass in grams and a positive metabolic rate in watts, which is 627 species, each appearing exactly once. Nothing was excluded and no outlier was trimmed. No value was entered by hand, which matters, because hand-entering data is where fabrication becomes possible without anyone intending it.

The lightest animal in the set is a red-backed salamander at 0.76 g burning 99 microwatts. The heaviest is an Asian elephant at 3,672 kg burning 2,336 W. Between them is a factor of 4.8 million in mass. As a unit check we looked up ourselves: AnAge gives Homo sapiens a mass of 70 kg and a metabolic rate of 82.78 W, which is 1.18 W/kg, or 1,709 kcal per day. That is the textbook resting figure, so the units are what we think they are.

Two fields we deliberately left alone. AnAge carries a body temperature, sparsely and with inconsistent definitions, and temperature is known to move metabolic rate a great deal [2][3]; we did not correct for it. AnAge carries no phylogeny, and we did not supply one. Both omissions come back in §10, where they do real damage.

Two Lines Through One Cloud

Take logarithms and the power law becomes a straight line. Write \(x = \log_{10} M\) in grams and \(y = \log_{10} B\) in watts, and \(B = aM^b\) becomes

The exponent is now a slope, and fitting a slope to a cloud of points is a question with more than one answer. Ordinary least squares minimises the vertical distances. It is the right estimator when all the error lives in \(y\) and none in \(x\), which here would mean that the metabolic rates are uncertain and the body masses are exact. They are not exact. Body masses in a cross-species compilation carry real error, both from measurement and from the question of which individual of a species you happened to weigh.

Reduced major axis regression, also called standardised major axis, answers the question symmetrically. It minimises the sum of the areas of the triangles the points make with the line, and its slope is simply the ratio of the two standard deviations:

That second equality is the whole story of half the dispute, so it deserves to be read slowly. The reduced major axis slope is the least squares slope divided by the correlation coefficient. Since \(|r| \le 1\), the RMA slope is always at least as steep. It is never less steep. And the size of the gap depends on nothing except how much scatter the data happen to have.

Look at the scatter and you can see the trap. Six decades of mass compress everything. A line of slope 0.667 and a line of slope 0.75 diverge by half a decade in metabolic rate over the full span, which on that figure is a few millimetres. The eye cannot referee this. Only the arithmetic can, and the arithmetic depends on which line you asked for.

Now put the two estimators side by side on every group and read the gap.

The last row is the one to stare at. On the same 627 species, the same day, the same file, least squares says 0.7132 and reduced major axis says 0.7883. Both intervals are tight. Both exclude three quarters. They exclude it from opposite sides. A reader handed only the first number would conclude that Kleiber was too high; a reader handed only the second would conclude he was too low; and neither reader would have any way of knowing that the difference between them was a choice made in a line of code.

Why is the pooled gap so much larger than the mammal gap? Because the gap is \(1/|r| - 1\) and nothing else. Pooling ectotherms with endotherms mixes two thermal strategies into one cloud, the scatter roughly triples, \(r^2\) falls from 0.95 to 0.82, and the RMA slope inflates accordingly. The pooled RMA figure of 0.7883 is not a discovery about metabolism. It is a measurement of how badly we mixed our sample, wearing the costume of a scaling exponent.

Which raises a question worth asking backwards. How clean would a dataset have to be for the two methods to agree to within, say, 0.01? At \(b = 0.70\) you would need \(|r| = 0.98592\), which is \(r^2 = 0.972\). To agree to within 0.001 you would need \(r^2 = 0.99715\). No interspecific compilation in biology is that clean, which means the choice of estimator will always matter at the scale of the dispute, forever, for everybody.

None of this is news to the people who work on line-fitting. Warton and colleagues [4] set it out clearly in 2006 and recommended reporting both estimators with their intervals. Smith [5] argued nearly the opposite, that reduced major axis is usually the wrong tool and should not be reached for by default. We report both and take no side, which is the coward's position and also, we think, the correct one for a referee.

The Arithmetic, Done on Paper

Nothing in this section needs a computer. Here are the five sums that the mammal fit is made of, as the script prints them.

Divide the fifth by the fourth and you have the exponent.

Take the square root of the sixth over the fourth and you have the other one.

The correlation is \(r = S_{xy}/\sqrt{S_{xx}S_{yy}} = 0.974920\), and \(0.709107/0.974920 = 0.727349\), which is the identity from §3 confirming itself on real data to six decimal places. The residual sum of squares is \(S_{yy} - b\,S_{xy} = 14.6539\) on 420 degrees of freedom, giving a residual variance of 0.034890 and

Now the two tests, which are one subtraction and one division each.

Multiply the standard error by the critical value 1.9657 and the interval is [0.6936, 0.7246]. Three quarters is outside it. Two thirds is outside it. That is the answer to the question this article was given, at least for mammals fitted this way, and it took six sums and four divisions.

Working Notes From the Club Table

The rejected plan in Meeting 1 was the best decision in the study and it was made for the wrong reason. We dropped it because it was tedious. The right reason is that a hand-typed dataset cannot be checked against its source by a reader, and an unverifiable dataset is worth nothing regardless of how carefully it was typed.

Birds Are Not Mammals, and Rodents Are Not Carnivores

If there were one metabolic exponent, every well-sampled group would find it. Test that directly: fit separate slopes to the four classes, fit one common slope, compare the residual sums of squares. The common slope is 0.7000 with a standard error of 0.0066, which is a pleasingly round number and a fiction. Allowing the slopes to differ improves the fit by \(F_{3,619} = 3.371\), \(p = 0.018\). Restricting to the two classes with real sample sizes sharpens it: \(F_{1,589} = 10.38\), \(p = 0.0013\).

Mammals 0.7091. Birds 0.6549. The difference is 0.0542, with \(t = 4.35\). That single difference between two classes of endotherm is 65% of the entire distance that Kleiber and Rubner's successors have been arguing over.

One level down, inside the mammals, it gets worse.

Eleven mammalian orders, eleven exponents, spread over 0.33 from shrews to opossums. Nine of the eleven intervals contain three quarters. Five contain two thirds. Three contain both, which is the polite statistical way of saying they contain no information. Rodentia, with 155 species and three and a half decades of mass, lands on 0.6666, and if the study had happened to include only rodents we would be publishing an article confirming Rubner.

Soricomorpha is instructive in the other direction. Nineteen shrews and moles across 1.31 decades give an exponent of 0.4772 with an interval nearly half a unit wide. That is not a finding about shrews. It is a finding about what happens when you ask for a slope from a group that barely varies in size, and it is a preview of §8.

Move the Window, Watch the Exponent Walk

Here is the demonstration the club found most persuasive, and the one that changed what we thought the article was about. Take the mammals. Take a window two decades of mass wide. Fit inside it. Slide the window up by a quarter decade and fit again. Repeat.

The exponent walks. It starts at 0.5911, below two thirds, in the window covering 1.8 g to 178 g. It ends at 0.8701, above three quarters, in the window covering 3.2 kg to 316 kg. Every one of those fits uses the same estimator on the same database. The range of answers is 0.2790, which is 3.35 times the entire quantity under dispute.

The nested version is even blunter. Keep only mammals under 100 g and the exponent is 0.5959, interval [0.5125, 0.6794]. Keep only mammals over 316 g and it is 0.7959, interval [0.7636, 0.8282]. Those two intervals do not overlap. They do not come close to overlapping.

This is not a new complaint. Dodds, Rothman and Weitz [10] made it carefully in 2001, and the range-dependence they described is what Figure 3 draws. What surprised us was the size. We expected the exponent to wobble. We did not expect it to traverse the whole disputed interval three times over.

How Many Decades Does It Take to Settle Anything?

Turn the problem into a design question. Suppose you are planning a study and you want it to be able to distinguish 0.75 from 0.6667. Your 95% interval has to be narrower than 0.0833, so its half-width has to be under 0.0417. How much mass range buys that?

We answered it empirically. For each window width from half a decade to six, draw 400 random windows of that width from the mammal mass axis, fit inside each, record the spread of the fitted exponents and the typical interval half-width. Seeded at 20260615 so it repeats.

Read the SD column. At half a decade of mass the fitted exponent has a standard deviation of 0.272 and has produced answers ranging from −0.126 to 1.514. Negative. A sample of 47 real mammals with real respirometry behind them, fitted correctly, can return a negative metabolic scaling exponent, purely because they all weigh about the same. At one decade the spread is 0.144, still comfortably larger than the disputed gap. Only at two and a half decades does the typical interval get narrow enough to be worth reporting.

That single table explains a great deal of the history. A physiologist working on one family of small mammals, or on one clade of birds, is working inside one or two decades of mass. Their exponent is real, their respirometry may be immaculate, and their estimate is still dominated by which animals they happened to be able to catch.

The Uncomfortable Part: It Is Not a Straight Line

By this point the club had stopped believing in the question. Figure 3 does not look like noise around a constant. It looks like a slope that changes with mass. So we did the obvious thing and added a quadratic term to the mammal fit.

The quadratic coefficient is \(c_2 = 0.03646 \pm 0.00544\), \(t = 6.70\), \(p = 6.6 \times 10^{-11}\), \(F_{1,419} = 44.93\). It earns its degree of freedom many times over. And because the local slope of a quadratic is \(c_1 + 2c_2(x - \bar{x})\), the exponent is now a function of where you stand on the mass axis.

At 10 g the local slope is 0.5721. At 1 kg it is 0.7179. At 100 kg it is 0.8637. A mouse and an elephant are not obeying the same exponent, and the single number 0.709 that we spent §4 computing is an average over a curve, weighted by wherever the species happen to sit on the mass axis.

Birds do not do this. The quadratic term for the 171 birds is \(-0.0047 \pm 0.0093\), \(t = -0.50\), \(p = 0.62\), which is nothing at all. Birds are straight and shallow; mammals are curved. That asymmetry is itself a result, and one we did not go looking for.

Kolokotrones and colleagues [9] reported mammalian curvature in 2010 from a much larger and better-curated dataset, with a body-temperature correction we cannot match. Our confirming it from 422 species in a public compilation is worth something as a replication and nothing as a discovery. The reason it matters here is narrower: if the relation is curved, then "is the exponent 3/4 or 2/3" is a question with a false presupposition, and ninety years of answers to it have been answering something else.

The Strongest Objection, Which We Think Is Right

Everything above assumes the 422 mammals are 422 independent observations. They are not, and this is not a technicality. It is the single largest problem with the study, and a careful referee would stop here.

Species share ancestry. Two mouse species resemble each other because they are both mice, and counting them as two independent data points overstates how much you know by roughly the amount by which they are related. Felsenstein [11] made this argument in 1985 and it has been standard ever since. The effect on our numbers is not on the slope so much as on the interval: our standard error of 0.0079 is too small, probably by a considerable factor, and every "excludes both" ruling in this article is written with more confidence than the data can support.

We cannot fix it. We have no phylogeny and no branch lengths, and inventing either would be worse than admitting the gap. What we can do is show what happens when the pseudoreplication is reduced by brute force. Collapse the mammals to one point per order, eighteen points, each the mean of its order in log space. Eighteen orders is closer to eighteen independent observations than 422 species is to 422.

That is the honest answer to the honest objection. Buy back independence and you lose the resolving power to say anything. Published phylogenetic analyses land in between: Capellini, Venditti and Barton [12] recovered an exponent close to three quarters once phylogeny was modelled properly, White and Seymour [13] found values near two thirds with a temperature-corrected and phylogenetically informed treatment, and Sieg and colleagues [14] showed explicitly that the answer depends on the regression model and the correction in about equal measure. Three careful groups, three answers, one law.

Two smaller objections deserve a hearing too. The first is that AnAge is a compilation of compilations and its "metabolic rate" field papers over real differences in what was measured: whether the animal had been fasted, and at what ambient temperature it was held. McNab [15] has argued for years that a substantial part of the scatter in mammalian basal metabolic rate is explainable by diet and climate rather than by mass, and if he is right then our residuals are not noise. The second is that log transformation and least squares assume multiplicative error, an assumption Packard and Birchard [16] have attacked and Kerkhoff and Enquist [17] have defended. We took the log-transform side because that is what the literature we are comparing against did, and because the alternative changes the question rather than answering it.

A Verdict, Delivered Without Enthusiasm

The question was whether the exponent is three quarters, two thirds, or neither. Our answer, on this compilation, with these methods, is: neither, and also both, and the honest version of the sentence is longer than either camp would like.

For mammals fitted by least squares across six decades, 0.7091, rejecting both candidates at better than five standard errors each. For the same mammals fitted by reduced major axis, 0.7273, still rejecting both, and closer to Kleiber. For birds, 0.6549, which is Rubner's number. For rodents, 0.6666, which is Rubner's number to four decimal places. For carnivores and primates, 0.78, which overshoots Kleiber. For a two-decade slice of small mammals, 0.59, which is below anybody's theory. And for eighteen mammalian orders treated as eighteen observations, 0.6838 with an interval so wide it contains every position anyone has ever held.

Set our 0.7091 beside the published values and the same picture appears. Savage and colleagues [24] reported 0.737 from binned mammalian data, which sits 3.53 of our standard errors above us. White and Seymour [13] reported 0.69 from 619 mammals, 2.42 standard errors below us, and their interval also excluded three quarters and two thirds together. Clarke, Rothery and Isaac's 0.70 [3] is within 1.15 standard errors and counts as agreement. Our number is unremarkable, and that is the most reassuring thing about it.

Kleiber was right that metabolism scales sublinearly and that the exponent is well above two thirds in the large-bodied mammals he mostly worked with [6][18]. Rubner's successors were right that small endotherms and birds come in near two thirds. West, Brown and Enquist [19] were right that a transport-network argument predicts something close to what the large mammals do, and their critics [10][20] were right that the data never supported a universal constant to the precision the theory claimed. Everyone involved was looking at a real feature of a curved relation, from a different part of the mass axis, through a different estimator.

Where a different choice would have changed the answer

The estimator. Least squares against reduced major axis is worth 0.018 on the mammals and 0.075 on the pooled set. On the pooled set it changes which side of three quarters the answer falls. If we had reported RMA alone, this article would have been titled "Metabolism Scales Steeper Than Three Quarters" and every number in it would have been correct.

The taxonomic scope. Mammals only gives 0.709. Birds only gives 0.655. Everything pooled gives 0.713 by OLS. Rodents only gives 0.667. We chose mammals as the headline because that is what the dispute has historically been about, and that choice is worth up to 0.05.

The mass range. Worth up to 0.28, which dwarfs everything else. Had we restricted to animals under 100 g, as a small-mammal physiologist naturally would, we would have found 0.596 and reported that Rubner was closer than anyone thought.

The functional form. Allowing curvature dissolves the question rather than answering it. We report the straight-line fit first because that is the fit the dispute is conducted in, and the curved fit second because it is the better model. A reader who prefers the curve should read §9 as the real result and §4 as a historical exercise.

The unit of analysis. Species against order. Worth a factor of 6.2 on the interval width, which is to say worth the entire conclusion.

What this study cannot tell you

It says nothing about invertebrates and nothing about unicellular life, both of which have their own literatures and their own exponents [21][22]. It says nothing about intraspecific scaling, because AnAge holds one number per species, and Heusner's old argument [23] that the interspecific exponent is an artefact of pooling intraspecific relationships cannot be tested here at all. It says nothing about mechanism. A fitted exponent is a description, and no amount of care in fitting it turns it into an explanation of why the number is what it is.

Reproducing this

One file and one command. Python 3.12 and numpy, and nothing else is imported.

python allometry-scaling.py > allometry-scaling-output.txt

Ours ran in 16.0 seconds on a laptop, on numpy 2.4.2 and Python 3.12.3. The data file must be present at analysis/data/anage_data.txt; if it is missing the script says so and stops rather than proceeding without it. The seed 20260614 governs the bootstrap and the random window sampling and nothing else, so the fitted exponents, the intervals and the hypothesis tests are all seed-independent and will reproduce exactly. The script prints the SHA-256 of the cached archive (e3ddb66e…3018d630) and of the extracted table (98867969…aa1c1519) so you can confirm you are fitting the same bytes. Numbers that differ from ours by more than the printed standard errors mean something is wrong and we would like to hear about it. The interactive model runs the same two estimators and the same window experiment in your browser, on the same 627 species.

References