INTERACTIVE COMPANION · VOLUME 2, ISSUE 4 · SUMMER 2026

The Referee's Bench

A live model accompanying “Does Metabolism Really Scale to the Three Quarter Power”

Two things move the answer to a ninety-year argument: which formula you fit with, and which animals you let into the room. Below is the same regression the article runs, rebuilt so you can drive it. Nothing here is a new dataset. It is the club's own fitted lines redrawn live from the same 627-species compilation, so every number you see matches a number already printed in the article.

Model 1. Two lines through the same cloud

Ordinary least squares assumes all the scatter sits in the metabolic-rate axis and finds the line that minimises vertical distance. Reduced major axis assumes both axes carry error and finds the line that minimises triangle area instead. The two slopes are related by one exact identity, \(b_{\text{RMA}} = b_{\text{OLS}} / |r|\), which is why RMA is always the steeper of the two.

Group Switching groups is the article's second axis of disagreement: the exponent is not one number, it is a family of numbers that depends who you let vote. Fitting method Mass window, log₁₀ grams full range Drag either handle to restrict the sample by mass. Narrowing the window is the manoeuvre the historical critics used, and it moves the fitted exponent further than switching method does.

n in window: - b (OLS): - b (RMA): - r²: - 3/4 excluded: - 2/3 excluded: -

Model 2. How far the identity carries you

Because \(b_{\text{RMA}} = b_{\text{OLS}}/|r|\) exactly, the gap between the two fitted lines is not a second measurement. It is arithmetic on the first one. A looser cloud (\(r^2\) closer to 0) inflates the RMA slope relative to OLS by more than a tight cloud does, regardless of what the true exponent is. This model lets you see that inflation happen without any real data at all, so you can tell how much of the OLS-versus-RMA argument in the literature is really an argument about method rather than about biology.

True OLS slope b 0.75 Scatter, r² 0.85 The mammal fit in the article sits near r² = 0.85. Drag this down toward the amphibian value near 0.3 and watch the RMA line pull far away from OLS on identical underlying biology.

b (RMA) = b (OLS) / |r|: - gap: - gap %: -

What the two models say together

Model 1 uses the real, fitted numbers from the article's own compilation. Model 2 strips those numbers away and asks a purely mathematical question: given only a slope and a scatter, how big must the OLS-RMA gap be. When the two agree, as they do for the mammal fit, the gap you see in Model 1 is exactly what Model 2 predicts from \(r^2\) alone. That is the article's central point in a form you can turn a dial on: two labs staring at identical biology can report meaningfully different exponents for a reason that has nothing to do with which animals are shrews and which are elephants.