The above graphs are from Regression and Other Stories, as a demonstration of the log-log transformation, but there are some questions of where this 0.74 slope comes from. Simple geometry would suggest a slope of 2/3 (if animals are spheres of constant temperature, they will radiate heat in proportion to their surface area), but there’s this idea that larger animals are more sphere-like and run colder, compared to smaller animals.

Dodds, Rothman, and Weitz look into some of this in a paper from 2001, “Re-examination of the 3/4-law of metabolism”:

I don’t like the whole “null hypothesis” thing at all–but the data and models are interesting, and overall I like the paper. It would be an interesting example for someone to go back and analyze the data more directly using hierarchical modeling.

Related is this paper from 2004 by Brown, Gillooly, Allen, Savage, and West:

Metabolism provides a basis for using first principles of physics, chemistry, and biology to link the biology of individual organisms to the ecology of populations, communities, and ecosystems. Metabolic rate, the rate at which organisms take up, transform, and expend energy and materials, is the most fundamental biological rate. We have developed a quantitative theory for how metabolic rate varies with body size and temperature. Metabolic theory predicts how metabolic rate, by setting the rates of resource uptake from the environment and resource allocation to survival, growth, and reproduction, controls ecological processes at all levels of organization from individuals to the biosphere. Examples include: (1) life history attributes, including development rate, mortality rate, age at maturity, life span, and population growth rate; (2) population interactions, including carrying capacity, rates of competition and predation, and patterns of species diversity; and (3) ecosystem processes, including rates of biomass production and respiration and patterns of trophic dynamics.

They talk a lot about that 3/4 power too:

It’s an interesting topic!