Isometric and allometric scaling relationships for raw values, logged values, and proportions of both types of values.

Both x and y are of the same dimensionality, as in a brain mass ~ body mass relationship. In an isometric relationship, the scaling exponent does not change when raw values in arithmetic space (A) are plotted compared to raw proportions in arithmetic space (B). When data are log-transformed (C), the scaling exponent becomes the slope of the line. The same is true when proportions are plotted in log-space (D). For raw data (E) or raw proportions (F), an allometric relationship can be recognized by a scaling exponent that is not equal to 1 (again assuming equivalent dimensionality of x and y). For logged data (G) or logged proportions (H), allometry is characterized by a slope not equal to 1. In all iterations, isometric relationships remain isometric and allometric relationships remain allometric. There is no “collapse” of allometry into isometry as suggested by Smaers et al. [6]. The authors either misidentify (A) as an allometric relationship, or do not appropriately convert logged data (C) into logged proportions (D).