Interpretation of the Coefficient in the Allometric Equation

Interpretation of the coefficient, b, in the allometric equation, y = b xk, is much disputed, and many presume it devoid of biological meaning. Difficulties impeding the interpretation of b are resolved: 1) biological significance cannot be dismissed on dimensional grounds; 2) the interval over which the measurements are valid must be stated as part of the equation; 3) b considered as the y-value at x = 1, in general has no biological meaning; 4) relations between b and k in a family of allometric equations have no biological significance. Examples which illustrate the biological meaning of b are presented in three categories: ontogenetic examples; b as a taxonomic indicator; phylogenetic and evolutionary examples. For a general interpretation of b in y = b xk where k is held constant, b is expressed in terms of a linear scale factor "s" which denotes relative magnitudes between corresponding sets of data having the same dimensionless properties (y/x, dy/dx). b can thus be interpreted as a factor dependent upon scale; and the allometric equation re-written y = s1-k xk, xi ≤ x ≤ xj, k ≠ 1. It is concluded that b is not only biologically interpretable, but that expressed in terms of s, it offers a means of quantitatively considering similarity of biologic forms which exists when corresponding measurements are in the same scale ratio (s-ratio), and when corresponding dimensionless numbers are equal.