On the Interpretation of the Normalization Constant in the Scaling Equation - Niklas, Karl J.; Hammond, Sean T. (Frontiers in Ecology and Evolution, 2019)
The scaling equation, Y1 = Y2, has been used empirically and explored theoretically primarily to determine the numerical value and meaning of the scaling exponent, . The mathematical interpretation of  is clear –– it is the quotient of the relative rate of change of Y1 with respect to the rate of change of Y2. In contrast, the interpretation of the normalization constant, , is obscure, so much so that some workers have rejected the idea that it has any biological importance. With the notable exception of Steven J. Gould’s early work, Huxley’s dismissal of  largely relegated the study of its biological role to that of an academic afterthought. Here, we attempt to clarify the meaning of  by using examples from plant biology to illustrate the four primary difficulties that have obscured its importance: (1) the consistency of the units and the metrics of measurement (e.g., meters and body length, respectively), (2) the relationship between  and , (3) the interpretation of scaling equations, and (4) detecting if the numerical value of  has changed and if the change is biologically meaningful. Using examples, we show that  is biologically interpretable and offers a way to quantitatively consider similarities of biological form if (1) it is expressed in terms of the relative magnitudes of Y1 or Y2 for corresponding data points in a set of Y1 = Y2 equations, (2) the units of measurements are in the same scale, and (3) the corresponding dimensionless numbers are established based on the same units of measurement. We provide examples of where the numerical value of or differences in the values of are important, and we propose a research agenda examining the meaning of  values in terms of trait-based ecology.