What are the units of the parameters in the power function for thelength-weight relationship?

摘要

One of the most widely used mathematical models in science Is the power function y = ax(b) with variables x and y and constants a and b. For example, such a model characterizes the relationship between fish length and weight. Yet, no units have been assigned to either a or b, partly because of a claim that b has a dimension. If b has a unit, almost every piece of work on allometric analysis in the biological sciences will be of limited value, with little hope of finding invariance in nature. Here, I give not only the correct units for a and b in the power function but also units for constants in three more general functions, thereby providing a basis for unit conversion between these constants in many sciences. In y = ax(b), a has a unit of yx(-b); and b is a dimensionless constant. In y = a Pi(i=1)(n)x(i)(bi),with variables x(i) and y and constants a and b(i), i = 1,2, ..., n, a has a unit of y(Pi(i=1)(n)x(i)(-bi)); and b(i), i = 1,2, ..., n, is a dimensionless constant. In y = c + a(x - d)(b), with variables x and y and constants a, b, c and d, a has a unit of yx(-b); b is a dimensionless constant; c has the same unit as y; and d has the same unit as x. In y = c + a Pi(i=1)(n)(x(i )- d(i))(bi), with variables x(i) and y and constants a, b(i), c and d(i), i = 1,2, ..., n, a has a unit of y(Pi(i=1)(n)(x(i) - d(i))(-bi)); b(i) is a dimensionless constant; c has the same unit as y; and d(i) has the same unit as x(i).