Relative-Growth Law with a Threshold

RELATIVE-GROWTH LAW WITH A THRESHOLD GEORGE M. ANGLETON* and DAVID PETTUSf Even before Huxley [i] formulated the relative-growth law, it was undoubtedly a matter of common observation that the relative growth of individuals and populations was frequently proportional to the relative time over which such measurements were made. The growth law is a solution to the differential equation dM/dt= ßM/t. The variables Mand (represent mass and time, respectively; ß is theproportionality constant which characterizes the mass growth being described by the law. The equation may alternately be put in the form dM/M= ßdt/t; that is, the relative-growth law states that the relative change in the mass is proportional to the relative change in time. The relative-growth law, and thus the solution to this differential equation, is M = at0, a being the value ofM at a time t equal to one. According to this model, the mass at time t equal to zero also has to be zero. The estimation ofthe two parameters in the model is facilitated ifit is assumed that the observations have a log-normal distribution [2]. This type ofvariation is frequently observed in biological studies; namely, the variations ofthe observations are proportional to their mean or expected value. When this occurs, maximum-likelihood estimates ofthe parameters can be obtained, and tests ofhypotheses can be conducted using analysisof -variance techniques. Estimates ofthe parameters may be obtained using the following computational formulas where ? is the number of observations,M¿ is the observed mass ofthe ith observation corresponding to the z'th value oft. * Departments ofRadiology and Radiation Biology, Mathematics and Statistics, Colorado State University, Fort Collins. I Department ofZoology, Colorado State University, Fort Collins. 42I Xi=In(U); Fi = In(Mi). ? = ^X' . ? — 5_í_i ? ' ? Sxv = 2XiYi-nXY. Sxx = S?] - nX* . S ß = estimate of ß = -^ . O ?? A = Y — ß?; à = estímate of a = exp( A ) . The computational formulas for obtaining the confidence limits on the estimates ofthe parameters are as follows: „ S(G?-?-;3?;)2 c¿ rr ----------------------------------- ¦ (n-2) ß-