Relative-Growth Law with a Threshold

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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- <u-i) VÎ^ 0 ^ 4+ft«-,) v£; ???Oxa Confidence limits on the regression line may then be obtained using the following equations: Lwbxx J Mu = exp {? + ß? + tin_W[V(y)]} ; ML = exp {A + ß? - hn-WlV(y)]} . The expressions forMj/ andMz, can be used to construct the upper and lower boundaries, respectively, on the confidence region for the regression line. The discussion up to this point has been based on the hypothesis that the growth of a mass commences at a time t equal to zero. However, some instances exist where the relative-growth law could apply when the production ofa mass commences at a time greater than zero. For example, the production oftotal egg mass in fishes or frogs should be proportional to the size of the species but does not start until the female has reached a degree ofdevelopment proportional to size or age (time). The mass ofeggs produced may then commence to increase from zero 422 George M. Angleton and David Pettus · Growth Law with Threshold Perspectives in Biology and Medicine · Spring 1966 according to the relative-growth law after this threshold is reached. Now, the differential equation becomes dM/M = ßdt/(t — r„), where t0 is the threshold value. The solution ofthis differential equation is M = a(t — t0)e. For a given value off0 estimates of a, ß, and M may be made using the same techniques as before. However, now X{ = In (t¡ — í0). The maximum-likelihood estimates of f0, a, ß, and M may be obtained using an iteration of the above procedure whereby the estimate of t0 is the one for which the following expression is a minimum: SSD = S [In Mi - In ? - ß In (U - h)]2 . The construction ofconfidence limits for the parameters and the regression relationship cannot be performed in the exact manner as when i„ is known. However, conditional confidence limits can be constructed that approach the true limits when (( — t0) is large. In the previous computational formulas the quantity (n — 2) is now replaced by (n — 3). The computations for performing the analysis are relatively simple for the special case of t0 equal to zero, the form of the relative-growth law presented first. However, when i„ must be estimated, the iterative calculations become tedious and are best done by a computer. A Fortran II program for use on an IBM 1620 computer was developed to perform these computations. The output of this first program is the input for a second Fortran II program which controls the plotting ofdata and construction of graphic and regression relationships on a Calcomp plotter attached to the computer. An initial evaluation ofthe plausibility ofthe model was performed in an attempt to describe the dependence oftotal egg mass on body length in chorus frogs, Pseudacria triseriata. The assumption was made that these frogs must reach a certain stage ofdevelopment which would be proportional to their body length before egg production would be observed. Then, after this threshold is reached, the postulates ofthe relative-growth law would apply. The maximum-likelihood estimates of the parameters of the model were calculated using the previous formulas and are tabulated in Table 1. The data, the estimated regression line (solid line), and the conditional 95 per cent confidence limits (broken line) on the regression line are given 423 in Figure ?. It is a copy ofa graph prepared by the plotter under program control on the computer. Since the variance in these data is rather large, a critical evaluation ofthe plausibility ofthe model cannot be performed. 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