On Growth Parameter Estimation for Early Life Stages

Abstract

The mathematical model that has most closely approximated observed growth has been the asymmetric sigmoid curve. Historically, this curve was given in the Gompertz form (Brody [1945]; Gompertz [1825]; Thompson [1948]; Weymouth and McMillin [1930]). The estimation of the parameters of such a curve so far has defied traditional statistical approaches, e.g. maximum likelihood. One method was developed by me [1960], but applies only when a set of many measurements with known time between each is available for a single individual or population. Under other circumstances, e.g. capture-recapture data, this technique cannot be used. To simplify estimation, simpler models are often used, e.g. a half-parabola or a symmetric sigmoid curve (cf. Brody [1945]; Rao [1958]; et at.). More recently, simple forms of the von Bertalanify curve (Beverton and Holt [1957]; von Bertalanffy [1938; 1949]) have given good results. Such a simplification is useful for comparing differential growth under varying environmental circumstances, but is inappropriate when the purpose is to describe the complete growth pattern. Walford [1946] presents a clever transformation which permits estimation, under usual assumptions (violated only in cases of pathologic growth patterns), of the asymptotic maximum of an organism's (or population's) growth, coupled with a mechanical means ofdrawing the upper part of the growth curve. Walford's technique, however, provides no information on the inflection point location, on the lower and central parts of the curve, or on the age at any time (i.e. on the time of birth). This paper extends Walford's method so that estimates of the birth and inflection points as well as the asymptotic maximum are obtained. From these extremers, the parameters of growth curves are estimated, and Walford's mechanical means of drawing the upper part of curves is extended to entire curves.