Statistical Properties of Alternative Parameterizations of the von Bertalanffy Growth Curve

Abstract

The von Bertalanffy growth curve is often used in fisheries research to describe the relationship between the weight or length of a fish and its age. The equation is also encountered in various other branches of science and applied science in a variety of different parameterizations and names; for example, it is also known as the asymptotic regression equation or the three-parameter exponential equation. Since these equations are all nonlinear regression models, the properties of the least squares estimators of the parameters of these models may be very different from their large-sample properties, where the estimators are unbiased, have the minimum attainable variance, and are normally distributed, the conditions that pertain in a linear model. Different parameterizations will have estimators which approximate the asymptotic properties to varying degrees of closeness. My study of eight parameterizations shows that one of them, a generalization which allows unequal age increments of a parameterization proposed by Schnute and Fournier, is far superior to any of the other models, which include the most commonly used parameterization, in that it exhibits close-to-linear behavior. Two of the three parameters in this model represent the expected mean lengths corresponding to the youngest and oldest ages, respectively, in the sample, and thus have a ready biological interpretation. I discuss why it is important to have a close-to-linear model when one wishes to make comparisons between two or more data sets. Methodology is briefly described for carrying out such comparisons, and some further remarks are made about why biologists should be concerned about the statistical properties of the models that they use. Although most data sets I used for illustration are obtained from marine animals, the conclusions are general and apply to all disciplines which make use of the von Bertalanffy model in whichever guise or form it appears.