Using nonlinear functional relationship regression to fit fisheries models

Abstract

Functional relationship regression refers to that class of statistical model where a functional relationship is assumed to exist between two arithmetic variables, but the two arithmetic variables can only be viewed with measurement error and (or) natural variability. The goal is to estimate the underlying functional relationship when observing only the variables containing error or natural variability. While statistical details need to be worked out, some general approaches to this class of problem can be recommended. First of all, nonlinear least squares can provide maximum likelihood estimates when the error variance ratio, λ;, is assumed known. Furthermore, the usual estimates of standard errors from nonlinear least squares, while theoretically flawed for this class of model, appear to provide usable estimates for many practical problems. Next, a K-sample F test, for testing the equality of nonlinear functional relationship regression curves, is proposed. Finally, computer memory-saving algorithms are suggested for situations where sample sizes are large. Methods proposed here are applied to three types of functional curves commonly estimated in fisheries biology: a stock-recruitment curve, allometry, and the von Bertalanffy growth curve.