Natural variability and the estimation of empirical relationships: a reassessment of regression methods

Abstract

Ecologists often rely on empirically defined statistical relationships to infer how variables might be related. However, the usual method of estimating such relationships (ordinary least-squares (OLS)) is generally inappropriate because of the substantial natural variability of most ecological variables. Natural error variability in the regressor variable can artificially create a significant empirical trend where no underlying or structural relationship exists, or fail to reveal a true structural relationship. In multivariate relationships, natural variability in one variable can induce statistical significance in collinear variables even if they bear no structural relationship. We propose a simple new method, based on instrumental variables, to detect and quantify natural error variability in the regressor variables and to estimate the parameters of the structural relationship. We apply this method to two examples: (1) we show that the structural relationship between adenosine triphosphate concentration (total planktonic biomass) and chlorophyll concentration (autotrophic biomass) does not vary latitudinally in the Southern Ocean despite a significant increase in the OLS slope relating the two at more southerly stations and (2) we demonstrate that the significance of nitrogen in nutrient–chlorophyll relationships in lakes probably reflects natural variability in phosphorus concentration, and not the fertilizing effect of nitrogen.