Weighted averaging, logistic regression and the Gaussian response model

The indicator value and ecological amplitude of a species with respect to a quantitative environmental variable can be estimated from data on species occurrence and environment. A simple weighted averaging (WA) method for estimating these parameters is compared by simulation with the more elaborate method of Gaussian logistic regression (GLR), a form of the generalized linear model which fits a Gaussian-like species response curve to presence-absence data. The indicator value and the ecological amplitude are expressed by two parameters of this curve, termed the optimum and the tolerance, respectively. When a species is rare and has a narrow ecological amplitude — or when the distribution of quadrats along the environmental variable is reasonably even over the species' range, and the number of quadrats is small — then WA is shown to approach GLR in efficiency. Otherwise WA may give misleading results. GLR is therefore preferred as a practical method for summarizing species' distributions along environmental gradients. Formulas are given to calculate species optima and tolerances (with their standard errors), and a confidence interval for the optimum from the GLR output of standard statistical packages.