The outer border and central border for species–environmental relationships estimated by non-parametric generalised additive models

In modelling species–environmental relationships, non-parametric generalised additive modelling (GAM) regression has become a popular approach, as it has been shown that the generalised linear model (GLM) approach includes a bias that may be biologically unrealistic. However, the more versatile GAM approach has the disadvantage that a simple numerical description of the estimated response and estimation of the range (2×tolerance) and tolerance (t=σ, sigma) are not possible. To try to overcome this, I introduce a procedure that estimates the outer border and central border numerically for non-parametric functions. The outer and central borders are given by the distance from the optimum that is needed for the response curve to drop a certain amount, i.e. these parameters represent the rate of decrease in the response in both directions from the optimum independent of each other. Thus, the species with a narrow niche will show a faster drop (shorter distance from the optimum) than the species with a broader niche. The outer and the central borders are defined as the predictor values which take the response value equal to a specified fraction of the maximum response: max(E(y∣x))×exp(−2) for the outer border and max(E(y∣x))×exp(−1/2) for the expected central border. These parameters are directly related to the estimated response values rather than, as both range and tolerance, related to a polynomial function defining the response curve. For logistic regressions there are increasing number of positive observations outside the classic range as the maximum response increases. In this, the expected borders differ from the range, as the interpretation of the former is independent of the maximum response. By reporting the lower and upper outer border and the central border combined with the optimum for the species, a simple unbiased numerical description of the species–environment response is obtained, which can be used in comparative studies.