Mahalanobis distances and ecological niche modelling: correcting a chi-squared probability error

The Mahalanobis distance is a statistical technique that can be used to measure how distant a point is from the centre of a multivariate normal distribution. By measuring Mahalanobis distances in environmental space ecologists have also used the technique to model: ecological niches, habitat suitability, species distributions, and resource selection functions. Unfortunately, the original description of the Mahalanobis distance technique for ecological modelling contained an error describing how Mahalanobis distances could be converted into probabilities using a chi-squared distribution. This error has been repeated in the literature, and is present in popular modelling software. In the hope of correcting this error to maximise the potential application of the Mahalanobis distance technique within the ecological modelling community, I explain how Mahalanobis distances are calculated, and through a virtual ecology experiment demonstrate how to correctly produce probabilities and discuss the implications of the error for previous Mahalanobis distance studies.