Moran’s Autocorrelation Coefficient in Comparative Methods

This document clarifies the use of Moran’s autocorrelation coefficient to quantify whether the distribution of a trait among a set of species is affected or not by their phylogenetic relationships. 1 Theoretical Background Moran’s autocorrelation coefficient (often denoted as I) is an extension of Pear-son product-moment correlation coefficient to a univariate series [2, 5]. Recall that Pearson’s correlation (denoted as ρ) between two variables x and y both of length n is: ρ = n∑ i=1 (xi − x̄)(yi − ȳ)[ n∑ i=1 (xi − x̄)2 n∑ i=1 (yi − ȳ)2]1/2, where x ̄ and y ̄ are the sample means of both variables. ρ measures whether, on average, xi and yi are associated. For a single variable, say x, I will measure whether xi and xj, with i 6 = j, are associated. Note that with ρ, xi and xj are not associated since the pairs (xi, yi) are assumed to be independent of each other. In the study of spatial patterns and processes, we may logically expect that close observations are more likely to be similar than those far apart. It is usual to associate a weight to each pair (xi, xj) which quantifies this [3]. In its simplest form, these weights will take values 1 for close neighbours, and 0 otherwise. We also set wii = 0. These weights are sometimes referred to as a neighbouring function. I’s formula is