EXPECTED VALUE AND VARIANCE OF MORAN BIVARIATE SPATIAL AUTOCORRELATION STATISTIC FOR A PERMUTATION TEST

Moran's I statistic has been used by ecologists and geographers alike to test for the presence of spatial autocorrelation in a single variable over a two-dimensional plane. In this paper, we provide the derivation of the expected value and variance of a bivariate version of Moran's I for use with multivariate data under the assumption of spatial independence. We also demonstrate that Moran's univariate I statistic is a special case of Moran's bivariate I(YZ). Results of an extensive Monte Carlo study show that the expected value and variance are reliable for several data sets with moderate sample sizes (n = 40 and 127) and varying degrees of correlation among different bivariate surfaces. For small sample sizes (n < 40) at least 5,000 simulations should be used to estimate the P-value to test the null hypothesis of no spatial autocorrelation between two variables.