A decomposition of Moran's I for clustering detection

The test statistics I_h, I_c, and I_n are derived by decomposing the numerator of the Moran's I test for high-value clustering, low-value clustering, and negative autocorrelation, respectively. Formulae to compute the means and variances of these test statistics are derived under a random permutation test scheme, and the p-values of the test statistics are computed by asymptotic normality. A set of simulations shows that test statistic I_h is likely to be significant only for high-value clustering, test statistic I_c is likely to be significant only for low-value clustering, and test statistic I_n is likely to be significant only for negatively correlated spatial structures. These test statistics were used to reexamine spatial distributions of sudden infant death syndrome in North Carolina and the pH values of streams in the Great Smoky Mountains. In both analyses, low-value clustering and high-value clustering were shown to exit simultaneously.