General saddlepoint approximations to the null distributions of Moran's I-type measures of spatial autocorrelation

Spatial autocorrelation is the tendency for neighboring observations in a spatial setting to display more or less similarity than expected under a hypothesis of independent observations. Moran's I is widely used as an index of spatial autocorrelation with a reference distribution that is asymptotically normal based on one of two assumptions: the observed data are realizations of independent identically distributed normal random variables; or the observed data are realizations of independent random variables from the same (possibly unknown) distribution, so that all random permutations of the data are equally probable. When either null assumption is met, asymptotic normality can still fail due to either inadequate sample size or non-standard neighborhood structures (e.g. star-shaped networks). Alternatively, asymptotic normality may fail under conditions where the null assumptions are not tenable, such as heterogeneous data from the same family of distributions (e.g. heterogeneous Poisson data common to disease mapping applications).; We propose a method that approximates the exact distribution of Moran's I and Moran's I-type indices, that accommodates all of the aforementioned scenarios. We base our approximation on general saddlepoint methods using generalized cumulants. We demonstrate the performance of this method for both independent identically distributed binary and independent heterogeneous Poisson data under varying parametric and spatial settings. We then use the general saddlepoint approximation to test for significant positive spatial autocorrelation in a disease mapping application based on the Glasgow mortality data. We demonstrate the accuracy of our method by comparing the general saddlepoint approximate reference distribution to that obtained via Monte Carlo simulation. We also compare the relative performance of our estimate with the reference distributions of other similar indices of spatial autocorrelation. We show that our method offers reliable upper tail significance tests that are at worst mildly conservative. We conclude that the general saddlepoint method provides a highly accurate approximation to the null distribution of Moran's I and Moran's I-type measures.