The analysis of spatial association on a regular lattice by join-count statistics without the assumption of first-order homogeneity

One of the widely used classical pieces of spatial statistics for the assessment of spatial association of nominal data, such as colors on a map, is the join-count statistic (JCS). Its application assumes first-order 'homogeneity, that is, the probability of colors is assumed to be uniform across the map. With recent developments in spatial analysis, particularly in remote sensing and landscape ecology, JCS and related measures are frequently applied in cases when this assumption is violated and can produce misleading conclusions. We present a new method with formulas and algorithms implemented in S-PLUS for handling first-order heterogeneity on a regular lattice. Based on the probability distribution of colors at each location (cell or pixel), we compute the expected value and variance of same-color neighbors. Using a stochastic simulation experiment we also confirm that the asymptotic Gaussian approximation holds. Environmental assessment and mapping application examples illustrate the impact of spatially heterogeneous probabilities of nominal variables on significance testing of their spatial association. (C) 2002 Elsevier Science Ltd. All rights reserved.