Tests of 'Randomness' for Spatial Point Patterns

SUMMARY Tests of "randomness" and methods of edge-correction for spatial point patterns are surveyed. The asymptotic distribution theory and power of tests based on the nearest-neighbour distances and estimates of the variance function are investigated. A MAP of small objects is often described as "random" if it is consistent with the null hypothesis of a binomial or Poisson process. The usual first step in the analysis of such a pattern is a test of this null hypothesis; indeed the analysis is often confined to quoting a test statistic or its significance level as a "measure of non-randomness". The aim of this paper is to investigate the power of such tests, particularly tests based on nearest-neighbour distances, interpoint distances and estimators of moment measures, and to assess the efficiency of various corrections for edge-effects. One interesting conclusion is that edge-correction such as applied in the k of Ripley (1977) can substantially reduce the sampling fluctuations of a statistic and so boost the power of a test based on it.