Nonparametric intensity bounds for the detection and delineation of spatial clusters

Abstract

There is considerable uncertainty in the disease rate estimation for aggregated area maps, especially for small population areas. As a consequence the delineation of local clustering is subject to substantial variation. Consider the most likely disease cluster produced by any given method, like SaTScan Kulldorff [2006], for the detection and inference of spatial clusters in a map divided into areas; if this cluster is found to be statistically signifcant, what could be said of the external areas adjacent to the cluster? Do we have enough information to exclude them from a health program of prevention? Do all the areas inside the cluster have the same importance from a practitioner perspective? We propose a criterion to measure the plausibility of each area being part of a possible localized anomaly in the map. In this work we assess the problem of finding error bounds for the delineation of spatial clusters in maps of areas with known populations and observed number of cases. A given map with the vector of real data (the number of observed cases for each area) shall be considered as just one of the possible realizations of the random variable vector with an unknown expected number of cases. In our methodology we perform m Monte Carlo replications: we consider that the simulated number of cases for each area is the realization of a random variable with average equal to the observed number of cases of the original map. Then the most likely cluster for each replicated map is detected and the corresponding m likelihood values obtained by means of the m replications are ranked. For each area, we determine the maximum likelihood value obtained among the most likely clusters containing that area. Thus, we construct the intensity function associated to each area's ranking of its respective likelihood value among the m obtained values. The method is tested in numerical simulations and applied for three different real data maps for sharply and diffusely delineated clusters. The intensity bounds found by the method re ect the geographic dispersion of the detected clusters. The proposed technique is able to detect irregularly shaped and multiple clusters, making use of simple tools like the circular scan. Intensity bounds for the delineation of spatial clusters are obtained and indicate the plausibility of each area belonging to the real cluster. This tool employs simple mathematical concepts and interpreting the intensity function is very intuitive in terms of the importance of each area in delineating the possible anomalies of the map of rates. The Monte Carlo simulation requires an effort similar to the circular scan algorithm, and therefore it is quite fast. We hope that this tool should be useful in public health decision making of which areas should be prioritized.