Exploring Goodness-of-Fit and Spatial Correlation Using Components of Tango's Index of Spatial Clustering

The ability to detect anomalies such as spatial clustering in data sets plays an important role in spatial data analysis, leading to interest in test statistics identifying patterns exhibiting significant levels of clustering. Toward this end, Tango (1995) proposed a statistic (and its associated distribution under a null hypothesis of no clustering) assessing overall patterns of spatial clustering in a set of observed regional counts. Rogerson (1999) observed that Tango's index may be decomposed into the summation of two distinct statistics, the first mirroring standard tests of goodness-of-fit (GOF), and the second an index of spatial association (SA) similar to Moran's I . In this article, we investigate the effectiveness of Rogerson's expression of Tango's statistic in separating GOF from SA in data sets containing clusters. We simulate data under the null hypothesis of no clustering as well as two alternative hypotheses. The first alternative hypothesis induces a poor fit from the null hypothesis while maintaining independent observations and the second alternative hypothesis induces spatial dependence while maintaining fit. Using Rogerson's decomposition and leukemia incidence data from upstate New York, we show graphically that one is unable to statistically distinguish poor fit from autocorrelation.