A framework for statistical inferential decisions in spatial pattern analysis

Introduction The continuing success and proliferation of geographical information science (GIS) is, in part, a recognition that pattern matters. Three of the recently posed ten 'big questions of geography' address spatial pattern in some form (Cutter et al. 2002). Spatial patterns are defined by the variability and arrangement of characteristics of phenomena in a geographic space (Cliff et al. 1975). They are the realisations of processes that operate over the geographic space (Getis and Boots 1978; O'Sullivan and Unwin 2002). What we observe (the data values) is, in turn, the result of these processes and out data collection systems. As the mapping and visualisation capabilities of the technology of GIS have become increasingly sophisticated, the desire of practitioners to determine the importance or significance of the spatial patterns that are revealed has burgeoned. Until relatively recently, this desire was frustrated by two sets of forces. The first was a lack of readily accessible and easily used software for undertaking spatial statistical analysis (Getis 2000). The second was the need to be an 'expert' (or collaborate with an expert) in the use of such analysis. Neither force is now a serious impediment. A host of software, which either works seamlessly in commercial GIS or is completely self-contained, is now accessible through organisations such as the Centre for Spatially Integrated Social Science (http://www. csiss.org/clearinghouse/). Pedagogic review papers aimed at de-mystifying analytical techniques and providing advice on which procedures to select are increasingly appearing. Typical examples of this genre are Perry et al. (2002) and Wiegand and Moloney (2004). However, before selecting specific software and techniques for spatial pattern analysis, it is important that practitioners recognise that all of these techniques operate within the more general context of spatial statistical inference. There are a number of issues involved here and we feel that an understanding of the nature of hypothesis testing and statistical inference for spatial data should be a requirement of good practice. Currently, there is a paucity of such material directed at GIS practitioners, and what there is conveys contradictory messages. For example, while Longley et al. (2001) suggest that statistical inference is quite constrained, Fotheringham and Brunsdon (2004) argue for a more liberal role. Interestingly, although much of the theory relating to spatial statistical inference was developed when GIS was only just emerging (Getis and Boots 1978; Cliff and Ord 1981; Unwin 1981), we feel that much of it remains fundamentally sound. In this article, we construct a spatial statistical framework, presented in the form of a decision tree, which is envisioned to guide practitioners in identifying the relevant inferential issues and selecting 'appropriate' answers to their questions relating to spatial patterns. As a clarification of this objective, it is important to emphasise that the focus of this framework is on the stochastic components of spatial pattern. There are numerous questions regarding spatial data that are not (or not necessarily) influenced by these components (e.g., 'Where is Budapest? ', 'Which way is the nearest hotel? ', 'When was the last snowfall in Idaho? ', 'Is this soil suitable for growing wheat? '), and, consequently, these are not part of this decision tree. Further, given the general nature of the framework and its intended role, and the practically infinite set of operations that can be performed on spatially referenced data, we deliberately avoid giving specific detailed illustrations. For readers interested in such specifics, one of the current textbooks on spatial data analysis, such as Fotheringham et al. (2000) or Haining (2003), provides an ideal starting point. The Decision Tree Prior to describing the decision tree, we need some technicalities for expressing and formalising some of the issues at hand. …