Clarifications on local and global data analysis

Jetz et al. (2004) have provided an interesting and thought provoking commentary on my earlier paper (Foody, 2004). Although their standpoint and interpretation is quite reasonable their commentary does not fully reflect either my own viewpoint or the content and intention of my paper. I am therefore taking this opportunity to highlight a few key issues in the hope that this will help to clarify some important features of geographically weighted regression (GWR) and its use in ecology. My earlier paper explicitly set out to explore spatial nonstationarity in the relationship between species richness and a set of determinants (e.g. see the statements made in the 'aim' section of the abstract and at the end of the introductory section). The implications of spatial nonstationarity, with particular regard to scale dependency, were then discussed. In their commentary, Jetz et al. have focused mainly upon issues connected with the strength of the relationships indicated by the R2 and some broad topics connected with the use of local and global statistical techniques. There are three points I would wish to stress in order to both emphasize the key thrusts made in my earlier paper and in response to the commentary: Jetz et al. comment that they see GWR as a supplement, and not an alternative, to standard regression. I could not agree more. GWR and standard regression are fundamentally different and aim to do different things — one does not replace the other but, consequentially, one can be more appropriate than the other depending on the application. GWR was not presented as an alternative to global regression in my earlier paper. Rather, GWR was used to do what it is appropriate for, namely to explore spatial nonstationarity in a relationship and to suggest that it would help in identifying additional variables to include in order to ensure a standard global model was appropriately specified. As such, GWR may be a useful technique to apply in the steps toward the development of an ultimate global model. The fundamental differences between global and local techniques should be recognized. In particular, it is important to note that conventional global regression fits with the view that a single parameter estimate applies over the entire region of study. With local statistics, such as GWR, the focus, however, is not on regularities but rather on differences, making it highly attractive for data exploration. The variations in parameter values observed in my earlier paper do cast doubt over the use and interpretation of standard regression analyses. As discussed in the earlier paper, the single estimate for a parameter derived from a conventional global regression may not represent conditions locally or even at any site within the study area. The value of the derived global model for descriptive and predictive purposes is therefore questionable. Although GWR is not problem-free, it does highlight that the model parameters may vary markedly, in sign and direction, while those from a global regression are constant. This is an issue noted by Jetz et al. who observe that local variation may arise in a global analysis because of missing variables or interaction terms. The key point to stress here, however, is that if the global model is not fully specified, its realism may vary over space, limiting the model's descriptive and predictive value. Again, the GWR results may be used to inform attempts to fully specify a global model. In the commentary, relatively little attention was paid to the issues of scale dependence raised in my earlier article. This issue was fundamental to the paper (see its title!) and is an issue on which standard regression yields little useful information. Jetz et al. comment that GWR does provide a framework for evaluating the effects of changing the scale of an analysis. This is an important attribute as scale dependency is often observed in ecological studies and Fig. 3 in my earlier paper highlights how spatial nonstationarity in a relationship can give rise to varying trends in scale dependence effects. Tools to explore spatial nonstationarity, such as GWR, should therefore be a welcome addition to the ecologist's armoury of techniques. Jetz et al. do raise some interesting issues, notably in relation to the spatial autocorrelation of residuals and magnitude of R2. Comparison of models using the R2 is difficult and may be more appropriately undertaken using the Akaike Information Criterion (AIC) or similar variable. Spatial autocorrelation of residuals is often observed in standard regression modelling as a function of forcing a global model when the relationship is actually spatially nonstationary. GWR offers the ability to model spatial nonstationarity and its effects directly rather than through some postanalysis interpretation of the manifestation of nonstationarity effects in terms of residuals (Fotheringham et al., 2002). Although some clustering of residuals may remain in a GWR analysis, the problem is usually reduced. However, this is an issue worthy of further thought and attention. I am grateful to Jetz et al. for their observations. Their perspective, which seems based on a standard global view, is common and appropriate when wishing to make global statements and ultimately laws. This is a perfectly valid perspective but it is not necessarily always superior to others and a local perspective has much to offer. I hope that the diversity of views, tools and motivations of researchers can be viewed positively and used together to advance understanding. I am very grateful to Walter Jetz for kindly forwarding a draft of the commentary and Stewart Fotheringham for some helpful comments on GWR.