Spatial nonstationarity is a condition in which a

Geographically weighted regression and the expansion method are two statistical techniques which can be used to examine the spatial variability of regression results across a region and so inform on the presence of spatial nonstationarity. Rather than accept one set of 'global' regression results, both techniques allow the possibility of producing 'local' regression results from any point within the region so that the output from the analysis is a set of mappable statistics which denote local relationships. Within the paper, the application of each technique to a set of health data from northeast England is compared. Geographically weighted regression is shown to produce more informative results regarding parameter variation over space.1 Spatial nonstationarity A frequent aim of data analysis is to identify relationships between pairs of variables, often after negating the effects of other variables. By far the most common type of analysis used to achieve this aim is that of regression, in which relationships between one or more independent variables and a single dependent variable are estimated. In spatial analysis the data are drawn from geographical units and a single regression equation is estimated. This has the effect of producing 'average' or 'global' parameter estimates which are assumed to apply equally over the whole region. That is, the relationships being measured are assumed to be stationary over space. Relationships which are not stationary, and which are said to exhibit spatial nonstationarity, create problems for the interpretation of parameter estimates from a regression model. It is the intention of this paper to compare the results of two statistical techniques, Geographically weighted regression (GWR) and the expansion method (EM), which can be used both to account for and to examine the presence of spatial nonstationarity in relationships.It would seem reasonable to assume that relationships might vary over space and that parameter estimates might exhibit significant spatial variation in some cases. Indeed, the assumption that such events do not occur, which until recently has been relatively unchallenged, seems rather suspect. There are three reasons why parameter estimates from a regression model might exhibit spatial variation: that is, why we might expect parameters to be different if we calibrated the same models from data drawn from different parts of the region (as shown by Fotheringham et al, 1996;1997). The first and simplest is that parameter estimates will vary because of random sampling variations in the data used to calibrate the model. The contribution of this source of variation is not of interest here but needs to be eliminated by significance testing. In this paper we want to concentrate on large-scale, statistically significant variations in parameter estimates over space, the source of which cannot be attributed solely to sampling. The second explanation is that, for whatever reason, some relationships are intrinsically different across space. Perhaps, for example, there ar...

In regression models where the cases are geographical locations, sometimes regression coef®cients do not remain ®xed over space. A technique for exploring this phenomenon, geographically weighted regression is introduced. A related Monte Carlo signi®cance test for spatial non-stationarity is also considered. Finally, an example of the method is given, using limiting longterm illness data from the 1991 UK census.

SUMMARYThis paper describes geographically weighted Poisson regression (GWPR) and its semi-parametric variant as a new statistical tool for analysing disease maps arising from spatially non-stationary processes. The method is a type of conditional kernel regression which uses a spatial weighting function to estimate spatial variations in Poisson regression parameters. It enables us to draw surfaces of local parameter estimates which depict spatial variations in the relationships between disease rates and socio-economic characteristics. The method therefore can be used to test the general assumption made, often without question, in the global modelling of spatial data that the processes being modelled are stationary over space. Equally, it can be used to identify parts of the study region in which 'interesting' relationships might be occurring and where further investigation might be warranted. Such exceptions can easily be missed in traditional global modelling and therefore GWPR provides disease analysts with an important new set of statistical tools. We demonstrate the GWPR approach applied to a dataset of working-age deaths in the Tokyo metropolitan area, Japan. The results indicate that there are signiÿcant spatial variations (that is, variation beyond that expected from random sampling) in the relationships between working-age mortality and occupational segregation and between working-age mortality and unemployment throughout the Tokyo metropolitan area and that, consequently, the application of traditional 'global' models would yield misleading results.