Spatial nonstationarity is a condition in which a

In this paper the examination of the modifiable areal unit problem is extended into multivariate statistical analysis. In an investigation of the parameter estimates from a multiple linear regression model and a multiple logit regression model, conclusions are drawn about the sensitivity of such estimates to variations in scale and zoning systems. The modifiable areal unit problem is shown to be essentially unpredictable in its intensity and effects in multivariate statistical analysis and is therefore a much greater problem than in univariate or bivariate analysis. The results of this analysis are rather depressing in that they provide strong evidence of the unreliability of any multivariate analysis undertaken with data from areal units. Given that such analyses can only be expected to increase with the imminent availability of new census data both in the United Kingdom and in the USA, and the current proliferation of GIS (geographical information system) technology which permits even more access to aggregated data, this paper serves as a topical warning.

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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...