Bivariate problems: The modifiable areal unit problem and correlation between processes

Abstract

In Chapter 5 we looked at a number of problems arising in the statistical analysis of a single variable measured in a cross-section such as a map, using the methodological framework put forward in Chapter 4. In this chapter, and in the next, we are interested in an extension of the previous results to the case of statistical analysis of spatial data with more than one variable. In particular the aim of this chapter is to consider the different results obtained when measuring the dependence between processes recorded at different level of spatial resolution, while in the next chapter we will deal with the ecological fallacy problem. For this reason throughout this chapter we will uniformly assume the local covariance hypothesis, as employed in Chapter 5. Furthermore we will restrict ourselves, consistent with the rest of the book, to Gaussian processes, so that the focus will be on the cross-correlations between processes. Finally, we will consider only bivariate spatial processes; the more general multivariate case involves a more complicated notation and theory beyond the scope of the present discussion (See Mardia, 1988; Wartenberg, 1985).