## ABSTRACT

The previous chapter introduced the de–nition of spatial autocorrelation as a nonzero covariance between spatial proximity and attribute value proximity (i.e., nearby values are more similar or more dissimilar than they would be if they were arranged randomly). This de–nition imposes the need for a means to measure, based on a sample of data values, the covariance between nearby points and to decide whether or not this covariance is consistent with a random spatial arrangement of values. We have to de–ne a statistic and establish a null hypothesis concerning the value of that statistic when no spatial autocorrelation exists and an alternative hypothesis concerning the value when spatial autocorrelation does exist. Consider the soil sand content data of Section 3.6. In that section, a model called the spatial error model was –t to the detrended sand content data of Field 4.1 of Data Set 4 (denoted Field 4.1). The parameter λ of the model was estimated in this process. One could measure and test autocorrelation using this parameter, and, indeed, we shall have occasion to do something like this in Chapter 13. However, the validity of this test requires that the data satisfy the spatial error model. In many applications, there is considerable advantage in having a statistic that does not depend on a particular model of the autocorrelation structure. In this chapter, we introduce a collection of such statistics that are used to measure the strength of spatial autocorrelation. After some preliminary discussion in Section 4.2, Section 4.3 discusses tests for spatial autocorrelation of categorical data, and Section 4.4 discusses tests for spatial autocorrelation of quantitative data. Section 4.5 discusses measures of autocorrelation structure within a data set. Each of these sections is concerned with areal data. Section 4.6 provides a brief discussion of measures of autocorrelation of continuous (geostatistical) data.