One measure of spatial autocorrelation provided by PROC VARIOGRAM is Moran’s I statistic, which was introduced by Moran (1950) and is defined as

where , and .

Another measure of spatial autocorrelation in PROC VARIOGRAM is Geary’s c statistic (Geary, 1954), defined as

These expressions indicate that Moran’s I coefficient makes use of the centered variable, whereas the Geary’s c expression uses the noncentered values in the summation.

Inference on these two statistic types comes from approximate tests based on the asymptotic distribution of I and c, which both tend to a normal distribution as n increases. To this end, PROC VARIOGRAM calculates the means and variances of I and c. The outcome depends on the assumption made regarding the distribution . In particular, you can choose to investigate any of the statistics under the *normality* (also known as *Gaussianity*) or the *randomization* assumption. Cliff and Ord (1981) provided the equations for the means and variances of the I and c distributions, as described in the following.

The normality assumption asserts that the random field follows a normal distribution of constant mean () and variance, from which the values are drawn. In this case, the I statistics yield

and

where and . The corresponding moments for the c statistics are

and

According to the randomization assumption, the I and c observations are considered in relation to all the different values that I and c could take, respectively, if the n values were repeatedly randomly permuted around the domain D. The moments for the I statistics are now

and

where , . The factor is the coefficient of kurtosis that uses the sample moments for . Finally, the c statistics under the randomization assumption are given by

and

with , , and .

If you specify LAGDISTANCE= to be larger than the maximum data distance in your domain, the binary weighting scheme used by the VARIOGRAM procedure leads to all weights , . In this extreme case the preceding definitions can show that the variances of the I and c statistics become zero under either the normality or the randomization assumption.

A similar effect might occur when you have collocated observations (see the section Pair Formation). The Moran’s I and Geary’s c statistics allow for the inclusion of such pairs in the computations. Hence, contrary to the semivariance analysis, PROC VARIOGRAM does not exclude pairs of collocated data from the autocorrelation statistics.