Moran (Moran 1950) proposed a measure of spatial autocorrelation (the degree of correlation among points in space) given by
\[ I = \frac{N}{\Sigma _{i}\Sigma _{j}w_{ij}} \frac{\Sigma _{i}\Sigma _{j}w_{ij}(y_i-\bar{y})(y_j-\bar{y})}{\Sigma _{i}(y_i-\bar{y})^2} \]
where \(w\) is a neighbor weight matrix such that \(w_{ii}=0\). In this case, we use neighbor weights as calculated for nearest neighbor analysis, based on trial dimensions and distance between plot centers.
The expected value of Moran’s I, under no spatial correlation, is \(E(I) = \frac{-1}{N-1}\). If data are perfectly dispersed (i.e. checkerboard), then I = -1, while a balanced arrangement (large values to one side, small values to the other) approaches 1. To Illustrate, consider the data from Cochran and Cox, Table 3.1. The ARM trial Cochran
contains First and Second as published. Column 3 has been modified by segrating large First values to one side of the field, while Column 4 has had First values evenly dispersed over plots. See the associated trial maps. The values listed are Moran I as calculated in Column Diagnostics (check Include spatial models)
The original data show a high degree of correlation; even higher than the segregated example.
When we are consider spatial models, we look for models that account for spatial correlations, thus, we look for models that reduce Moran’s \(I\) to near 0 (or slighly negative) values.
Consider, for example, the diagnostics for Column 2 PDF. The non-spatial diagnostics (Levene’s, Shapiro-Wilks, Skewness, Kurtosis) do not suggest anything other than the design (IID) model, while spatial analysis might select a cubic spatial trend (see Report and Report Set ).
However, the improvement in AIC is relatively small (~ 6), while residual Moran \(I\) is further from 0, suggesting the CST model adds anti-correlation to the residuals. We would not prefer a spatial model in this case.
Moran, P. A. P. 1950. “Notes on Continuous Stochastic Phenomena.” Biometrika 37 (1/2): 17–23.