3.4 Spatial diagnostics
As discussed earlier, the MESS model can employ standard leverage statistics, unlike many spatial models. Figure 2 identifies tracts that have the lowest and highest (one-half percentile) degrees of leverage on a map of the US. Note the clustering of high leverage points along the southeast coast of Florida, the coast of California, and in scattered interior points in the western US. Conversely, low leverage points seem concentrated in Pennsylvania, North Carolina, and in some of the major cities.
Truly random outliers should reduce the degree of estimated spatial dependence, so case deletion estimates of the autoregressive parameter may provide interesting results. Examination of the one-out autoregressive estimates, , shows some striking features. For example, the range of is extremely small (min()=-1.6744, max()=-1.6726).
If the estimated spatial dependencies showed a spatial pattern, this might suggest possible ways of improving the spatial dependencies component of the model. Figure 3 illustrates the lowest and highest percentile of the one-out autoregressive parameters plotted against latitude and longitude. The smallest indicate where the degree of spatial dependence increased ( became more negative) upon deletion of the th observation. These observations exhibit weaker spatial dependence with their neighbors than the typical observation. From the figure, we see some regular patterns. For example, the southeast coast of Florida has a number of tracts whose deletion increases the degree of estimated spatial dependence. This may indicate the need for a variable measuring contiguity with the ocean.
In contrast, the largest indicate that the degree of spatial dependence decreased ( became more positive) upon deletion of the th observation. For these observations we see stronger spatial dependence with neighbors than that associated with the typical observation. Almost all of these large values occur in urban areas. It required less than 5 minutes to compute the MESS spatial diagnostics presented here, despite the large number of observations used in our example.