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Background

I have 50 service areas, each service area has a number of points associated with. In order to compare the areas to each other, one idea was to describe the degree to which its service area's set of points were clustered or dispersed.

The service areas vary in size and shape. The number of points vary in each service area, too.

Question

Is it possible and is it meaningful to compare the Moran's I of one set of points in a service area to another?

The end goal would be to make a map and rank service areas by levels of clustering/dispersion.

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As there is no answer here, and really no reliable answer in the searchable internet, let me provide some information, that would be at least partly useful in this context.

  1. Global Moran's I statistic is shape dependent:

    • The expected value of this statistic is dependent on the number of polygons in area. See Wikipedia article about Moran's I. Still I do not know what spatial weights matrix was used to calculate this expected value. Maybe it holds for all popular types.

      This already means, that if we wanted to be mathematically strict, we could not use it for comparison regions with different polygon count. Nevertheless as the number of polygon is higher, this problem diminishes.

    • We can also perform an exercise to calculate Moran's I in controlled environment. Let's use such area (black = 1, white = 0).

enter image description here

When calculating The Moran's I for this one, using contiguity spatial weights matrix, the value of I is: -0.3333 (equal to expectation).

Then if we divide one of polygons by half this way:

enter image description here

the Moran's I value is very close to 0 (much above expectation, which is -0.25 for 5 polygons).

So the values of variable in space did not change, only shape of spatial borders was changed, and this changed value of Moran's I. This property is much more troubling when trying to compare different areas. It might result in strong differences just for shape dependency. But maybe it is not that problematic in some cases, like for example lots of polygons, similar dispersion of them and so on. Unfortunately, I can not provide answer about asymptotic properties, possibly such articles help: On the asymptotic distribution of the Moran I test statistic with applications.

Please consider, that this shape-dependency might look different under different matrices of spatial weights, and comparing across regions might be reasonable for some matrices, while problematic for others.

  1. I did some paper search and I did not find any articles making comparisons of Moran's I for countries (I guessed it would be most reasonable). This does not mean that such thing do not exist - finding one could prove, that somebody had similar idea and published. If found, this would be one of best hints here, but I found nothing, which is also negative hint.

  2. Some sources say, that Moran's I is purely inferential statistic and is used only to test for the presence and sign of spatial autocorrelation. (See this site.). But this might be only partly true and I have seen under some circumstances comparisons between time periods and between variables. So it is used for some comparisons. Also, it is a spatial correlation coefficient. It is multidimensional and so on, but it is standardised and therefore appears to be somehow interpretable. Sadly not in the strong way as standard correlation, could not be strict statistical inference.


At this point let me suggest some solutions:

  1. Not doing it for formal reasons.

  2. If the temptation is strong you can do it as colorful, amateur exercise, hoping that the problem is "not that bad", possibly because shapes of whatever you use "do not differ so much" or dispersion of points is "somehow" similar.

  3. You can increase the effort and search for the articles, that if they exist, test Moran's I or derive some its properties. You can also try to simulate some Monte-Carlos to see how big would be magnitude of the problem.

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