Black/White Statistics

Andy Long

This note is based on the technical appendix on page 35 of [1]. It assumes that there are n regions on a map, and that some regions are colored white and some black. Let be the ``connection matrix'' for which if the and regions are joined, and otherwise (Queen's contiguity). Let if the area is black, and if it is white. Then

where

Assume that BW is asymptotically normally distributed for large enough n. The first two moments of BW then approximate E(BW) and . Those moments may be evaluated under two different assumptions:

• free sampling (with replacement): the individual areas are independently coded B or W, with probabilities p and q=1-p respectively; or
• nonfree sampling (without replacement): each unit has the same probability a priori of being B or W, but coding is subject to the constraint that there are B areas, W areas, and .

In the first case,

and

in the second,

and

where A is the total number of joins on the map, and is the total number of areas joined to the area. Thus

while

and

Finally,

may be treated as a standard normal score (N(0,1)), and tested for significance.

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