The Contiguity Ratio and Statistical Mapping

TLDR: In this article, the authors considered the problem of determining whether statistics given for each "county" in a "country" are distributed at random or whether they form a pattern.

Abstract: The problem discussed in this paper is to determine whether statistics given for each "county" in a "country" are distributed at random or whether they form a pattern. The statistical instrument is the contiguity ratio c defined by formula (1.1) below, which is an obvious generalization of the Von Neumann (1941) ratio used in one-dimensional analysis, particularly time series. While the applications in the paper are confined to oneand two-dimensional problems, it is evident that the theory applies to any number of dimensions. If the figures for adjoining counties are generally closer than those for counties not adjoining, the ratio will clearly tend to be less than unity. The constants are such that when the statistics are distributed at random in the counties, the average value of the ratio is unity. The statistics will be regarded as contiguous if the actual ratio found is significantly less than unity, by reference to the standard error. The theory is discussed from the viewpoints of both randomization and classical normal theory. With the randomization approach, the observations themselves are the "universe" and no assumption need be made as to the character of the frequency distribution. In the "normal case," the assumption is that the observations may be regarded as a random sample from a normal universe. In this case it seems certain that the ratio tends very rapidly to normality as the number of counties increases. The exact values of the first four semi-invariants are given for the normal case. These functions depend only on the configuration, and the calculated values for Ireland, with number of counties only 26, show that the distribution of the ratio is very close to normal. Accordingly, one can have confidence in deciding on significance from the standard error.