ABSTRACT
Many problems that involve a possible relationship between two or more variables can be thought of in terms of an association between distance matrices, often with one of the matrices relating to spatial distances. The following examples show the wide range of situations that can be approached in this way:
In Example 1.2, a matrix of distances between the continents based on the present-day earwig distribution (A) was compared with a matrix that reflects the present location of the continents (B) and with a matrix that reflects assumed positions of the continents before continental drift (C). A comparison of the level of association between A and B with the level of association between A and C showed that the latter association is stronger. This suggests that the present-day distribution of earwigs reflects evolution in Gondwanaland rather than evolution with the continents in their present positions.
In studying evidence of disease contagion, information on n cases of a disease can be obtained. An n × n matrix of geographical distances between these cases can then be constructed and compared with another n × n matrix of time distances apart. If the disease is contagious, then the cases that are close in space will have some tendency to also be close in time. Generally, it can be expected that cases will tend to be clustered in space around areas of high population. They may also be clustered in time because of seasonal effects. However, in the absence of contagion, the space and time clustering can be expected to be independent. Hence, the hypothesis of no contagion 204becomes the hypothesis of no association between the elements of the spatial distance matrix and the corresponding elements of the time distance matrix. If it is desirable, the distance matrices can be simplified so that they consist of zeros for adjacent cases and ones for nonadjacent cases (with a suitable definition of adjacent), as was done by Knox (1964) in his study of childhood leukemia in Northumberland and Durham. See Besag and Diggle (1977), Robertson and Fisher (1983), Marshall (1989), Robertson (1990), and Besag and Newell (1991) for further discussions of this type of medical application.
If an animal or plant population is located in n distinct colonies in a region, then there may be some interest in studying the relationship between environmental conditions in the colonies and genetic or morphometric variation, with a positive association regarded as evidence of adaptation (Sokal, 1979; Douglas and Endler, 1982; Dillon, 1984). One approach involves constructing a matrix of environmental distances between the colonies and comparing this with a matrix of genetic or morphometric distances. The construction of measures of distance for a situation like this is discussed by Manly (2005, Chapter 5). Inevitably, the choice will be arbitrary to some extent. One of the measures commonly used in cluster analysis can be used for environmental and morphometric distances. A range of measures is available for genetic distances. Two examples of this type of situation are discussed by Manly (1985, p. 182). The first (which is also the subject of Example 9.3) concerns a comparison between environmental and genetic distances for the 21 colonies of the butterfly Euphydryas editha studied by McKechnie et al. (1975) in California and Oregon. The second concerns a comparison between morphological distances (based on color and banding frequencies) and habitat distances (0 for the same habitat type, 1 for a different habitat type) for 17 colonies of the snail Cepaea nemoralis studied by Cain and Sheppard (1950) in southern England.
Suppose that birds are ringed at a number of locations, and at a later date, n of the birds are recovered in another area that they have migrated to. Here the n × n matrix of distances between the recovered birds prior to migration can be compared with the same size matrix of distances between 205the birds on recovery. If the distances in the two matrices appear to match to some extent, then this is evidence of pattern transference. Besag and Diggle (1977) describe a comparison of this type for 84 blackbirds ringed in Britain in winter and recovered in northern Europe in a subsequent summer.
An area is divided into n equal-size quadrats, and in each quadrat a count is made of the number of individuals present for a plant species. The question of interest is whether there is any evidence that similar counts tend to occur in clusters. In this case, an n × n matrix of differences between quadrat counts can be constructed and compared with a matrix of distances between the quadrats. If the matrices show a similar pattern, then it appears that quadrats that have similar counts tend to be close together.
