Testing association patterns: issues arising and extensions

Abstract

A reasonable necessary condition for a population to be socially structured is that individuals associate in a nonrandom fashion. Thus, testing for preferred or avoided companions is a fundamental step in analyses of social structure (Whitehead & Dufault 1999). Cluster analyses, sociograms and their ilk (e.g. Morgan et al. 1976) all assume nonrandom associations, and if associations are random, none of these mean anything. Unfortunately testing nonrandom association on real data is not simple. A suitable test statistic is the standard deviation of the association indexes, which will be higher than expected if individuals have preferred or avoided associates. But what is ‘expected’ in the case of no preference or avoidance? The distributions of the standard deviation of association indexes, or any other suitable test statistic, are not analytically tractable under the null hypothesis. A solution is to use permutation tests in which the association data are randomized many times subject to certain constraints, each time calculating the test statistic (Manly 1997). The mean of these randomized test statistics can be considered its ‘expected value’, and a P value is then calculated as the proportion of times that the permuted statistics are more extreme than the real test statistic. A number of analyses of animal social structure have taken this approach (e.g. Whitehead et al., 1982, Smolker et al., 1992, Slooten et al., 1993 and Pepper et al., 1999). However, it was not easy to design an efficient computational routine that randomly permutes the records of which individuals were found in which groups in such a way that the number of individuals in each group and the number of groups containing each individual are both held constant. There are conceptual difficulties with such tests (Manly 1997).