Multiple regression and correlation extensions of the mantel test of matrix correspondence

It is often necessary in population biology to compare two sets of distance measures. These measures can be based on genetic markers, morphological traits, geographic separation, ecological divergence, and so on. The distance measures can take various forms and frequently have unknown distributional properties. Many different procedures have been developed to compare the correspondence of one set of distances with another set. Prominent among them are the: (1) matrix correlation techniques of Sokal and Rohlf (1962), and Sneath and Sokal (1973); (2) networkmatching techniques of Spielman (1973); (3) matrix dilation and rotation techniques of Gower (1971) and of Schonemann and Carrol (1970); and (4) smallest-space techniques of Lingoes (1965) and Guttman (1968). Each of these strategies has strong points, but all suffer from a difficulty in assessing the statistical significance of attained correspondence. The problem is that a set of all possible pairwise distances between k units (populations, taxa, habitats, etc.) cannot be independent. More recently, a test of matrix correspondence-originally developed by Mantel (1967) and widely applied in geography (Hubert and Golledge, 1982) and psychometrics (Hubert, 1979a, b)-has caught the attention of population biologists (Sokal, 1979; Sokal et al., 1980; Douglas and Endler, 1982; Dow and Cheverud, 1985; Schnell et al., 1985, 1986; Sokal et al., 1986, 1987; O'Brien, 1987; Smouse and Wood, 1987). Attractions of the Mantel procedure are its wide applicability and computational simplicity. Mantel (1967) presented a formal analysis of matrix correspondence based on the assumption of asymptotic normality for a particular test criterion. Later workers (Mielke et al., 1981) developed more general procedures for Mantel statistics that assume a Pearson Type III distributional form. The most widely used evaluation procedure, however, involves the construction of a null distribution by Monte Carlo randomization, whereby one of the matrices is held rigid and the other has its rows and corresponding columns randomly permuted (Cliff and Ord, 1981). Dietz (1983) evaluated the Mantel test as one of several permutational tests for association between distance matrices. When dimensions of two matrices are small (say K ' 7), it is customary to evaluate the Mantel test criterion for all K! equally likely permutations. If K is large, then a large number of random permutations are sampled with replacement. Although the test has been useful in its present form, there are some simple modifications and extensions that would encourage even wider deployment in population work. Our purpose here is to sketch these changes and to illustrate them with an example drawn from the Yanomama Indians of lowland South America.