Complex clines: the predictivity of complicated patterns of geographic variation portrayed by multivariate analysis

The stability of complex patterns of geographic variation was investigated by assessing the congruence between multivariate ordinations derived from randomly chosen real characters. Two series of populations were analysed representing two situations with complex patterns of geographic variation. The first, a ‘Eurasian' series of populations, showed a strongly structured hierarchical pattern, the second, an ‘eastern' series of populations, showed a more subtle complex pattern of smooth clines and steps. The characters were selected from a total of 81 (Eurasian) or 61 (eastern) within-population independent characters from six different systems. The congruence between ordinations of the geographical populations was measured by the rotational fit statistic, R². Three procedures were used to compare ordinations based on from two to up to 80 characters randomly chosen to give: A, completely independent character sets; B, subsets compared to the total set; and C, potentially overlapping sets. All three procedures showed that congruence between the ordinations was asymptotic in relation to character number. This relationship was described by one of two mathematical models (procedure B did not result in a hyperbolic model as found with simple patterns of geographic variation). Generally speaking, once a sufficient number of characters are used, the complex patterns of geographic variation are stable, reliable and predictive and not substantially influenced by character choice. The strongly structured hierarchical pattern required 15 or so characters to achieve reliability whilst the more subtle patterns required 20 or so characters. However, the addition of further characters does improve reliability in both cases. The greater percentage of variance portrayed by three-dimensional ordinations compared to two-dimensional ordinations is achieved at the cost of lower congruence when a sufficient number of characters are used. If case studies of geographic variation were to adopt these procedures (preferably using completely independent character sets; procedure A) the reliability of their results would be indicated).