Character Number and the Multivariate Analysis of Simple Patterns of Geographic Variation: Categorical or "Stepped Clinal" Variation

Computer selection of random sets of real characters was used to elucidate the relationship between character number and the reliability of multivariate patterns of geographic variation. The simple pattern of geographic variation was a distinct pair of parapatric races (categorical or clinal variation) in the grass snake (Natrix natrix) in Europe. The characters used were 71 significant characters of low within-group correlation drawn from six different character systems (e.g., scalation, color pattern and internal morphology). The pattern was assessed by principal coordinate analysis, and the congruence was taken as the absolute correlation between the principal coordinates. The investigation was based on two procedural models: (A) the congruence between com- pletely independent character sets using from 1 to 35 characters; and (B) the congruence be- tween the analysis based on the total 71 characters and analyses based on from 1 to 65 characters. The relationship between congruence and character number is expressed by one of two math- ematical models depending on the procedural model employed. Congruence is clearly asymp- totic with respect to character number. This indicates the existence of a stable, overall pattern of racial variation that, once established, is unlikely to be influenced by the addition of further characters. The asymptotes are reached at an early stage and, between 8 to 10 characters, reveal the distinct patterns of racial variation with at least 90% confidence, depending upon whether one is characterizing patterns on the basis of a geographic transect or a scatter on the first two principal coordinates. It does not appear to be cost effective to use more than 18 significant characters, because each additional character increases the minimum congruence by very little beyond this point. Univariate characters, and those patterns based on too few characters, can misrepresent this stable pattern. The observed failure of univariate/conventional studies to reveal this pattern, or to agree with one another, is related to the low probability of selecting the appropriate individual characters, even though they occur. (Geographic variation; races; character number; congruence; predictivity; multivariate morphometrics; stepped cline.)