Measuring Differences of Trait Distributions Between Populations

A measure of difference between populations for a trait should reflect not only the differences in the relative frequencies of the trait states but also the trait differences between the states. Common approaches to measuring differences between populations rely on distance, probability, or variance concepts. To overcome conceptual problems of these approaches, a new difference measure Δ is presented that is based on both frequency and trait differences. For two populations, Δ expresses the degree to which the frequency distribution of the trait states within one population must be transformed in order to make it match the distribution in the other population. This is done by shifting the relative frequency excesses of trait states to other trait states of deficient frequency, where shifts occur between as similar states as possible. Δ equals the minimum sum of the shifted frequencies weighted by the respective trait differences. Its bounds are functions of the difference measure d 0 , which considers only differences in relative frequency. The computer program DeltaS applies an algorithm from operations research to calculate Δ. The effect of including trait differences is demonstrated by the topological differences observed between Δ‐ and d 0 ‐dendrograms constructed from microsatellite allele frequencies in four riparian stands of black poplar ( Populus nigra ), where the trait difference between two alleles equals the difference in numbers of tandem repeats. Δ is applicable to all traits for which trait differences are measurable, and it is shown to have elementary linearity properties that considerably simplify its interpretation.