The mean measure of divergence: its utility in model-free and model-bound analyses relative to the Mahalanobis D(2) distance for nonmetric traits.

The mean measure of divergence (MMD) distance statistic has been used by researchers for nearly 50 years to assess inter-sample phenetic affinity. Its widespread and often successful use is well documented, especially in the study of cranial and dental nonmetric traits. However, the statistic has accumulated some undesired mathematical baggage through the years from various workers in their attempts to improve or alter its performance. Others may not fully understand how to apply the MMD or interpret its output, whereas some described a number of perceived shortcomings. As a result, the statistic and its sometimes flawed application(s) have taken several well-aimed hits; a few researchers even argued that it should no longer be utilized or, at least, that its use be reevaluated. The objective of this report is to support the MMD, and in the process: (1) provide a brief history of the statistic, (2) review its attributes and applicability relative to the often-used Mahalanobis D(2) statistic for nonmetric traits, (3) compare results from MMD and D(2) model-free analyses of previously-recorded sub-Saharan African dental samples, and (4) investigate its utility for model-bound analyses. In the latter instance, the ability of the D(2) and other squared Euclidean-based statistics to approximate a genetic relationship matrix and Sewall Wright's fixation index using phenotypic data, and the inability of the MMD to do so, is addressed. Three methods for obtaining such results with nonlinear MMD distances, as well as an assessment of the fit of the isolation-by-distance model, are presented.