LASR 2007 — Systems

Bioinformatics sample summaries often involve distances or the conversion of similarities to dissimilarities, while Procrustes methods can proceed either via distances or via the shape coordinates underlying those distances. There is a distance–distance statistical method, currently increasing in popularity in some of my applications domains, that is at once simple, intuitive, and quite wrongheaded: the method of Mantel matrix correlations. This note explains the error when the test is applied in most of our biometric contexts, and, in view of the unsuitability of the principal-coordinates model to most bioinformatic domains other than the morphometric, suggests a hybrid statistic bridging Procrustes analysis to bioinformatics that taps the power of Procrustes principal coordinates to the maximum feasible extent. At a recent international meeting in one of the fields I work in (the American Association of Physical Anthropologists congress, last March in Philadelphia), Mantel tests of Procrustes distance against some other “distance” or “dissimilarity” appeared surprisingly often in the course of presentations about the relation of shape to other sorts of biological processes such as evolution or ecophenotypy as captured in geographical distance, cladistic distance (counts of evolutionary branching events), or “molecular distance” (e.g., some measure of sequence dissimilarity). I found the Mantel test to be a really poor choice in every example I saw. That suggested that its use was not the fault of the presenters independently but instead owed to some systematic misunderstanding of biometric principles. An argument might then be possible that would persuade a potential user to abandon the idea of exploiting this test prior to wasting any time computing things. Through the application to molecular distances/dissimilarities, the argument becomes relevant to much of bioinformatics: hence this talk at this meeting. The Mantel test (named for Nathan Mantel, 1919–2002, American epidemiologist) applies to data in the form of two (or more) distance matrices on the same sample of cases (in the original application, cases of a disease under study). Nowadays it takes the form of a permutation computation for assessing the observed correlation of entries between the distance matrices. The test is said to be “statistically significant” just when the correlation of the off-diagonal part of two distance matrices is greater than the 0.95 fractile, or whatever you choose, in the appropriate permutation distribution of the correlations of the off-diagonal parts of the matrices. 1. A Mantel test is not a “test of association” as we biometricians usually use that phrase. “Association” is between variables measured on one case, specimen, or group at a time, and the Mantel doesn’t refer to those; it talks about distances or dissimilarities relating pairs of cases or specimens or groups. So, no matter what the permutation procedure is doing, it shouldn’t be thought of as a significance test. For a null-hypothesis significance test to make sense, there need to be two probabilistic models, not one, and the models have to generate the data (i.e. the values we observe case by case). In the first, “null” model, usually some parameter (scalar or vector) of the population is zero; in the second model, it is not. In Mantel’s original application of 1967, one distance was geographical and the other temporal. The alternative