Some results on vector correlation

Measuring multivariate association and beyond

Simple correlation coefficients between two variables have been generalized to measure association between two matrices in many ways. Coefficients such as the RV coefficient, the distance covariance (dCov) coefficient and kernel based coefficients are being used by different research communities. Scientists use these coefficients to test whether two random vectors are linked. Once it has been ascertained that there is such association through testing, then a next step, often ignored, is to explore and uncover the association's underlying patterns. This article provides a survey of various measures of dependence between random vectors and tests of independence and emphasizes the connections and differences between the various approaches. After providing definitions of the coefficients and associated tests, we present the recent improvements that enhance their statistical properties and ease of interpretation. We summarize multi-table approaches and provide scenarii where the indices can provide useful summaries of heterogeneous multi-block data. We illustrate these different strategies on several examples of real data and suggest directions for future research.

Resampling-based approaches to study variation in morphological modularity

Modularity has been suggested to be connected to evolvability because a higher degree of independence among parts allows them to evolve as separate units. Recently, the Escoufier RV coefficient has been proposed as a measure of the degree of integration between modules in multivariate morphometric datasets. However, it has been shown, using randomly simulated datasets, that the value of the RV coefficient depends on sample size. Also, so far there is no statistical test for the difference in the RV coefficient between a priori defined groups of observations. Here, we (1), using a rarefaction analysis, show that the value of the RV coefficient depends on sample size also in real geometric morphometric datasets; (2) propose a permutation procedure to test for the difference in the RV coefficient between a priori defined groups of observations; (3) show, through simulations, that such a permutation procedure has an appropriate Type I error; (4) suggest that a rarefaction procedure could be used to obtain sample-size-corrected values of the RV coefficient; and (5) propose a nearest-neighbor procedure that could be used when studying the variation of modularity in geographic space. The approaches outlined here, readily extendable to non-morphometric datasets, allow study of the variation in the degree of integration between a priori defined modules. A Java application – that will allow performance of the proposed test using a software with graphical user interface – has also been developed and is available at the Morphometrics at Stony Brook Web page (http://life.bio.sunysb.edu/morph/).

Morphometric integration and modularity in configurations of landmarks: tools for evaluating a priori hypotheses

Identifying the modular components of a configuration of landmarks is an important task of morphometric analyses in evolutionary developmental biology. Modules are integrated internally by many interactions among their component parts, but are linked to one another only by few or weak interactions. Accordingly, traits within modules are tightly correlated with each other, but relatively independent of traits in other modules. Hypotheses concerning the boundaries of modules in a landmark configuration can therefore be tested by comparing the strength of covariation among alternative partitions of the configuration into subsets of landmarks. If a subdivision coincides with the true boundaries between modules, the correlations among subsets should be minimal. This article introduces Escoufier's RV coefficient and the multi-set RV coefficient as measures of the correlation between two or more subsets of landmarks. These measures can be compared between alternative partitions of the configuration into subsets. Because developmental interactions are tissue bound, it is sensible to require that modules should be spatially contiguous. I propose a criterion for spatial contiguity for sets of landmarks using an adjacency graph. The new methods are demonstrated with data on shape of the wing in Drosophila melanogaster and the mandible of the house mouse.