A statistical framework for testing modularity in multidimensional data.

Modular variation of multivariate traits results from modular distribution of effects of genetic and epigenetic interactions among those traits. However, statistical methods rarely detect truly modular patterns, possibly because the processes that generate intramodular associations may overlap spatially. Methodologically, this overlap may cause multiple patterns of modularity to be equally consistent with observed covariances. To deal with this indeterminacy, the present study outlines a framework for testing a priori hypotheses of modularity in which putative modules are mathematically represented as multidimensional subspaces embedded in the data. Model expectations are computed by subdividing the data into arrays of variables, and intermodular interactions are represented by overlapping arrays. Covariance structures are thus modeled as the outcome of complex and nonorthogonal intermodular interactions. This approach is demonstrated by analyzing mandibular modularity in nine rodent species. A total of 620 models are fit to each species, and the most strongly supported are heuristically modified to improve their fit. Five modules common to all species are identified, which approximately map to the developmental modules of the mandible. Within species, these modules are embedded within larger "super-modules," suggesting that these conserved modules act as building blocks from which covariation patterns are built.