The conceptual and statistical relationship between modularity and morphological integration.

The modular variation of organismal form during evolution or development can be viewed as the result of dissociated local developmental processes. Whereas studies of modularity often are experimental, morphological integration is a more descriptive concept applying to groups of correlated phenotypic characters. Using simple path models, this paper shows that the classic underlying assumption of modularity (high correlations within modules, lower correlations between modules) is met only when local developmental factors have high effects on the traits relative to all factors' variations of effect (i.e., allometry). Accordingly, many classic approaches to morphological integration are meaningful only when local as well as common growth factors are nearly isometric. We show that morphometric modules might instead be defined more generally as sets of variables with non-zero within-module covariances, even when the covariances due to common factors have been removed, so that the residual between-module covariances are all near zero. Although it is still inherently unreliable to identify modules from phenotypic covariances in nonexperimental data, patterns of integration can sometimes be estimated based on prior identification of the modules themselves. We outline a simple approach for this case using Wright-style factor analysis and demonstrate the relation of its algebra to the more familiar approach via partial least squares.