Statistical methods for studying modularity: a reply to Mitteroecker and Bookstein.

A major focus in studies of multivariate evolution is on trying to understand the biological and evolutionary forces that shape patterns of covariance between organismal traits. This is particularly truewith respect to studies concerned with the topics of phenotypic integration and modularity (see Chernoff and Magwene 1999, for a review). In a recent paper published in Systematic Biology on the topic of modularity and integration, Mitteroecker and Bookstein (2007) reviewed the use of “Wrightstyle” factor analysis to characterize and study patterns of morphological integration. These authors compared their preferred approach with a number of frameworks that previous investigators have employed to address similar questions. Among the methods that Mitteroecker and Bookstein comment on is an approach, described in Magwene (2001), for studying modularity and integration using Gaussian graphical models (GGMs). Mitteroecker and Bookstein (hereafter M&B) assert that the methods described in Magwene (2001) are unsuitable for the study of morphological integration because they are only applicable to modules with 2 variables. In their words “Magwene’s example . . . accidentally worked because all modules consisted of two variables only.” Below I demonstrate that M&B’s assertion is incorrect and there is no inherent limitation in terms of the size of the modules that can be distinguished using the GGM approach. I discuss conditional independence relationships and briefly describe what might be termed a “Wright-style” approach based on graphical modeling. I also highlight a number of recent biological applications of graphical modeling.