Examining Modularity via Partial Correlations: A Rejoinder to a Comment by Paul Magwene

We work, as Paul Magwene does, within the general tradition of path models set out nearly a century ago by Sewall Wright. In this approach, when 2 variables each depend linearly on the same cause, the part of their covariance that explanation accounts for is sim ply the product of the 2 path coefficients (regressions on the common cause) times the variance of that cause. Any additional covariance observed is attributed to ad ditional common causes and is conveniently quantified by the partial correlation of the 2 outcome variables con ditional on the first common cause (which is the cor relation between the two residuals from the regression on this cause). A partial correlation thus combines a test for the adequacy of one particularly simple explanation of the covariances among observed variables together with an assessment of the maximum covariance possible that any additional factor, however hypothesized, might contribute. The notion of correlations among residuals extend easily from the case of 3 variables to many. Pearl (2000) is a fine summary of the way in which this sort of logic can be ramified to incorporate fairly sophisticated statistical analyses about alternate causal interpretations of a data set. The first appearance of this elaboration in the con text of morphological integration and modularity was apparently due to Paul Terenrjev in the astoundingly early year of 1933. He investigated partial correlations among measurements on frogs conditioned on 1 mea sure of overall size together with 1 estimated more lo cal morphogenetic factor. Similarly, in Mitteroecker and Bookstein (2007, 2008), we demonstrated interpretation of a pattern of covariances within and between mod ules after partialling out 1, 2, or more common factor estimates.