Comparing the strength of modular signal, and evaluating alternative modular hypotheses, using covariance ratio effect sizes with morphometric data

The study of modularity is paramount for understanding trends of
phenotypic evolution, and for determining the extent to which covariation
patterns are conserved across taxa and levels of biological organization.
However, biologists currently lack quantitative methods for statistically
comparing the strength of modular signal across datasets, and a robust
approach for evaluating alternative modular hypotheses for the same
dataset. As a solution to these challenges, we propose an effect size
measure (Z_CR) derived from the covariance ratio, and develop
hypothesis-testing procedures for their comparison. Computer simulations
demonstrate that Z_CR displays appropriate statistical properties and low
levels of misspecification, implying that it correctly identifies modular
signal, when present. By contrast, alternative methods based on likelihood
(EMMLi) and goodness of fit (MINT) suffer from high false positive rates
and high model misspecification rates. An empirical example in
sigmodontine rodent mandibles is provided to illustrate the utility of
Z_CR for comparing modular hypotheses. Overall, we find that covariance
ratio effect sizes are useful for comparing patterns of modular signal
across datasets or for evaluating alternative modular hypotheses for the
same dataset. Finally, the statistical philosophy for pairwise model
comparisons using effect sizes should accommodate any future analytical
developments for characterizing modular signal.