Correcting for complexity: incorporating trait-numbers enhances the performance of EMMLi in investigating modularity

Adams and Collyer (2019) evaluated the statistical performance of several approaches for quantifying morphological modularity and found that EMMLi had inflated type I error rates and a bias toward more complex models compared to the Covariance Ratio (CR) approach. They suggested that this may have been at least partly driven by the fact that AICc values from EMMLi do not incorporate trait numbers, but this was not verified. Here, I present a performance analysis of a trait-number corrected EMMLi approach ("EMMLip"), showing that this ameliorates rates of false discovery and produces conservative results that favor less complex models. The corrected EMMLi approach was effective at differentiating models of modularity with varying between- and within-module covariation especially when effect size or dataset size were sufficiently large. While CR tests remained more effective at specifically detecting overall modularity, I found that CR tests are sensitive to varying within/between module covariation, and in some cases had inflated model misspecification between 2- and 3-module hypotheses. With this minor correction (albeit incomplete), the combination of EMMLip and CR tests becomes the best available toolkit for detecting and contrasting modularity hypotheses. This toolkit is however still imperfect, and I discuss future avenues for improvements.