Data from: EMMLi: a maximum likelihood approach to the analysis of modularity

Identification of phenotypic modules, semi-autonomous sets of highly-correlated traits, can be accomplished through exploratory (e.g., cluster analysis) or confirmatory approaches (e.g., RV coefficient analysis). While statistically more robust, confirmatory approaches are generally unable to compare across different model structures. For example, RV coefficient analysis finds support for both two- and six-module models for the therian mammalian skull. Here, we present a maximum likelihood approach that takes into account model parameterization. We compare model log-likelihoods of trait correlation matrices using the finite-sample corrected Akaike Information Criterion, allowing for comparison of hypotheses across different model structures. Simulations varying model complexity and within- and between-module contrast demonstrate that this method correctly identifies model structure and parameters across a wide range of conditions. Empirical analysis of a dataset of 61 3-D landmarks from macaque (Macaca fuscata) skulls, representing five age categories, tested 31 models, from no modularity to 2, 3, 6, and 8 modules. Our results clearly support a complex six-module model, with separate within- and inter-module correlations, across all ages, demonstrating that this complex pattern of integration in the macaque skull is highly conserved throughout postnatal ontogeny. Subsampling analyses further demonstrate that this method is robust to relatively low sample sizes.

表型模块（phenotypic modules）指高度相关性状的半自主集合，可通过探索性方法（如聚类分析）或验证性方法（如RV系数分析）进行识别。尽管验证性方法在统计学上更为稳健，但通常无法对不同模型结构开展比较。例如，RV系数分析在兽亚纲哺乳动物头骨的研究中，同时支持两模块模型与六模块模型。本研究提出一种考虑模型参数化的极大似然方法：我们利用有限样本校正的赤池信息准则（Akaike Information Criterion）对比性状相关矩阵的模型对数似然值，从而实现不同模型结构下的假设比较。通过开展改变模型复杂度、模块内与模块间对比度的模拟实验，本研究证明该方法可在广泛的条件范围内准确识别模型结构与参数。我们对来自日本猕猴（Macaca fuscata）头骨的61个三维地标数据集开展实证分析，该数据集涵盖5个年龄组，共测试了从无模块化到2模块、3模块、6模块及8模块的共计31种模型。研究结果清晰支持包含独立模块内与模块间相关性的复杂六模块模型，且该模型在所有年龄组中均成立，这表明猕猴头骨的复杂整合模式在产后个体发育过程中高度保守。子采样分析进一步证实，该方法对较低样本量具有良好的稳健性。