Comparing morphological integration patterns across groups using angular distances between PLS covariation axes

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Abstract Morphological integration describes coordinated covariation among traits arising from shared developmental, functional, or genetic interactions and plays an important role in shaping evolutionary trajectories. In geometric morphometrics, two‐block partial least squares (PLS) analysis is widely used to identify dominant axes of covariation between anatomical modules. However, existing approaches primarily quantify the strength of integration and provide limited tools for formally evaluating and comparing the orientation of integration patterns across groups. In this article, I introduce Integration_Patterns , a statistical framework implemented in R for comparing group‐specific integration patterns based on the angular relationships among PLS covariation axes. The method accepts Procrustes‐aligned landmark coordinates or other matched multivariate descriptors, estimates group‐specific covariation axes, and returns pairwise angular distances and permutation‐based p ‐values. These procedures allow formal statistical testing of whether integration patterns differ between groups, enabling explicit evaluation of hypotheses about the conservation or divergence of covariation structure across taxa. Using simulated datasets, I evaluate the statistical properties of the estimator across a wide range of sample sizes and true angular differences between integration axes. These simulations show that the permutation procedure maintains appropriate Type I error, statistical power increases with sample size and angular separation, and angular estimates are reliable when the selected PLS axis captures a substantial proportion of total between‐block covariation. An empirical example further illustrates the application of the approach and its ability to visualise and interpret species‐specific integration patterns. Overall, these results demonstrate that angular comparisons of group‐specific PLS covariation axes provide a simple and interpretable framework for evaluating differences in morphological integration patterns across taxa. Integration_Patterns therefore extends existing approaches to morphological integration and offers a general method for formally comparing the orientation of covariation structure in multivariate datasets analysed using two‐block PLS.

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