Quantifying between-species differences in morphological integration patterns using two-block PLS: an R implementation and sensitivity analysis

Abstract

Morphological integration is commonly quantified within species using covariance or correlation structures, and two-block partial least squares (PLS) has become a standard tool for assessing covariation between anatomical modules. However, most applications focus on a global PLS axis for a pooled sample, and there is no simple, widely used procedure to compare whether integration patterns differ between species in the same PLS space. Here I present an angle-based method and an accompanying R function, Integration_Patterns, that quantify between-species differences in integration patterns based on the orientation of the first eigenvector of the within-species covariance matrix in PLS1-PLS1 space. For each species, the method extracts PLS1 scores for two modules, computes the covariance matrix, retrieves the principal eigenvector representing the main axis of covariation, and then measures the angle (0-90{degrees}) between these axes across species. Small angles indicate nearly parallel integration patterns, whereas larger angles indicate divergent or orthogonal patterns. To evaluate the behaviour and robustness of the estimator, I conducted a simulation-based sensitivity analysis using species-specific PLS1 scores. Pairs of species were simulated under known angular separations (0{degrees}, 30{degrees}, 45{degrees}, 60{degrees}, 90{degrees}) and a range of per-species sample sizes (n = 3 to 200). For each combination, 100 replicate datasets were generated and the between-species angle was estimated. The results show that the estimator is biased and highly variable for very small samples (n [≤] 10), but converges rapidly to the true angle with increasing sample size. For n {approx} 20-30 individuals per species, the mean estimated angle is close to the true value and the standard deviation is substantially reduced across all scenarios, including orthogonal patterns (90{degrees}). These findings provide practical guidance on sample-size requirements when comparing integration patterns across species.