# Fluctuating Asymmetry and Stress in _Macaca fuscata_: Does Captivity Affect Morphology?
## 2 Material and Methods
### The Analytical Approach  
Several procedures have been proposed to test for asymmetry in biological structures using Geometric Morphometrics and multivariate statistics. As a preliminary step of all these approaches, it is necessary to identify the type of symmetry under study. Symmetry can be defined as 'object symmetry' (the symmetry within a structure) or'matching symmetry' (two separate mirrored copies of a structure). Our analysis focuses on 'object symmetry' as this is the case of the mandible. In the 'object symmetry' case, the asymmetry is quantified by comparing the object with its mirrored (reflected) counterpart. After the step of reflection, the mirrored landmarks were relabelled to maintain homology with the original configuration [6; 35]. The original and the mirrored configurations were superimposed in a common Generalized Procrustes Analysis (GPA) [36], including the midline unpaired landmarks (along the mandibular symphysis). Following these preliminary transformations of the landmarks, two approaches were adopted to quantify asymmetry. The first approach follows [35; 37]. First, the differences between each landmark configuration and its reflection were computed, obtaining the individual asymmetric component. To explore the asymmetry visually, a modified PCA was performed, in which the origin of the axes represents perfect symmetry. The principal components of the asymmetry vectors were calculated around the symmetric origin; therefore, the asymmetry vectors represent the magnitude and direction of asymmetry starting from an origin of perfect symmetry. This method allows immediate visualisation of the degree (length of the vectors) and type (orientation of the vectors) of asymmetry (whether it is directional or fluctuating). Fluctuating asymmetry can then be estimated as the Procrustes distance between each individual and its relabelled mirrored version. The proportions of directional and fluctuating asymmetry were then estimated using a sum-of-squares decomposition of total asymmetry. Finally, following [35], we used the density distribution of the PC1 and PC2 scores to assess the degree of directional and fluctuating asymmetry. Additionally, PCA of the symmetric component was performed to visualise the symmetric shape variations within the sample and the potential differences between the captive and wild groups in the symmetric space. The symmetric shape was obtained by averaging each landmark configuration and its superimposed relabelled counterpart [35]. The shape variations of the first and second Principal Component axes (PC1 and PC2) were visualised using Thin-Plate-Splines (TPS) [38]. First, the 3D mesh of the mandible showing the smallest Procrustes distance from the mean shape of the symmetric shape space was chosen for the visualisation, and warped onto the mean shape. To visualise the shape variations of the symmetric shape space, the mesh was warped onto the maximum and minimum scores of the symmetric PC1 and PC2. Then, to visualise the asymmetric shape variations, the maximum and minimum scores of the asymmetric PC1 and PC2 were added to the symmetric mean shape, using again TPS to warp the symmetric mean shape mesh. Multivariate Procrustes ANOVA was performed to test for symmetric and asymmetric shape differences between the mandibles of captive and wild _M.fuscata_. We used the shape (symmetric and asymmetric, in different ANOVA tests) as dependent variable and group (captive/wild) and sex as independent variables, allowing for the interaction of group and sex. In the second approach, we performed a Multivariate Procrustes ANOVA using the overall shape as dependent variable (original and mirrored landmarks with no further transformations). In this analysis, the overall shape was tested against individual, reflection (a factor identifying the original and mirrored datasets), group (captive/wild groups), size, and sex. Interactions were allowed among individual, reflection and group [39]. The variation among individuals was quantified by the net effect of the variable \"individual\".