Trait covariance

We studied the patterns of trait covariance by generating a phenotypic matrix of variance–covariance (P matrix) for each cross type. P matrices are reliable surrogates of genetically based patterns of trait covariances (i.e. of the G matrices) when no pedigree is available [64, 92]. P matrices are especially likely to be good proxies in our particular study because the effects of the environment were mitigated by the use of common-garden conditions, and because the parental effects were accounted for by including in the subsequent models the family of origin (i.e. the egg clutch) of all individuals. We estimated the components of the three matrices by running three separate Multi-Response Generalized Mixed models [89]. All three models contained seven variables as a response (Table ​(Table1).1). The family was included as a fixed effect while the identity of the individual was included as a random factor. All the traits were mean-standardized by dividing the raw values by their group means [93].

The P matrices of each cross type were first compared on the basis of their size, shape and orientation [94]. The matrices sizes (V_tot) were used to compare the types of crosses in the overall phenotypic variance and were calculated as the sum of their eigenvalues (Eq. 2 in [95]) [94, 95]. Eccentricity (Ω) was used as a measure of the shape of the matrices and was calculated as the ratio of the first two eigenvalues [94]. Differences in overall matrix orientation were assessed using the angles (θ) between the first eigenvector of each P matrix. Briefly, if the patterns of trait covariances were not conserved but have rapidly evolved among the two morphs, we expected the two types of pure-morph offspring to show differences in the overall size of P (V_tot), which should suggest a response to two selective regimes eroding genetic variations to different extents. Similarly, differences in eccentricity (Ω) between the two purebred offspring were expected (for example, correlational selection, which can produce more constrained, “cigar shaped”, G matrices [94], might differ among the respective habitats of each morph). The orientation of G can also be subjected to changes because of the effects of correlational selection, among other evolutionary forces [94, 96, 97]. Thus, differences between purebred offspring in the orientation of P (θ) were also expected [68]. Regarding the hybrids, breakdowns in their trait covariance structure should be indicated by P matrices with larger sizes and reduced eccentricity [38]. Meanwhile, differences in the orientation of P between the hybrid and the purebred offspring should indicate whether the remaining constraints on the hybrid phenotypes are intermediate, under dominance and conserved relative to one morph, or transgressive (i.e. biased toward a unique direction of the phenotypic space).

Next, we assessed which part of P (i.e. which suits of covarying traits) differed the most among cross types in their variance by using Krzanowski’s common subspaces method [60]. This method produces a set of vectors (H) that can be used to determine the groups’ similarities in parts of the trait space. Eigenvalues of H indicates the degree of resemblance between principal components of the trait subspaces of each group while the eigenvectors are informative of the variables associated with this resemblance. We used the approach of [98], which implements the subspace method in a Bayesian framework. Eigenvalues tending towards the number of measured variables would indicate highly similar subspaces. Significance was assessed through a comparison with eigenvalues generated by randomized P matrices (by randomly assigning individuals of each cross types to three groups).

For visualisation purposes, P matrices were projected into a subspace composed by the first three eigenvectors P matrix of the PL × PL offspring by modifying the plotsubspace() function from [89]. Because angles between eigenvectors are necessarily positive, we compared the angles between the first eigenvectors of P with the angles between the first eigenvector of one cross type (depending on the comparison) and the first eigenvector of a “random” P matrix. The simulated matrix was generated by sampling 150 individuals from the two cross types being compared.