Among individual covariance in behavioural traits
Next, we fitted a multivariate mixed model to estimate the among-individual behavioural co-variance matrix (ID ) for the full set of 8 traits. Fixed and random effects on each trait were as described above for the univariate models. ID contains estimates of VI for each trait on the diagonal, with off-diagonal elements corresponding to COVI, the among-individual covariance for each pair of traits. Residual within-individual (co)variance was partitioned to the corresponding matrix R . However residual covariance (COVR) is only identifiable between trait pairs observed simultaneously (i.e. in the same trial), so was fixed to zero between OFT and FST traits. To test the presence of among-trait covariance in ID , we compared the full model to one in which all COVI were fixed to zero by LRT assuming twice the difference in model log-likelihoods as distributed as χ 228.
Having estimated ID , we then wanted to assess whether it was qualitatively consistent with a dominant underlying axis of shy-bold variation as we predicted. To do this we (i) standardised among-individual covariance terms to the more intuitive correlation scale (where, for any pair of traits x,y the among-individual correlation  r I(x,y) = COVI(x,y) / √(V Ix × V Iy)); and (ii) subjected our estimated matrix to eigen decomposition (principal component analysis). Since all traits were transformed such that high values indicated bolder behaviour, we predict correlations should be uniformly positive. We also predict that the leading eigen vector of ID (subsequently referred to asidmax ) will explain a large proportion of among-individual variance and have same-sign loadings on all traits. We used a parametric bootstrap approach, following with a bootstrap sample size of 1000, to generate approximate 95% CI on the eigen values ofID and trait loadings on idmax.