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.