Consider the covariance matrix V (among traits) for a random effect, e.g. additive genetic effects. Assume we have analyses of subsets of traits, with the th analysis comprising traits. Further, let denote the matrix of estimates of covariance components for the random effect from analysis . The pooled matrix V is constructed by iterating on
Other components of (A.8) are , the weight given to analysis , I, an identity matrix of size , and transformation matrix for analysis , of size . has elements of unity, , if the -th trait overall is the -th trait in analysis , and zero otherwise.
Starting values (V) are obtained initially by decomposing covariance matrices into variances and correlations, and calculating simple averages over individual analyses. If the resulting correlation matrix is not positive definite, it is modified by regressing all eigenvalues towards their mean, choosing a regression factor so that the smallest eigenvalue becomes , and pre- and post-multiplying the diagonal matrix of modified eigenvalues with the matrix of eigenvectors and its transpose, respectively. Using the average variances, V is then obtained from the (modified) average correlation matrix. WOMBAT is set up to perform up to iterates. Convergence is assumed to be reached when
with denoting the th element of V.