A.6 Iterative summation of expanded part matrices

Consider the q × q  covariance matrix V (among q  traits) for a random effect, e.g. additive genetic effects. Assume we have S  analyses of subsets of traits, with the s− th analysis comprising ks  traits. Further, let Cs  denote the ks × ks  matrix of estimates of covariance components for the random effect from analysis s  . The pooled matrix V is constructed by iterating on

       S
Vt+1 = ∑  w {Vt (P P ′VtP  P ′)− P C  P ′(P P ′VtP P  ′)− Vt
       s=1  s      s  s    s s    s  s s   s s     s s
                                                     }   S
                      + [(I − P P ′)(Vt)−1(I − P P ′)]−  ∕∑  w
                              s s              s s      s=1  s
(A.8)
until Vt   and Vt+1    are virtually identical, with Vt   the value of V for the t− th iterate.

Other components of (A.8) are w
 s  , the weight given to analysis s  , I, an identity matrix of size q× q  , and transformation matrix P
  s  for analysis s  , of size q× k
    s  . P
 s  has k
 s  elements of unity, p  = 1
 ij  , if the i  -th trait overall is the j  -th trait in analysis s  , and zero otherwise.

Starting values (V0   ) are obtained initially by decomposing covariance matrices Cs  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 10−6  , and pre- and post-multiplying the diagonal matrix of modified eigenvalues with the matrix of eigenvectors and its transpose, respectively. Using the average variances, V0    is then obtained from the (modified) average correlation matrix. WOMBAT is set up to perform up to 100000  iterates. Convergence is assumed to be reached when

---2--- ∑q ∑q ( t+1    t)2    −7
q(q+ 1)        vij  − vij  ≤ 10
        i=1 j=i

with  t
vij  denoting the ij− th element of Vt  .