A.4 Approximation of sampling errors

A.4.1 Sampling covariances

At convergence, the inverse of the AI matrix gives estimates of lower bound sampling covariances among the parameters estimated. These are used to approximate sampling errors of covariance components and genetic parameters.

For full rank analyses parameterising to the elements of the Cholesky factors of the corresponding covariance matrices, the AI matrix is obtained by first calculating the AI matrix for the covariance components. This is then transformed to the AI matrix for the parameters to be estimated by pre- and postmultiplying it with the corresponding Jacobian and its transpose, respectively. Full details are given in Meyer and Smith [23]. This implies that sampling covariances among the covariance components can be obtained directly by simply inverting the corresponding AI matrix.

For reduced rank analyses, however, the AI matrix for the parameters to be estimated are calculated directly, as outlined by Meyer and Kirkpatrick [20]. Hence, sampling covariances among the corresponding covariance components need to be approximated from the inverse of the AI matrix. WOMBAT estimates the leading columns of the Cholesky factor (L) of a matrix to obtain a reduced rank estimate, Σ = LL ′ . Let lir  (for i ≤ r  ) denote the non-zero elements of L. The inverse of the AI matrix then gives approximate sampling covariances Cov (lir,ljs)  . The ij− th covariance component, σij  , is

     q∑(i,j)
σij =     lir ljr
     r=1

with q(i,j) = min(i,j,t)  and t  the rank which the estimate of Σ  is set to have. The covariance between two covariances, σij  and σkm  is then

             q(i,j)q(k,m)
Cov (σ  ,σ ) = ∑   ∑   Cov (l l ,l l  )
     ij  kl   r=1 s=1      irjr ksms

Using a first order Taylor series approximation to the product of two variables, this can be approximated as

Cov (σij,σkl) r=1q(i,j) s=1q(k,m) [ljrlmsCov (lir,lks) + ljrlksCov (lir,lms) (A.4)
+lirlmsCov (ljr,lks)+ lirlksCov (ljr,lms)]
A.4 extends readily to two covariance components belonging to different covariance matrices, Σ1  and Σ2  , and their respective Cholesky factors.

A.4.2 Sampling errors of genetic parameters

Let σ denote the vector of covariance components in the model of analysis and V (σ)  its approximate matrix of sampling covariances, obtained as described above (A.4.1). The sampling covariance for any pair of linear functions of σ is then simply

      ′   ′       ′
Cov (w 1σ,w2σ ) = w 1V (σ)w2
(A.5)

with wi  the vector of weights in linear function i  .

Sampling variances of non-linear functions of covariance components are obtained by first approximating the function by a first order Taylor series expansion, and then calculating the variance of the resulting linear function. For example, for a variance ratio

   (  2)
Var  σ12- ≈ [σ42Var(σ21)+ σ41Var(σ22)− 2σ21σ22Cov(σ21,σ22)]∕σ82
     σ2
(A.6)

Similarly, for a correlation

Var(∘σ12---
  σ21σ22)[ 4σ14σ 24 Var(σ 12) + σ122σ 24 Var(σ 12) + σ 122σ 14 Var(σ 22)
4σ12σ12σ 24 Cov(σ 1212) 4σ 12σ14σ 22 Cov(σ 1222)
+2σ122σ 12σ 22 Cov(σ 12 22)](     )
 4σ61σ62 (A.7)