Approximation of sampling errors

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 [19]. 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 [18]. 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 $q(ij) = min (i,j,t)$ 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 ≈∑ _r=1^q(i,j) ∑ _s=1^q(k,m)		(A.4)

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≈	4σ₁⁴σ₂⁴ Var(σ₁₂) + σ₁₂²σ₂⁴ Var(σ₁²) + σ₁₂²σ₁⁴ Var(σ₂²)
	− 4σ₁₂σ₁²σ₂⁴ Cov(σ₁₂,σ₁²) − 4σ₁₂σ₁⁴σ₂² Cov(σ₁₂,σ₂²)
	+2σ₁₂²σ₁²σ₂² Cov(σ₁²,σ₂²)∕	(A.7)

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