Scaling traits for numerical analyses and de-scaling results such as variance component estimates can be a headache. Here it is how to proceed in a systematic manner.
Assume that we have a trait in a scale of very large variance (milk yield in liters) and other traits in a very small scale (points). We want to scale the traits so that the algorithms are numerically stable. So for a single record we have, say, 7 traits.
We transform each row of record, y, pre-multiplying by a matrix with a scale factor, S. For instance S can contain 1/10 for milk yield (assume that this is the first trait) and 1 otherwise. Or it can contain the inverse of the phenotypic standard deviation of each trait. Scaled records y^s that we fed to, say, airemlf90 is
$latex \bf{y}^s=\bf{Sy} $latex
Then the variances are scaled such that $latex Var({\bf y}^s )={\bf S}Var({\bf y}){\bf S}' $latex. If the original variances of y are, for instance for the genetic component, $latex {\bf G}_0 $latex, they are scaled such that $latex {\bf G}^s_0 $latex =$latex {\bf S G_0 S} $latex'. For instance
G0 = [ 10 2 3
2 4 0
3 0 5]
S=[1/10 0 0
0 1 0
0 0 1]
Then
G0s=S*G0*S’
3×3 Array{Float64,2}:
0.1 0.2 0.3
0.2 4.0 0.0
0.3 0.0 5.0
The 1st col and row go multiplied by 10 so the [1,1] element is multiplied by 100.
To transform back from, say, REML estimates of $latex \bf G_0^s $latex we multiply by the inverse of S:
$latex \bf \hat{G}_0=S^{-1} \hat{G}_0^s (S' )^{-1} $latex
For instance
inv(S)*G0s*inv(S’)
3×3 Array{Float64,2}:
10.0 2.0 3.0
2.0 4.0 0.0
3.0 0.0 5.0
In the particular case of genetic correlation and heritabilities, they are invariant to the transformation. This is because they are multiplied by the same numbers in the numerator and the denominator.
