coefficients is the list of coefficients for the linear function.
Linear combinations of components
First ASReml extracts the variance components from the .asr file
and their variance matrix from the .vvp file.
Each linear function formed by an F line is added to the list of
components. Thus, the number of coefficients increases by one each
F line.
We seek to calculate k + c'v, cov(c'v, v)
and var(c'v) where v is the vector of existing variance
components, c is the vector of coefficients for the linear
function and k is an optional offset which is usually omitted
but would be 1 to represent the residual variance in a probit analysis and
3.289 to represent the residual variance in a logit analysis.
The general form of the directive is
F
label a + b*c_b+c+d+m*k
where a,
b,
c
and d are subscripts to existing
components v_a, v_b, v_c and v_d and
c_b is a multiplier
for v_b.
m is a number bigger than the current
length of v to flag the special case of
adding the offset k.
Where matrices are to be combined the form
F
label a:b * k + c:d
can be used, as in the Coopworth data example, see UserGuide.
For example, after fitting a model like
ywt ~ mu sex age !r sire
ASReml will have estimated two variance components, Sire and Residual
respectively. To calculate heritability, the .pin file
is
F phenvar 1 + 2 # pheno var
F genvar 1 * 4 # geno var
H herit 4 3 # heritability
F phenvar 1 + 2
gives a third component which is the sum of the variance components, that is, the phenotypic
variance, and
F genvar 1 * 4
gives a fourth component which is the sire variance component
multiplied by 4, that is, the genotypic variance.
Heritability
Heritabilities are requested by lines in the .pin file
beginning with an H.
The specific form of the directive in this case is
H label n d
This calculates v_n/v_d and its standard error
where n
and d are integers
pointing to the variance components to be used as the
numerator and denominator respectively in the heritability calculation.
In the example above
H herit 4 3
calculates the heritability as the ratio of component 4 (from second line
of .pin) to component 3
(from first line of .pin), that is,
genetic variance / phenotypic variance.
Correlation
Correlations are requested by lines in the .pin file
beginning with an R.
The specific form of the directive is
R label a ab b
This calculates the correlation r =
v_ab/sqrt(v_a v_b) and the associated standard
error. a,
b and
ab are integers indicating the
position
of the components to be used.
Alternatively,
R label a:n
calculates all the correlations
in the
lower triangular row-wise matrix represented by components a to
n and the associated standard errors.
For example, after fitting a trivariate sire model
gfw fd lwt ~ Trait Tr.sex !r Tr.sire,
the Residual matrix is in positions 1:6 and the Sire matrix
is in positions 7:12. A .pin file is
F phenvar 1:6 + 7:12 # defines 13:18
F addvar 7:12*4 # defines 19:24
H heritgfw 19 13
H heritfd 21 15
H heritlwt 24 18
R phencorrAB 13 14 15
R phencorrAC 13 16 18
R phencorrBC 15 17 18
R gencorr 7:12
In the example
R phencorrAB 13 14 15
calculates the phenotypic correlation for the first two traits
from components 13, 14 and 15, and
R gencorr 7:12
calculates the set of three genetic correlations.
Another example
This example relates to the bivariate sire
model in bsiremod.as:
Bivariate sire model
sire !I
ywt fat
bsiremod.asd
ywt fat ~ Trait !r Trait.sire
1 2 1
0 # ASReml will count units
Trait 0 US
3*0
Trait.sire
2
Trait 0 US
3*0
sire
which estimates six variance components
Source Model terms Gamma Component Comp/SE % C
Residual UnStruct 1 26.2197 26.2197 18.01 0 U
Residual UnStruct 1 2.85090 2.85090 9.55 0 U
Residual UnStruct 2 1.71556 1.71556 18.00 0 U
Tr.sire UnStruct 1 16.5262 16.5262 2.69 0 U
Tr.sire UnStruct 1 1.14422 1.14422 1.94 0 U
Tr.sire UnStruct 2 0.132847 0.132847 1.88 0 U
A .pin file to calculate heritabilities and correlations is
F phenvar 1:3 + 4:6
F addvar 4:6 * 4
H heritA 10 7
H heritB 12 9
R phencorr 7 8 9
R gencorr 4:6
Labelling the initial six components
| V1 | error variance for ywt
|
| V2 | error covariance for ywt
and fat
|
| V3 | error variance for fat
|
| V4 | sire variance component for ywt
|
| V5 | sire covariance for ywt
and fat
|
| V6 | sire variance for fat
|
F phenvar 1:3 + 4:6
calculates
| V7 | phenotypic variance for ywt
|
| V8 | phenotypic covariance for ywt
and fat
|
| V9 | phenotypic variance for fat
|
F addvar 4:6 * 4
calculates
| V10 | additive genetic variance for ywt
|
| V11 | additive genetic covariance for ywt
and fat
|
| V12 | additive genetic variance for fat
|
H heritA 10 7
calculates the heritabilty for ywt
H heritB 12 9
calculates the heritabilty for fat
R phencorr 7 8 9
calculates the phenotypic correlation
R gencorr 4:6
calculates the genetic correlation
The result is:
7 phenvar 1 42.75 6.297
8 phenvar 2 3.995 0.6761
9 phenvar 3 1.848 0.1183
10 addvar 4 66.10 24.58
11 addvar 5 4.577 2.354
12 addvar 6 0.5314 0.2831
h2ywt = addvar 10/phenvar 7= 1.5465 0.3574
h2fat = addvar 12/phenvar 9= 0.2875 0.1430
phencorr = phenvar /SQR[phenvar *phenvar ]= 0.4495 0.0483
gencor 2 1 = Tr.si 5/SQR[Tr.si 4*Tr.si 6]= 0.7722 0.1537
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