GIV  General Inverse Variance structure
Function
For the more general situation where the pedigree based inverse
relationship matrix is not the appropriate/required matrix, the user can
provide a particular general inverse variance (
GIV
) matrix explicitly
in a
.giv
file.
The user can prepare a
.giv
file containing this matrix and use it in the analysis.
Alternatively, the user can prepare the relationship matrix in a
.grm
file and ASReml will invert it to form the GIV matrix.
The syntax for specifying a G matrix file (say name.grm) or
the G inverse file (say name.giv)
is
name.grm [ !SKIP n ] [ !DENSEGRM [ o] ]
[ !GROUP n ] [ !ND  !PSD  !NSD ]
or
name.giv [ !SKIP n ] [ !DENSEGIV [ o] ]
[ !GROUP n ] [ !SAVEGIV f ]
the named file must have a .giv or .grm extension,
the G (inverse) files must be specified on the line(s)
immediately prior to the data file line after any pedigree file,
up to 98 G (inverse) matrices may be defined,
the file must be in SPARSE format
unless the !DENSE qualifier is specified,
a dense format file has the whole matrix presented lower triangle
rowwise, with each row beginning on a new line (rows may be split over several lines),
a sparse format file must be free format with three numbers per line, namely
row column value
defining the lower triangle rowwise of the matrix,
the file must be sorted column within row,
every diagonal element must be represented; missing offdiagonal elements are assumed to be zero cells,
1 1 1
2 2 1
3 3 1
4 4 1
5 5 1.0666667
6 5 0.2666667
6 6 1.0666667
7 7 1.0666667
8 7 0.2666667
8 8 1.0666667
9 9 1.0666667
10 9 0.2666667
10 10 1.0666667
11 11 1.0666667
12 11 0.2666667
12 12 1.0666667
The
.giv
file presented above gives the following G inverse matrix
1  0  0  0  0  0  0  0  0  0  0  0

0  1  0  0  0  0  0  0  0  0  0  0

0  0  1  0  0  0  0  0  0  0  0  0

0  0  0  1  0  0  0  0  0  0  0  0

0  0  0  0  1.067  .267  0  0  0  0  0  0

0  0  0  0  .267  1.067  0  0  0  0  0  0

0  0  0  0  0  0  1.067  .267  0  0  0  0

0  0  0  0  0  0  .267  1.067  0  0  0  0

0  0  0  0  0  0  0  0  1.067  .267  0  0

0  0  0  0  0  0  0  0  .267  1.067  0  0

0  0  0  0  0  0  0  0  0  0  1.067  .267

0  0  0  0  0  0  0  0  0  0  .267  1.067 
the
!SKIP n
qualifier tells ASReml to skip n header lines in the file.
!GROUP n
allows adjustment of degrees of freedom when the GIV matrix
incorporates fixed effects.
Positive and Negative Semidefinite matrices
If the file has a .grm file extension, ASReml will invert it.
If it is not Positive Definite, the job will abort unless an
appropriate qualifier !ND, !PSD or !NSD is supplied.
!ND ( !NSD) allows the matrix to be Negative (Semi)Definite.
!PSD allows the matrix to be Positive SemiDefinite.
If the matrix is Negative (Semi)Definite, the iteration sequence may fail
as some parameter values will generate 'Negative Residual Sum of Squares'.
If SemiDefinite is permitted and the matrix is singular,
ASReml forms an expanded 'Singular' representation of the inverse which
allows the REML algorithm to proceeds. The effects for the extra equations
have no natural interpretation.
Using the GIV matrix
The GIV matrix is used by associating it with a factor in the model.
number and order of the rows must agree with the
size and order of the associated factor.
The
.giv
file can be associated with a factor in two ways:
the
first is to declare a G structure for the model term and to refer to
the
.giv
file with the corresponding identifier
GIV1, GIV2, GIV3 ...
for example,
animal 1
animal 0 GIV1 0.12
for a onedimensional structure put the scale parameter (
0.12
in this case) after the
GIVg
identifier,
site.variety 2
site 0 CORUH 0.5
8*1.5
variety 0 GIV1
for a twodimensional structure.
the second is for onedimensional structures; in this case the
.giv
structure can be directly associated with the term using the
giv(factor,i)
model function which associates the
i th
.giv
file with factor factor, for example,
giv(animal,1) 0.12
is equivalent to the first of the preceding examples.
The !GROUP qualifier
A !GROUP n qualifier has been added to the GIV matrix
specification line to allow for adjustment of any fixed degrees of freedom
incorporated into the GIV matrix. This enables a GIV
matrix generated using a pedigree with the !GROUPS
qualifier to be used again as a GIV matrix.
The value of the argument is the number of degrees of freedom
fitted by the GIV matrix which will normally be the number of groups.
When groups are constrained, then it will be the number of groups
less number of constraints. For example, if the pedigree file
qualified by !GROUPS 7 begins
A 0 0
B 0 0
C 0 0
ABC 0 0 # ABC is not present in the subsequent pedigree lines
D 0 0
E 0 0
DE 0 0 # DE is not present in the subsequent pedigree lines
there are actually only 5 genetic groups and two constraints
so that the fixed effects for A, B and
C
sum to zero, and for D
and E sum to zero
leaving only 3 fixed degrees of freedom fitted.
Therefore if the A inverse for this pedigree was
saved, and subsequently used as a GIV matrix, it should be
declared as !GROUPDF 3.
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