BENDOPT and BENDPDF V1.1 User's Manual
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= B E N D O P T =
=====================
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Fortran-Program for two optional bending approches:
Use of: (i) an "optimized" procedure, or
(ii) a "fixed" bending factor.
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Copyright by A. Essl, Vienna; April, 1996.
Some general remarks:
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For the theoretical background of this program see Essl (1991):
"Choice of an appropriate bending factor using prior knowledge of the
parameters". J.Anim.Breed.Genet. 108, 89-101.
The cited bending approach was originally meant for optimizing index
selection when there are only small-sample parameter estimates available.
But the recommended procedure can be used quite generally for an
appropriate adjustment of any poor genetic variance-covariance (VC) matrix
to get more reliable parameter estimates (although biased, mean squared
errors (MSE) are lower on average; see Essl (1991), page 95).
The use of the "optimized" procedure requires the entry of some priors
in the input file for those heritabilities and genetic correlations
which should be re-estimated by bending.
However, if there is no useful prior knowledge about the desired
parameters available, a bending procedure with a freely chosen
"fixed" bending factor can also be carried through by this program.
For an appropriate choice of that factor in dependence of some sample
properties see Essl (1991), Table 2.
Warning: Useful results from this bending procedure can be expected only
if the available reference matrix (i.e. residual VC-matrix) is positive
definite in a sufficient degree (some problems in this respect may arise
when the data analysis is based on an animal model and the sample size is
very small).
Program input:
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This program is directed as well by the entries of an input-file
as by some additional screen input informations via program dialogue.
The common form of the input file (with freely chosen name) must be
as following (no specific format of all figures is required):
---------------------------------------------------------------------------
First line: Title of the problem (mandatory, max. length 79 col.).
Second line: Dimension of VC-matrix which should be adjusted by bending
(equal to the number of traits involved).
Next lines: - Lower triangle of the estimated additive genetic VC-matrix
(animal model) or "variance among" VC-matrix (Henderson 3).
- Lower triangle of the respective residual VC-matrix
(= reverence matrix for bending).
- Optional: Lower triangle of the matrix with priors for all
heritabilities (diagonals) and genetic correlations (off
diagonals) in question (mandatory for "optimized" procedure).
---------------------------------------------------------------------------
All three triangles have to be written 'line by line'.
An example for a possible input file is given below (see file "in.opt"):
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Test Example for program "bendopt.for"
3
.00151
-.00312 .00797
.01507 .11564 14.05649
.12961
.00745 .61631
.22806 .21539 103.05261
.05
-.2 .05
.0 .1 .25
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Coefficient of relationsship
----------------------------
The approriate "coefficient of relationship" (cr-value) is asked via
program dialogue and is equivalent to that relationship-constant
by which an "among" VC-matrix has to be divided to get an estimate of
the full additive genetic VC-matrix (e.g. halfsib-analyses: cr = 0.25).
Any procedure on the base of an "animal model" delivers an estimate of
the full additive genetic VC-matrix so that cr = 1 in all such situations.
Choice of different trait weights (optional):
---------------------------------------------
According the initial objective of the underlying bending approach
(maximizing the response to index selection) the "optimized" bending
procedure allows also the use of different trait weights. In this context
these weights are to be interpreted as the relative economic
values of a genetic standard deviation of the respective traits.
Program output:
===============
The results (including all relevant input terms) are written on an output
file whose name can be freely chosen by the user (see file "out.opt"
for example, which is according to the input file "in.opt" using the
"optimized" procedure and a coefficient of relationship (cr) of 0.25.
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= B E N D P D F =
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Fortran-Program for a stepwise bending procedure applied
to (e.g. negative-definite) variance-covariance matrices
until their lowest eigenvalue exceeds a preset limit
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Copyright by A. Essl, Vienna; January, 1996.
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This fortran program can be used either stand-alone or
- after some suitable adaptions by the user -
for instance as a subroutine in simulation programs requiring
positive-definite variance-covariance matrices in multiple
trait approaches.
Program input
=============
If the program is used stand-alone, the e.g. negative-definite
variance-covariance (VC) matrix (e.g. a "between" VC-matrix) and
a positive-definite reference VC-matrix to which the
negative-definite VC-matrix should be bended (e.g. an according
"within" VC-matrix) must be written on an input file, which name
is asked via program dialogue.
Warning: This bending procedure will only work satisfactoriely
if the used reference VC-matrix is distinct positive definite
(the eigenvalues of the used reference VC-matrix are calculated
and presented in the output file!).
If there is no proper reference VC-matrix available an identity matrix
of suitable dimension can be used instead. This identity matrix will
be generated by the program itself and don't need to be written on
the input file (selection of this possibility via program dialog).
The general form of the input file must be as following:
(no specific format of all figures is required)
-----------------------------------------------------------------------
First line: Title of the problem (mandatory, max. length 79 col.).
Second line: Dimension of VC-matrix which should be adjusted by bending.
Next lines: - Lower triangle of the VC-matrix which should be bended.
- Lower triangle of the reference VC-matrix (only necessary
if the reference VC-matrix is not an identity matrix).
Both triangles have to be written 'line by line'.
An example for an possible input file is given below (see file "in.pdf"):
-------------------------------------------------------------------------
Test Example for program bendpdf.for
3
1.3911
-.2646 1.0572
-.9265 -.8812 1.3219
1.0026
.0819 .9215
-.1097 -.1806 .8871
-------------------------------------------------------------------------
Input of preset limit of lowest eigenvalue
------------------------------------------
The preset limit of the lowest eigenvalue for the VC-matrix after
bending is asked via program dialogue.
For an appropriate choice of this value it should be noticed that
there are theoretical limits (the lowest eigenvalue of a bended VC-matrix
can never exeed the average of all eigenvalues of the original VC-matrix;
and it must not be negative to get for instance, the original VC-matrix
positive-definite).
Dimension features
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If the dimension of the VC-matrix exceeds 30, the concerning
parameter statement in the program and the dimension entries in the
subroutines "MINV" and "MPRO" have to be adapted correspondingly.
Program output
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The results (eigenvalues of original VC-matrix and corresponding
reference matrix as well as the adjusted values of VC-matrix after
bending with new eigenvalues) are written on an output file (name is
asked via program dialogue) like "out.pdf", which is according to the
input test example given above.
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