Visitor number 11003 since June 13, 2007.

- Quantitative genetic theory for continuous traits assumes a multivariate normal distribution with (semi-) positive definite covariance or correlation matrix, i.e. a matrix which has non-negative eigenvalues.
- However,
*estimates*of these matrices frequently have negative eigenvalues. - 'Bending' and related techniques which modify the eigenvalues of such matrices are widely used to 'correct' covariance matrices out of bounds of the parameter space.
- However, estimates of individual covariances or correlations often originate from different analyses and can be based on greatly differing amounts of information.
- 'Bending' etc. do not allow for differential weighing of individual estimates.

PDMATRIX is a collection of - currently two - programs to help make invalid, i.e. non-positive definite covariance matrices positive (semi-) definite, allowing individual estimates to be given differential emphasis. The programs are :

- FLBEND written by John Henshall (
John.Henshall@ csiro.au)
- FLBEND modifies a non-positive definite matrix comprised of pair wise or block covariance estimates, where estimates may or may not be based on experimental results.
- FLBEND finds a positive definite covariance matrix at 'minimum distance' from a non-positive definite matrix, from of pair- or or block-wise covariance estimates. The distance function is the weighted Euclidean norm, i.e. the elementwise sum of squared differences, weighted by standard error estimates. Finding the positive definite matrix is a constrained minimisation problem with the boundary of the constraint not a simple function. A reparameterisation is used to simplify the minimisation problem.
- FLBEND has been used to obtain covariance matrices for use in Australian sheep genetic evaluations. Variance components for groups of 42 traits were combined into genetic and residual matrices which were close to, but not positive definite. Using FLBEND, positive definite matrices were obtained in which the largest deviations from the composite matrices were for those correlations with highest standard errors.
- ITSUMCOV written by Karin Meyer (kmeyer.agbu@ gmail.com)
- ITSUMCOV deals with specific case of multiple, multivariate analyses of different subsets of a number of traits. Say we have q traits in total, and S analyses involving k traits each.
- ITSUMCOV implements the 'iterative summing of expanded part matrices' approach of Mantysaari (1999; 50th Annual Meeting Europ. Ass. Anim. Prod.), modified to allow for differential weights for different analyses.
- ITSUMCOV has been used to combine estimates from 231 bivariate analyses for 22 traits recorded for Hereford cattle, resulting in an estimate of the complete genetic covariance matrix for traits in BREEDPLAN, the Australian genetic evaluation scheme for beef cattle.
- New: Pooling of estimates from part analyses using a penalized likelihood approach - provides an alternative with better properties; see: http://didgeridoo.une.edu.au/km/pool.php

Programs are written in standard Fortran 95 and self-contained, except for some public domain random number generation routines used in FLBEND. Compiled versions are available for Linux and Compaq Alpha stations.

- While every effort
has been made to ensure that programs in PDMATRIX do what they
claims to do, there is absolutely
**no**guarantee for its correctness. - You are using PDMATRIX entirely at your own risk, and
there is
**no**user-support service.

Henshall, J.M. and Meyer, K. (2002). "PDMATRIX - Programs to make matrices positive definite".
*Seventh World Congress on Genetics Applied to Livestock Production,
Montpellier, France, August 19-23, 2002*, Communication No. 28-12.

- FLBEND has been packaged into a Unix 'tape archive' and compressed using 'gzip'. It can be downloaded as : flbend.tar.gz (305 KB, 1425 downloads since 18/6/2007), last update: 05/03/05.
- Unpacking this file will create
a directory
**FLBEND**with 4 subdirectories :**doc**contains the file**flbend.txt**with the documentation for FLBEND and a worked example.**source**contains the Fortran source code for FLBEND.**bin**contains the compiled exectables for FLBEND :**flbend_linux**is a compiled version for a PC running Linux.**flbend_compaq64**is a compiled version for a Compaq True64 work station

**example**contains the files for an example run of FLBEND.

- ITSUMCOV has been packaged into a Unix 'tape archive' and compressed using 'gzip'. It can be downloaded as : itsumcov.tar.gz (583 KB, 1088 downloads since 18/6/2007), last update: 05/03/05.
- Unpacking this file will create
a directory
**ITSUMCOV**with 4 subdirectories :**doc**contains the file**itsumcov.txt**with the documentation for ITSUMCOV.**source**the Fortran source code for ITSUMCOV :**itsum.f**

**bin**contains the compiled exectables for ITSUMCOV :**itsumcov_linux**is a compiled version for a PC running Linux.**itsumcov_compaq64**is a compiled version for a Compaq True64 work station

**example**contains the files for an example run of ITSUMCOV.