WOMBAT – A program for Mixed Model Analyses by Restricted Maximum Likelihood

### FAQ: Accuracy of breeding values

I am using WOMBAT to obtain breeding values for animals. How can I get the corresponding accuracies?

This depends on what exactly you are doing. WOMBAT does not provide approximations as needed for large models - however, if the inverse of the coefficient matrix can be calculated, there are several scenarios and ways to obtain the standard errors or sampling variances which you can then use to calculate the corresponding accuracies.

1. If you have used the −−blup runtime option, WOMBAT calculates the inverse of the coefficient matrix in the mixed model equations to obtain solutions for all fixed and random effects fitted, and provides corresponding standard errors, obtained from the diagonal elements of the inverse coefficient matrix.

2. If you have obtained solution for animals' genetic effects as a by-product of a run to estimate variance components, chances are that you have used the AI-REML algorithm (the default) in the final iterates. This uses a computing strategy which does not require the inverse of the coefficient matrix, and standard errors for fixed and random effects fitted are thus not available automatically'.
• Simplest: You can add the option FORCE-SE in a SPECIAL block at the end of the parameter file. For cases where standard errors are not obtained automatically, this will force WOMBAT to invert the coefficient matrix at the end of the analysis and report standard errors, for both fixed and random effects fitted. NB: This can be rather time consuming!
SPECIAL
FORCE-SE
END
• Standard errors for the effects fitted can be obtained by carrying out a continuation run, specifying one round of the EM-algorithm, i.e. wombat −c −−emalg1.
Note though that this algorithm does not provide sampling errors of parameter estimates and will overwrite the file SumEstimates.out, i.e. you may wish to rename your results file before this additional run.
• Alternatively, you could use run options −c −−blup for this purpose; note though that −−blup switches off pruning of pedigrees, i.e. that this may cause WOMBAT to recalculate inbreeding coefficients and the inverse of the numerator relationship matrix.

3. If you have used WOMBAT to set up and solve a set of mixed model equations iteratively, chances are that there are too many equations to invert the corresponding coefficient matrix. You may then want/need to approximate accuracies.
• Most approximation procedures described in the literature utilize selected parts of the coefficient matrix.
• Hence, to facilitate such calculations, WOMBAT provides the runtime option −−mmeout which acts like −−solvit but writes out the complete mixed model equations (non-zero elements of one triangle of the coefficient matrix only) to file (there is no equivalent for −−s1step).
• WOMBAT does not provide any options to do this type of calculations for you.

P.S. You may encounter a column with heading xxxxx in your file with solutions and standard errors for random effects - the content of which looks like an accuracy. Yes, for some (simple) models this gives estimates of the accuracy. However, for other (more complicated) models the results can be wrong (bug) – so don't use this column until you have convinced yourself (by calculating what the accuracy should be for a few levels and making sure the values agree) that the numbers are indeed correct.

P.P.S When calculating accuracies, make sure you account for inbreeding - otherwise you may underestimate it and may even get a negative number (which is clearly wrong)!

Please check your favourite textbook/lecture notes/paper for documentation on how to calculate accuracies once you have an (approximate) inverse of the coefficient matrix in the mixed model equations.

@ARTICLE{crh75,
author = {Henderson, C. R.},
title = {Best linear unbiased estimation and prediction under a selection model},
journal = {Biometrics},
volume = {31},
year = {1975},
pages = {423--447}
}

@ARTICLE{bruce03,
author = {Tier, B. and Meyer, K.},
title = {Approximating prediction error covariances in multiple-trait and random regression models},
journal = {Journal of Animal Breeding and Genetics},
volume = {121},
year = {2004},
pages = {77--89},
doi = {10.1111/j.1439-0388.2003.00444.x}
}

@article{hickey2009,
title={Estimation of prediction error variances via Monte Carlo sampling methods using different formulations of the prediction error variance},
author={Hickey, J.M. and Veerkamp, R.F. and Calus, M.P.L. and Mulder, H.A. and Thompson, R.},
journal={Genet Sel Evol},
volume={41},
pages={23},
year={2009}
}`