Approximation of reliabilities for random-regression single-step genomic best linear unbiased predictor models

Random-regression models (RRM) are used in national genetic evaluations for longitudinal traits. The outputs of RRM are an index based on random regression coefficients and its reliability. The reliabilities are obtained from the inverse of the coefficient matrix of mixed model equations (MME). The reliabilities must be approximated for large data sets because it is impossible to invert the MME. There is no extensive literature on methods to approximate the reliabilities of RRM when genomic information is included by single-step GBLUP. We developed an algorithm to approximate such reliabilities. Our method combines the reliability of the index without genomic information with the reliability of a GBLUP model in terms of effective record contributions. We tested our algorithm in the 3-lactation model for milk yield from the Czech Republic. The data had 30 million test-day records, 2.5 million animals in the pedigree, and 54,000 genotyped animals. The correlation between our approximation and the reliabilities obtained from the inversion of the MME was 0.98, and the slope and intercept of the regression were 0.91 and 0.02, respectively. The elapsed time to approximate the reliabilities for the Czech data was 21 min.