MOTIVATION: The amount of genomic data is increasing exponentially. Using many genotyped and phenotyped individuals for genomic prediction is appealing yet challenging. RESULTS: We present SLEMM (short for Stochastic-Lanczos-Expedited Mixed Models), a new software tool, to address the computational challenge. SLEMM builds on an efficient implementation of the stochastic Lanczos algorithm for REML in a framework of mixed models. We further implement SNP weighting in SLEMM to improve its predictions. Extensive analyses on seven public datasets, covering 19 polygenic traits in three plant and three livestock species, showed that SLEMM with SNP weighting had overall the best predictive ability among a variety of genomic prediction methods including GCTA's empirical BLUP, BayesR, KAML, and LDAK's BOLT and BayesR models. We also compared the methods using nine dairy traits of ∼300k genotyped cows. All had overall similar prediction accuracies, except that KAML failed to process the data. Additional simulation analyses on up to 3 million individuals and 1 million SNPs showed that SLEMM was advantageous over counterparts as for computational performance. Overall, SLEMM can do million-scale genomic predictions with an accuracy comparable to BayesR. AVAILABILITY AND IMPLEMENTATION: The software is available at https://github.com/jiang18/slemm.

Abstract Background Traditional selection in livestock and crops focuses on additive genetic values or breeding values of the individuals. While traditional selection utilizes variation between individuals, differences between gametes within individuals have been less frequently exploited in selection programs. With the successful implementation of genomic selection in livestock and crops, estimation and selection for gametic variation is becoming possible. Results The gamevar.f90 software is designed to estimate individual-level variance of genetic values of gametes for complex traits in large populations. The software estimates the (co)variances of gametic diversity as well as other diversity parameters that are useful for selection programs and mating designs. The calculation is carried out chromosome by chromosome and can be easily parallelized. The gamevar.f90 program is written in Fortran with efficient computing algorithms in a user-friendly software package with easily-handled input and output files. Finally, we applied the program to estimate gametic variance for hundreds of bulls for lifetime net merit, productive life, and livability. The RPTA (relative predicted transmitting ability), assuming a future selection intensity (if) of 1.5, showed larger variance than GEBV/2, indicating that greater future genetic gains can be obtained using an index that includes gametic variances. We also used the relative coefficient of variation to estimate with 95% confidence the sample sizes required to observe 90% variability of the progeny for lifetime net merit (or to allow at maximum 10% of change in the EBV predicted from progeny data). Conclusions Collectively, we develop an efficient computer program package, gamevar.f90, for estimating gametic variance for large numbers of individuals. The novel information on gametic variation will be useful in future animal and crop breeding programs.