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.

Genetic linkage analysis is a recombinant technology used for mapping disease genes on the genome, based on genotypic and phenotypic data collected from families that have affected members. The LOD score is a commonly used statistic in genetic linkage analysis. LOD scores are computed assuming specific values for genetic parameters. However, for complex disorders the specified parameter values are often unknown. One way to address this issue is to maximize the LOD score over all genetic parameters to get a maximum LOD score, or MOD score. Another way is to integrate the LOD score across the genetic parameters to form a posterior probability of linkage, or PPL. Both methods require calculation of large numbers of LOD scores under different sets of parameter values. These calculations may be very time-consuming and can form a significant bottleneck in disease gene mapping. The motivation for this work is to speed up the computation of large numbers of LOD scores in linkage analysis. Instead of the usual LOD calculation where the likelihood of a pedigree under each set of parameter values is computed based on traversing the pedigree, the likelihood of the pedigree is computed here as an algebraic expression that can be optimized and reused. This optimized likelihood expression can be evaluated an arbitrary number of times for LOD scores under different values of the genetic parameters, resulting in much faster speeds. Our initial results show that this approach can speed up the traditional genetic linkage computation by 10~1200 times.