Background

A random multiple-regression model that simultaneously fit all allele substitution effects for additive markers or haplotypes as uncorrelated random effects was proposed for Best Linear Unbiased Prediction, using whole-genome data. Leave-one-out cross validation can be used to quantify the predictive ability of a statistical model.

Methods

Naive application of Leave-one-out cross validation is computationally intensive because the training and validation analyses need to be repeated n times, once for each observation. Efficient Leave-one-out cross validation strategies are presented here, requiring little more effort than a single analysis.

Results

Efficient Leave-one-out cross validation strategies is 786 times faster than the naive application for a simulated dataset with 1,000 observations and 10,000 markers and 99 times faster with 1,000 observations and 100 markers. These efficiencies relative to the naive approach using the same model will increase with increases in the number of observations.

Conclusions

Efficient Leave-one-out cross validation strategies are presented here, requiring little more effort than a single analysis.

Background

The mixed linear model employed for genomic best linear unbiased prediction (GBLUP) includes the breeding value for each animal as a random effect that has a mean of zero and a covariance matrix proportional to the genomic relationship matrix ([Formula: see text]), where the inverse of [Formula: see text] is required to set up the usual mixed model equations (MME). When only some animals have genomic information, genomic predictions can be obtained by an extension known as single-step GBLUP, where the covariance matrix of breeding values is constructed by combining the pedigree-based additive relationship matrix with [Formula: see text]. The inverse of the combined relationship matrix can be obtained efficiently, provided [Formula: see text] can be inverted. In some livestock species, however, the number [Formula: see text] of animals with genomic information exceeds the number of marker covariates used to compute [Formula: see text], and this results in a singular [Formula: see text]. For such a case, an efficient and exact method to obtain GBLUP and single-step GBLUP is presented here.

Results

Exact methods are already available to obtain GBLUP when [Formula: see text] is singular, but these require working with large dense matrices. Another approach is to modify [Formula: see text] to make it nonsingular by adding a small value to all its diagonals or regressing it towards the pedigree-based relationship matrix. This, however, results in the inverse of [Formula: see text] being dense and difficult to compute as [Formula: see text] grows. The approach presented here recognizes that the number r of linearly independent genomic breeding values cannot exceed the number of marker covariates, and the mixed linear model used here for genomic prediction only fits these r linearly independent breeding values as random effects.

Conclusions

The exact method presented here was compared to Apy-GBLUP and to Apy single-step GBLUP, both of which are approximate methods that use a modified [Formula: see text] that has a sparse inverse which can be computed efficiently. In a small numerical example, predictions from the exact approach and Apy were almost identical, but the MME from Apy had a condition number about 1000 times larger than that from the exact approach, indicating ill-conditioning of the MME from Apy. The practical application of exact SSGBLUP is not more difficult than implementation of Apy.

Background

In whole-genome analyses, the number p of marker covariates is often much larger than the number n of observations. Bayesian multiple regression models are widely used in genomic selection to address this problem of [Formula: see text] The primary difference between these models is the prior assumed for the effects of the covariates. Usually in the BayesB method, a Metropolis-Hastings (MH) algorithm is used to jointly sample the marker effect and the locus-specific variance, which may make BayesB computationally intensive. In this paper, we show how the Gibbs sampler without the MH algorithm can be used for the BayesB method.

Results

We consider three different versions of the Gibbs sampler to sample the marker effect and locus-specific variance for each locus. Among the Gibbs samplers that were considered, the most efficient sampler is about 2.1 times as efficient as the MH algorithm proposed by Meuwissen et al. and 1.7 times as efficient as that proposed by Habier et al.

Conclusions

The three Gibbs samplers presented here were twice as efficient as Metropolis-Hastings samplers and gave virtually the same results.

This study aims at characterizing the asymptotic behavior of genomic prediction R2 as the size of the reference population increases for common or rare QTL alleles through simulations. Haplotypes derived from whole-genome sequence of 85 Caucasian individuals from the 1,000 Genomes Project were used to simulate random mating in a population of 10,000 individuals for at least 100 generations to create the LD structure in humans for a large number of individuals. To reduce computational demands, only SNPs within a 0.1M region of each of the first 5 chromosomes were used in simulations, and therefore, the total genome length simulated was 0.5M. When the genome length is 30M, to get the same genomic prediction R2 as with a 0.5M genome would require a reference population 60 fold larger. Three scenarios were considered varying in minor allele frequency distributions of markers and QTL, for h2 = 0.8 resembling height in humans. Total number of markers was 4,200 and QTL were 70 for each scenario. In this study, we considered the prediction accuracy in terms of an estimability problem, and thereby provided an upper bound for reliability of prediction, and thus, for prediction R2. Genomic prediction methods GBLUP, BayesB and BayesC were compared. Our results imply that for human height variable selection methods BayesB and BayesC applied to a 30M genome have no advantage over GBLUP when the size of reference population was small (<6,000 individuals), but are superior as more individuals are included in the reference population. All methods become asymptotically equivalent in terms of prediction R2, which approaches genomic heritability when the size of the reference population reaches 480,000 individuals.

Background

Two types of models have been used for single-step genomic prediction and genome-wide association studies that include phenotypes from both genotyped animals and their non-genotyped relatives. The two types are breeding value models (BVM) that fit breeding values explicitly and marker effects models (MEM) that express the breeding values in terms of the effects of observed or imputed genotypes. MEM can accommodate a wider class of analyses, including variable selection or mixture model analyses. The order of the equations that need to be solved and the inverses required in their construction vary widely, and thus the computational effort required depends upon the size of the pedigree, the number of genotyped animals and the number of loci.

Theory

We present computational strategies to avoid storing large, dense blocks of the MME that involve imputed genotypes. Furthermore, we present a hybrid model that fits a MEM for animals with observed genotypes and a BVM for those without genotypes. The hybrid model is computationally attractive for pedigree files containing millions of animals with a large proportion of those being genotyped.

Application

We demonstrate the practicality on both the original MEM and the hybrid model using real data with 6,179,960 animals in the pedigree with 4,934,101 phenotypes and 31,453 animals genotyped at 40,214 informative loci. To complete a single-trait analysis on a desk-top computer with four graphics cards required about 3 h using the hybrid model to obtain both preconditioned conjugate gradient solutions and 42,000 Markov chain Monte-Carlo (MCMC) samples of breeding values, which allowed making inferences from posterior means, variances and covariances. The MCMC sampling required one quarter of the effort when the hybrid model was used compared to the published MEM.

Conclusions

We present a hybrid model that fits a MEM for animals with genotypes and a BVM for those without genotypes. Its practicality and considerable reduction in computing effort was demonstrated. This model can readily be extended to accommodate multiple traits, multiple breeds, maternal effects, and additional random effects such as polygenic residual effects.

State-of-the-art health care includes genome sequencing of the patient to identify genetic variants that contribute to either the cause of their malady or variants that can be targeted to improve treatment. The goal was to introduce state-of-the-art health care to cats using genomics and a precision medicine approach. To test the feasibility of a precision medicine approach in domestic cats, a single cat that presented to the University of Missouri, Veterinary Health Center with an undiagnosed neurologic disease was whole-genome sequenced. The DNA variants from the cat were compared to the DNA variant database produced by the 99 Lives Cat Genome Sequencing Consortium. Approximately 25× genomic coverage was produced for the cat. A predicted p.H441P missense mutation was identified in NPC1, the gene causing Niemann-Pick type C1 on cat chromosome D3.47456793 caused by an adenine-to-cytosine transversion, c.1322A>C. The cat was homozygous for the variant. The variant was not identified in any other 73 domestic and 9 wild felids in the sequence database or 190 additionally genotyped cats of various breeds. The successful effort suggested precision medicine is feasible for cats and other undiagnosed cats may benefit from a genomic analysis approach. The 99 Lives DNA variant database was sufficient but would benefit from additional cat sequences. Other cats with the mutation may be identified and could be introduced as a new biomedical model for NPC1. A genetic test could eliminate the disease variant from the population.

Tabby patterns of fur coats are defining characteristics in wild and domestic felids. Historically, three autosomal alleles at one locus (Tabby): Abyssinian (T^a ; a.k.a. ticked), mackerel (T^m ; a.k.a. striped) and blotched (t^b ; a.k.a. classic, blotched) were thought to control these patterns in domestic cats and their breeds. Currently, at least three loci influence cat tabby markings, two of which are designated Tabby and Ticked. The Tabby locus is laeverin (LVRN) and affects the mackerel and blotched patterns. The unidentified gene for the Ticked locus on cat chromosome B1 was suggested to control the presence or absence of the ticked pattern (Tabby - Abyssinian (T^a ; a.k.a. ticked). The cat reference genome (Cinnamon, the Abyssinian) has the ticked phenotype and the variant dataset and coat phenotypes from the 99 Lives Cat Genome Consortium (195 cats) were used to identify candidate genes and variants associated with the Ticked locus. Two strategies were used to find the Ticked allele(s), one considered Cinnamon with the reference allele or heterozygous (Strategy A) and the other considered Cinnamon as having the variant allele or heterozygous (Strategy B). For Strategy A, two variants in Dickkopf Wnt Signaling Pathway Inhibitor 4 (DKK4), a p.Cys63Tyr (B1:41621481, c.188G>A) and a less common p.Ala18Val (B1:42620835, c.53C>T) variant are suggested as two alleles influencing the Ticked phenotype. Bioinformatic and molecular modeling analysis suggests that these changes disrupt a key disulfide bond in the Dkk4 cysteine-rich domain 1 or Dkk4 signal peptide cleavage respectively. All coding variants were excluded as Ticked alleles using Strategy B.