An Iterative Implementation of the Single Step Approach for Genomic Evaluation which Preserves Existing Genetic Evaluation Models and Software

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1 INTERBULL BULLETIN NO. 44. Stavanger, Norway, gst 6-9, 011 n Iterative Implementation of the Single Step pproach for enomic Evalation which Preserves Existing enetic Evalation Models and Software V. Dcrocq 1 and. Legarra 1 INR, UMR1313 nimal enetics and Integrative Biology (BI, 7835 Joy en Josas, France; INR, UR631 Station d mélioration énétiqe des nimax (S, F-3136 Castanet-Tolosan, France vincent.dcrocq@joy.inra.fr bstract The single step approach of Misztal et al (009 and Christensen et al (010 combines phenotypic, genomic and pedigree information into a single BLUP analysis. To adapt its implementation to most crrent genetic and genomic evalation models, we propose an iterative soltion procedre. Its main benefits are the fact that only moderate modifications of existing software are reqired and that it offers a framework for extension to more complex genetic or genomic models or their approximation. Keywords: genomic evalation, single step approach, iterative soltion Introdction The crrent genomic evalation models implemented worldwide sally involve mltiple steps: a classical genetic evalations from which b corrected phenotypes sch as daghter yield deviations (DYD or deregressed proofs (DP are obtained for a genotyped reference sbpoplation; c comptation of direct genomic vales (DV; d blending of genomic and phenotypic information. These sccessive steps may lead to biases, for example if genetic and genomic evalations are not expressed on consistent scales or even more importantly, when only partial information is sed. Patry and Dcrocq (011a showed that genomic preselection of yong blls leads to biased genetic breeding vales when candidates clled based on their DV are not inclded, which in trn leads to biased DYD for genomic evalations. Vitezica et al. (011 showed that in presence of family-based selection, genotyped animals were not a random sample of the poplation, leading to bias as well. nother drawback of mltiple step procedres is that non genotyped animals do not benefit directly from genomic information on relatives, except if another extra step is added (Patry and Dcrocq, 011b. To circmvent these limitations, Misztal et al. (009; see also Legarra et al., 009; gilar et al., 010 and Christensen and Lnd (010 proposed a single step approach in which a joint evalation of phenotypic and genomic information is performed simltaneosly for all animals. The approach appears as an extension of the sal mixed model eqations (and therefore as an extension of BLUP. The implementation of the single step approach is not complicated in the simplest cases bt may become mch more challenging for advanced models (test day, mltiple trait, threshold models reqiring demanding software adaptation. Or objective here is to develop an iterative procedre for the soltion of the single step mixed model eqations which reqires only moderate software modifications and which can be adapted to a broad variety of genetic and genomic evalation models. Methods The single step BLUP model Consider a single trait analyzed with the classical mixed model for genetic evalations: y = Xb + W + e (1 with the sal notations. Let the sbscripts 1 and correspond to non genotyped and genotyped animals respectively, 138

2 INTERBULL BULLETIN NO. 44. Stavanger, Norway, gst 6-9, =σe σ and let = be the 1 nmerator relationship matrix. ssme var( = σ, Note that may be a genomic relationship matrix (VanRaden, 009 bt can have other forms too, if another genomic evalation method is preferred; for instance, Zhang et al. (010 and Legarra et al. (011 sggested to (pre-estimate throgh, respectively, BayesB or the Bayesian Lasso, giving more weight to SNPs of large effect. The mixed model eqations for the single step approach are: XX b = + H Wy where 1 H = 1 + nother parameterization ( In (1, one can decompose the additive genetic vale into a strictly polygenic part and an independent deviation d de to genomic information, with d=-, so: y = Xb + W (+d + e (3 The deviation d 1 for non genotyped information is obtained by regression on genomic contribtion d from genotyped individals: d = d (4 Therefore, model (1 can be written as: y1 X1 W1 0 1 W 1 e1 = b+ + d + y X 0 W W e (5 If var( = σ and assming that cov(, d = 0, we have: var( = var( + d = σ + var( d It follows that var( d = ( σ The corresponding mixed model eqations are of the form M x=z, i.e.: M b + 1 = Wy Wy d Wy Wy with M eqal to: (6 XX + XX { 1 { } + +( } + (7 I I Define = 0 I 0. = 0 I 0 S S 0 0 I I 0 0 I I I I System (6 is eqivalent to: S -T M (S -1 S x=s -T z (8 The development of expression (8 leads to a system M x= z where: and: I b b x = = 0 I I I I d d (9 XX + XX M4 M = 1 + M34 0 M4 M43 M44 with: 139

3 INTERBULL BULLETIN NO. 44. Stavanger, Norway, gst 6-9, M4 = M4 = ( M34 = M43 = ( M44 = ( ( + + ( 11 1 Using the rles for the inverse of a partitioned matrix (Searle, 198, we have: M4 = M4 = 0, M34 = M43 = and M = + ( { } 44 Therefore, system (8 simplifies to: XX + XX 0 b + 1 = Wy Wy d { ( } (10 Solve for d : + ( d = (1 3 Iterate 1 and ntil convergence t convergence, eqation (1 is: + ( d = or: ( d = ( - d = fter maniplations, it follows that: d = (13 and therefore =. (14 Now, if we plg expression (13 into the second term of the right hand side of (11, we have: d = ( = ( t t pplying again the rles for the inverse of a partitioned matrix (Searle, 198, it can be shown that the absorption of the last eqation of system (10 into the first ones leads to system (: models (1 and (3 are eqivalent. n iterative soltion of the single step mixed model eqations Isolating the first three blocks of eqations in (10 from the last one sggests the following iterative soltion procedre: lgorithm : 0 ssme d = 0 1 Solve for b, 1 and : XX + XX b = Wy + 0 (11 Wy d The soltion algorithm becomes (lgorithm B: B0 ssme d = 0 B1 Solve (11 for b, 1 and B Compte t = or solve t = for t B3 Compte t = or solve t = for t B4 Iterate B1 and B ntil convergence Note that in the iterative algorithm, the genomic information is inclded only at one place, for the comptation of t in step B3. One can se here a matrix corresponding to any genomic evalation method, e.g., Bayes to Z, Lasso, PLS, Elastic Net or a BLUP on QTL as in the French genomic evalation (Boichard et al., 010. ll comptations remain nchanged as long as t and t are consistent, i.e., as long as is really eqal to var(. 140

4 INTERBULL BULLETIN NO. 44. Stavanger, Norway, gst 6-9, 011 daptation when the genomic evalation is not BLUP So far, we assmed that û in step B3 came from (11 bt it can also be obtained sing existing genomic evalation software. fter all, the soltions mst be the same at convergence, if exactly the same information is sed. There are several reasons to prefer sch strategy: first, we are not only interested in the EBV ( û bt also in the estimated (SNP or haplotype markers effects. Second, there may be some phenotypes from genotyped animals that we wold like to exclde from the genomic evalation part: for example, in most Eropean contries and Canada, own records from bll dams are exclded for the estimation of marker effects, becase of fear of preferential treatment. Third and perhaps even more importantly, there may also be some extra phenotypes from genotyped reference animals that we wold like to inclde in the genomic evalation part only. typical example is phenotypes (sally deregressed international EBV from foreign blls in mltinational reference poplations. Finally, genomic evalations software are already available and or goal is to minimize changes compared to the existing sitations. The iterative natre of lgorithms or B sggests to se (11 to compte at each iteration corrected phenotypes (DYD or DP to be sed as inpt data in the genomic evalation software. For example, sppose that we are interested in genotyped males. For each one of them, one can correct their daghters records for fixed effects and half their mates EBV and absorb the corresponding eqations as it is crrently done. This leads to a system: ( WΨ W + = WΨ ( DYD 1 d (15 for which it is hoped that, where Ψ is a diagonal matrix of EDC (Eqivalent daghter contribtion and DYD is a vector of DYD pdated at each iteration for the crrent vales of β and 1. Take as an example a BLUP evalation on QTL haplotypes (Boichard et al, 010, which can be written as: ( 1 y i =µ+ ai + hij + h ij +εi ( 16 j where y i is a corrected phenotype (DYD or DP, h ij1 and h ij are haplotype effects 1 and at QTL j for animal i and i is a residal polygenic effect. In matrix notation, y =µ 1ε + a + Nh + where N is the incidence matrix relating phenotypes to haplotypes. Let F be proportional to the (covariance matrix of haplotype effects and θ the proportion of the total genetic variance attribted to QTL. Then the genomic evalation software crrently sed can be rn at each iteration of lgorithm B to get estimates of a, h and the relevant elements of as well as t, sing: = Var( a + Nh = (1 θ Var( a +θvar( Nh = (1 θ +θnfn (17 Discssion The interest of an iterative approach is conceptal as well as comptational: Splitting the evalation into two separates the difficlty (and wealth of national evalations with their (trait dependent complex models (hge data sets, mltiple traits, heterogeneity of variances, nknown parent grops, threshold models, etc from that of the genomic evalation, with other problems (modeling, imptation of missing genotypes, comptations. It allows s to focs on one problem at a time. Modifying a national evalation software to inclde a correction of the right hand side shold be relatively easy. Frther, becase these are two separate procedres, changing one does not imply to change the other. Moving, for instance, from BLUP to BayesB or any other approach wold imply changing jst a part (B3 of the system. The fact that the method applies to any genomic evalation system, not only to BLUP, as far as describes covariances among breeding vales (as emphasized by Legarra et al., 009 mst be nderlined. 141

5 INTERBULL BULLETIN NO. 44. Stavanger, Norway, gst 6-9, 011 The comptations as described are rather simple. Matrices and (and their inverses can be stored in core for small nmbers of genotyped animals. For large nmbers, B and B3 can be solved by iteration on data, repeatedly compting the prodcts t and t. For example for BLUP, the first prodct can be compted as Z ( Λ ( Z t, where Z describes the genotypes of and Λ is a diagonal matrix at a cost of O(mp operations if m is the nmber of genotyped animals and p the nmber of markers; the second prodct can be calclated sing Collea s algorithm, at a cost of O(n operations where n is the nmber of animals related to the genotyped ones. This algorithm comptes the prodct w = t as the soltion to the system T w = T D T w = t, i.e., solving twice an extremely sparse trianglar system of eqations by reading the pedigree file twice. In order to compte t, some appropriate elements of t have to be set eqal to 0. Misztal et al. (009 and gilar et al. (011 describe the algorithm and provide a Fortran code. n nsolved problem is the estimation of reliabilities becase for large applications, the inverses of and will not be available. Bt the decomposition (5 isolating the contribtion of the genomic information independent from the rest sggests that an extension of the approach of Harris and Johnson (1998 cold perhaps be conceived. pertinent qestion is: why shold one consider a single step evalation if crrent mlti-step evalations work? One reason is the problem of bias above mentioned. Bias will plage national evalations if early selection based on genomic proofs becomes a rle. Bt the other reason is the elegance and power of a single, nified framework, sch as BLUP was in relation to contemporary comparison or Schenkel, F.S Reliability of genomic selection indexes. predictions for North merican Holstein (0 blls. J. cknowledgments Financing of the MSEN project (Joy-en- Josas, France by gence Nationale de la Recherche and PISENE is grateflly acknowledged. References gilar, I., Misztal, I., Johnson, D.L., Legarra,., Tsrta, S. & Lawlor, T.J nified approach to tilize phenotypic, fll pedigree, and genomic information for genetic evalation of Holstein final score. J. Dairy Sci. 93, gilar, I., Misztal, I., Legarra,. & Tsrta, S Efficient comptation of the genomic relationship matrix and other matrices sed in single-step evalation. J. nim. Breed. enet. 18, Boichard, D., illame, F., Bar,., Croisea, P., Rossignol, M.-N., Boscher, M.-Y., Dret, T., enestot, L., Eggen,., Jornax, L., Dcrocq, V. & Fritz, S enomic Selection in French dairy Cattle. In: Proceedings of the 9 th World Congress on enetics pplied to Livestock Prodction. Leipzig, ermany, gst 1-6, 010, Commnication 716 Christensen, O.F. & Lnd, M.S enomic prediction when some animals are not genotyped. enet. Sel. Evol. 4,. Collea, J.J. 00. n indirect approach to the extensive calclation of relationship coefficients. enet. Sel. Evol. 34, Legarra,., gilar, I. & Misztal, I relationship matrix inclding fll pedigree and genomic information. J. Dairy Sci. 9, Legarra,., Robert-ranié, C., Croisea, P., illame, F. & Fritz, S Improved Lasso for genomic selection. enet. Res. 93, Misztal, I., Legarra,. & gilar, I Compting procedres for genetic evalation inclding phenotypic, fll pedigree, and genomic information. J. Dairy Sci. 9, Patry, C. & Dcrocq, V. 011a. Evidence of biases in genetic evalations de to genomic preselection in dairy cattle. J. Dairy Sci. 94, Patry, C. & Dcrocq, V. 011b. cconting for genomic pre-selection in national genetic dairy cattle evalations. enet. Sel. Evol. 43, 30. Searle, S.R Matrix algebra sefl for statistics. John Wiley & Sons., 438p Van Raden, P.M., Van Tassell, C.P., Wiggans,.R., Sonstegard, T.S., Schnabel, R.D., Taylor, J.F. & Dairy Sci. 9, Vitezica, Z.., gilar, I., Misztal, I. & Legarra, Bias in genomic predictions for poplations nder selection. enet. Res. 93, Zhang, Z., Li, J., Ding, X., Bijma, P., de Koning, D-J. & Zhang, Q Best linear nbiased prediction of genomic breeding vales sing a trait-specific marker-derived relationship matrix. Plos ONE 5(9: e