New models and computations in animal breeding. Ignacy Misztal University of Georgia Athens

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1 New models and computations in animal breeding Ignacy Misztal University of Georgia Athens Introduction Genetic evaluation is generally performed using simplified assumptions and methodology. Usually, the only genetic effect is additive, crossbred data is analyzed ignoring most crossbred effects, traits that can be measured continuously are analyzed at discrete points, and categorical traits if present are analyzed as continuous. Now, theories exist for more complete models. With careful programming, many of these theories can presently be implemented with large data sets. However, it needs to be tested for particular cases whether the new methodology results in more accurate prediction of breeding values. Nonadditive models The genetic evaluations more of the following reasons: are currently performed with the additive model only for one or l) methodology for including the nonadditive effects is too expensive, 2) data is insufficient to estimate the variance component of a particular nonadditive effect and therefore the importance of that effect, 3) using the prediction of a nonadditive effect for practical purposes is too complicated. For most nonadditive effects, computations can be done only if corresponding relationship matrices can be stored as dense, i.e., up to about 5000 animals and only if inbreeding is accounted for only partially. With full accounting for inbreeding, variances of additive and nonadditive effects become very complicated (de Boer and Hoeschele, 1993). In the absence of inbreeding, Hoeschele and VanRaden (1991) developed methodology for inverting the dominance relationship matrix for very large data sets. VanRaden and Hoeschele (1992) developed a similar methodology for the additive x additive effect. To estimate nonadditive effects with sufficient precision requires data sets with relationships containing these nonadditive variances. Table 1 shows coefficients of relationships of various gene actions for common types of relationships (Van Raden et al., 1992). Full-sibs and 3/4 sibs contain information on the dominance variance while half-sibs do not. Some species contain large full-sib groups, e.g., fish, while some contain relatively few, e.g., cattle. Chang (1988) estimated for specific simulated data sets that about 20 times more data was required to estimate the dominance or additive x additive variance than the additive variance with the same accuracy. For the joint estimation of these two nonadditive variances, about 400 times larger data set was required. In poultry and fish, reasonable estimates of the dominance variance were 32

2 obtained with about 20,000 animals (Pante et al.,2001; Misztal and Besbes, 2000). In dairy cattle, that number was over 300,000 _Jiisztal et al., 1997). An effort was made at the University of Georgia to estimate the dominance variance for many traits in many species. The results in Table 2 for cattle and swine, and in Table 3 for laying hens. It seems that dominance is moderately important for growth but not for other traits. It is possible that for traits where dominance is important, the dominance variance was not estimated. If the dominance effect is included in the evaluation model, the benefit could be two-fold. First, the additive effects could be better predicted. Varona et al (1999) founded that the largest changes after adding the dominance effect to the model were for animals with no own records and having many progenies by a single mate, e.g. dams of a single embryo-transfer batch in cattle. Second, one can exploit specific combining abilities in the purebred population, by mating a particular male to a daughter of a particular grandsire (DeStefano and Hoeschele, 1992), or a particular male to a particular female (Varona and Misztal, 1999) A large mating system was implemented for Holsteins for type (misztal and Lawlor, 1999). It was not found beneficial because of small values of dominance deviations with a small number of full-sibs in the Holstein population (Lawlor et al., 1998). Dominance and epistatic interactions are found in various QTL studies. If these interactions are real and not artifacts of limited QTL methodology, it would mean that nonadditive effects may be important for some traits. Such traits would most likely have low additive heritability because otherwise not much variance is left for the nonadditive effects. Confounding between dominance and full-sib permanent environmental effects What is estimated as the dominance variance can be partly full-sib environmental covariance. When the number of dominance relationships is small, the dominance relationship matrix is almost diagonal, and the effects of parental dominance and full-sib environment are nearly confounded. In a study by Misztal and Besbes (2000), one and both of these effects were fit for several layer traits. Pante et al. (2000) did a similar study for several traits in fish. Both studies used populations with a large number of fullsibs, applied REML methodology, and determined superiority of models by likelihood ratio tests. For some traits, elimination of the dominance effect from the model resulted in almost the same likelihood as with the dominance effect included. For other traits, the same was true with the full-sib environment. Statistically significant differences were obtained when both the dominance and full-sib environmental effects were removed from the model. Even in medium-sized populations with a large number of fullsibs, there may not be enough information to partition the dominance and full-sib environmental variances. Modeling with and without dominance effects In studies in dairy (Misztal et al., 1997) and beef cattle (Duangjinda, 2000b), estimates of the

3 additive variance changed very little whether the dominance effect was fit or not. Few animals in these populations were full sibs. In fish, where large fuu-sib groups are typical, the additive variance is inflated about three times if either the dominance effect or the full-sib permanent environment effect is not fit (Pante et al., 2000). If the dominance effects are ignored, inflated estimates of additive variance may be obtained from data containing large fraction of full sibs. Mating systems to account for inbreeding and dominance One possible benefit from accounting for dominance in national genetic evaluation is the possibility of predicting special combining abilities for each pair of prospective parents through a mating system (Misztal and Lawlor, 1999). In general, this requires preparing the pedigree of the prospective parents, computing additive and dominance relationship matrices for this pedigree, and inverting these matrices (Varona et al., 1999). Tsuruta (personal communication, 1998) has implemented such a system for about 4 million Holsteins and 14 linear traits. The system was implemented as a network module, where the input of two IDs resulted in the output of predicted dominance deviation plus additional information such as inbreeding depression and additive predictions. In tests of the system, the predicted dominance deviations were found to be small and, thus, the system was not implemented (Lawlor et al. 1998). Small predictions of dominance deviations were partially due to small estimates of dominance variance for linear type traits, which were 10% of phenotypic variances or smaller. Inbreeding With substantial inbreeding in the population, a question arises of what evaluation model would be best. If the evaluation model contains no provision for inbreeding, the PBV will partly account for inbreeding depression of past animals. If the evaluation model contains regression for inbreeding depression, PBVs for inbred animals will be higher. The prediction of future merit would be correct only if it includes adjustment for inbreeding depression. VanRaden and Smith (1999) adjusted PBV of sires for average inbreeding of future daughters. More complicated approaches exist to slow the increase of inbreeding by minimizing the total additive relationship within the population, e.g., as described in by Mesuwissen and Sonesson (1998). Crossbreeding Femando and Grossman (1996) presented methodology using correct variance for crossbreds. For arbitrary crosses, his methodology is very complex. For pureline and terminal crosses (F 1), his methodology simplifies to a model where each pureline has one additive effect, each F 1 has two additive effects corresponding to parental purelines, and all the lines have a dominance effect. Lutaaya et al. (2001 a and 2000b) analyzed lifetime daily gain and backfat in swine using this model. The dominance effect was very important for lifetime daily gain but not for backfat. Genetic correlations between purelines and F 1 showed breed complementarity, i.e., correlation between purelines and F1 was high for one pureline and one trait, and it was high for the the

4 other pureline for the other trait. Models for Longitudinal Data In most models, evaluation is for very well defined traits. For example, weights are measured at certain days of age, and measurements taken outside these days are adjusted to those days. Lately, there is an interest in analyses of traits that can be measured on a (semi)continu0us scale. The most popular case is test-day model in dairy, where variances and covariances are defined for each day-in-milk (DIM). Another popular trait is live weight at any day. Traits measured on a continuous scale are usually analyzed by random regression models (Kirkpatrick et al., 1994). Such models assume that variances and covariances are described by polynomials and then parameters of some functions, which often are polynomials, and then parameters of these functions are estimated. There are many random regression models and selecting an appropriate one is still an art; functions with a larger number of parameters can potentially approximate (co)variances better but estimates of these parameters may be less accurate. Longitudinal data models allow to ask new types of research questions. Some questions asked at the University of Georgia include: l) which sire produces offspring that ages slower (Tsuruta et al., 2001), 2) what is the genetic correlation of a trait recorded this year with a trait of the same name but recorded 10 or 20 years ago (Tsuruta et al, 2001), 3) which sire produces optimal offspring at a given degree of temperature-humidity index (Ravagnolo et al., 2000) Software in Animal Breeding Parameters of new types of models cannot be estimated and breeding values cannot be predicted until appropriate software is available. Availability of such soft'ware is often a limiting factors in types of models used in analyses. Also different methodologies are appropriate for different problems. For example, in variance component estimation, derivative-free REML is simple to implement for linear models but is unreliable and very expensive in multiple-trait models. EM-REML is much more reliable but relatively slow. Average-Information REML is fast and reliable when it converges. For nonlinear models, Bayesian methods via Gibbs sampling are much easier to implement than REML but the run times may be very long, especially with many missing traits, incomplete pedigrees and not well connected subpopulations. When models are very large, Method R may be the only choice but its theoretical properties are not as good as in the previous methods and general formulas for multiple-traits do not exist. A choice of methodologies allows for optimum method for a particular problem. However, programming such methodologies for a variety of models is time consuming and prone to errors. Project BLUPF90 For small models, programs written in a matrix language are usually easy to write but not efficient. Programs in a programming language are usually much more efficient but harder to

5 write and debug. Currently, personal computers are as fast as mainframes of a few years ago and becoming faster every year. Therefore low-level software optimization that may responsible for most of program complexity is no longer as important as before. Also, object-oriented features in programming languages such as C++ or Fortran 90/95 facilitate writing simpler sorware. Project BLUPF90 is an attempt for general, easily extensible software written in Fortran 90 that can support a variety of models with reasonable efficiency, and can easily be modified for new types of models/methodology ( or Misztal, 1999). Software in this project consists of easy to use library modules (dense matrix computations, sparse matrix computations, fast input/output, statistical and Gibss sampling etc), and application programs. Table 4 lists current applications. While applications in the package can easily be modified, easy of use for standard models has not been a priority; for such models, software written with the goal of ease of use may be more appropriate. Applications in the BLUPF90 project have enabled the animal breeding group at UGA to become productive by reducing modeling limitations, decreasing programming load, and thus allowing to concentrate on science of projects. Final Remarks Several types of models have not been mentioned. A very active area of research is in models to exploit the molecular data. These models can be divided into those for association studies and those for the joint analysis of quantitative and molecular information. Models to estimate the competition effect (see to account for differences when animals are raised in groups rather than individually may provide larger benefits to the industry than any other model. It must be emphasized that any analysis consists of at least four steps: preparation of data, choice of estimation/prediction methodology, adjustments for fixed effects, and the choice of random effects. Improvements in one step with disregard for any other step may not result in an overall improvement. Summarizing, new theories with efficient computing environment to support them provide opportunity for research and implementation in topics relevant to the animal industry. References B.H.G. Boswerger, T.J. Lawlor, F.R. Allaire Expected progeny production gain by balancing inbreeding depression and selection. J. Dairy Sci. 77(Suppl. 1):201. H.A. Chang Studies on estimation of genetic variances under non-additive Ph.D. dissertation, University of Illinois, Urbana, USA. gene action. I.J.M. de Boer, I. Hoeschele Genetic evaluation methods for populations with dominance and inbreeding. Theor. Appl. Genet. 86:245. A.L. DeStefano, I. Hoeschele Utilization of dominance variance through mate allocation

6 strategies. J. Dairy Sci. 75:1680. Druet, T., I. Misztal, M. Duangjinda, A. Reverter, and N. Gengler Estimation of genetic covariances with Method R. J. Anita. Sci. 79: Femando, R. L. and M. Grossman Genetic evaluation in crossbred populations. Proc. Forty-fifth Annual National Breeders Roundtable. I. Hoeschele, P.M. VanRaden Rapid inversion of dominance relationship matrices for noninbred populations by including sire by dam subclass effects. J. Dairy Sci. 74:557. Kirkpatrick, M., W. G. Hill, and R. Thompson Estimating the covariance structure of traits during growth and ageing, illustrated with lactation in dairy cattle. Genet. Res. Cambridge. 64: Lawlor, T. J., L. Klei, I. Misztal, and L. Varona Managing inbreeding and utilizing dominance effects in a herd mating. J. Dairy Sci. (Suppl. 1) 81:67. Lutaaya, E. I. Misztal, J.W. Mabry, T. Short, H.H. Timm and R. Holzabauer. 2001a. Genetic parameter estimates from joint evaluation of purebreds and crossbreds in swine. J. Anim. Sci. (Submitted) Lutaaya, E., I. Misztal, J.W. Mabry, T. Short, H.H. Timm and R. Holzabauer. 2001b. Joint evaluation of purebreds and crossbreds in swine: II. Animal Rankings. J. Anim. Sci. (Submitted) Meuwissen, T. H. E., Sonesson, A. K Maximizing the response of selection with predefined rate of inbreeding: overlapping generations. J. Animal Sci. 76: Misztal, I Complex models, more data: simpler programming. Proc. Inter. Workshop Comput. Cattle Breed. '99, March 18-20, Tuusala, Finland. Interbull Bul. 20: Misztal, I. and B. Besbes Estimates of parental-dominance environment variances in laying hens. Anita. Sci. 71: and full-sib permanent Misztal, I., R.L. Femando, M. Grossman, T. J. Lawlor and M. _ukaszewicz Dominance and epistatic effects in genetic evaluation of farm animals. Animal Science Papers and Reports 13: Misztal, I., and T. J. Lawlor Supply of genetic information - amount; format and frequency. J. Dairy Sci. 82: Misztal, I. L. Varona, M. Culbertson,, J.K. Bertrand, J. Mabry, T. J. Lawlor, C. P. Van Tassell, and N. Gengler Studies on the value of incorporating effect of dominance in genetic evaluations of dairy cattle, beef cattle, and swine. Biotech. Agric. Soc. Environ. 2:227:233.

7 Pante, M. J. R., B. Gjerde, I. McMillan, and I. Misztal Estimation of additive and dominance genetic variances for body weight at harvest in rainbow trout, Oncorhynchus mykiss. Aquaculture (submitted) Ravagnolo, O., I. Misztal Genetic component of heat stress in dairy cattle - parameter estimation. J. Dairy Sei. 83: A. Reverter,B.L. Golden, R.M.Bourdon Method R variance components procedure: application on the simple breeding value model. J. Anim. Sci. 72:2247. F.A. Rodriguez-Almeida, L.D. Van Vleck, R.L. Wilham, S.L. Northcutt Estimation of non-additive genetic variances in three synthetic lines of beef cattle using an animal model. J. Anim. Sci. 73:1002. Tsuruta, S., I. Misztal, L. Klei and T. J. Lawlor Analysis of predicted transmitting abilities at different ages for final scores in Holsteins with a random regression model. J. Dairy Sci. (Submitted) P.M. VanRaden, I. Hoeschele Rapid inversion of additive by additive relationship matrices for noninbred populations by including sire-dam combination effects. J. Dairy Sci. 74:570. P.M. VanRaden, T.J. Lawlor, T.H. Short, I. Hoeschele Use of reproductive technology to estimate variances and predict effects of gene interactions. J. Dairy Sci. 75:2892. VanRaden, P. M., and L. A. Smith, L. A Selection and mating considering expected inbreeding of future progeny. J. dairy Sci. 82: Varona, L., and I. Misztal Prediction of Parental Dominance Combinations For Planned Matings. Methodology And Simulation Results. J. Dairy Sci 82: Varona, L., I. Misztal, J. K. Bertrand and T. J. Lawlor Effect of full-sibs on additive breeding values under the dominance model for stature in United States Holsteins. J. Dairy Sci. 81: Varona, L., I. Misztal, J. K. Bertrand and T. J. Lawlor Effect of full-sibs on additive breeding values under the dominance model for stature in United States Holsteins. J. Dairy Sci. 81:

8 Table 1. Coefficients (VanRaden et al., 1992) of relationships of various gene actions for common types of relationships Relationship Coefficient additive dominance additive-additive additivedominance clone full-sib parent-offspring /4 sib half-sib Table 2. Estimates of variance components by Method R and size of data sets for several traits and species from the study of Misztal and Besbes (2000) Percentage of Number Species Trait phenotypic variance (1000s) (breed) Additive Dominance Animals Records Dairy cattle Milk yield (Holstein) Fat yield ± Protein yield ll Stature Strength Body depth Dairy form Fore udder attachment Swine Number born alive (Yorkshire) 21-day litter weight Days to kg Baekfat at kg Beef cattle Post-weaning gain (Limousin)

9 Table 3. Estimates of variance components by REML for several traits of about 26,000 layer hens from the study of Misztal and Besbes al. (2000) Trait I1- EN EN EN EW 0.65 SS 0.23 d Table 4. Application programs in the BLUPF90 project; all programs use the same parameter file Program BLUPF90* BLUP90IOD Description BLUP solutions in memory BLUP by iteration on data; supports very large data sets (by Shogo Tsuruta) REMLF90* Accelerated EM REML AIREMLF90* Average-Information REML (by Shogo Tsuruta) GIBBSF90* Plain implementation of Bayesian method with flat priors on variances and joint sampling of correlated effects; coefficient matrix recreated every sample GIBBS 1F90* GIBBS2F90* As above but with coefficient matrix stored in single-trait version once; much faster for multiple traits As above but with :oint sampling of correlates effects; faster mixing for maternal and random regression models

10 POSTGIBBSF90 MR.F90 CBLUP1F90* CBLUP90IOD Post Gibbs analyses (by Shogo Tsuruta) Method R estimation of variance components with support for large single-wait models and multiple-trait models with zero residual covariances (by Tom Druet) Solutions for models with single categorical and multiple linear traits; in memory CoyBenoit Auvray) As above by iteration on data; supports very large data sets ACCF90 *Availableat Approximate accuracies of breeding values for selected models

11 50 TM Annual National Breeders Roundtable St. Louis, Missouri May 3-4, 2001 Speaker: Dr. lgnacy Misztal Question 1 From: Dr. Danny Lubritz If dominance variance is not included in your animal model and the heritability becomes inflated as a consequence, should we include some estimate of variance or common environment so that we reduce the heritability and put proper emphasis on family data relative to individual data? Response: It is hard to generalize without analyzing the data because the inflation can be large, as was found in fish, or small, as was found in dairy or beef. In any case, it is better to use inflated than deflated heritability. Incorrect heritability will result in a smaller selection response in the next generation. Inflated heritability may result in an increased selection response after many rounds of selection due to a smaller buildup of inbreeding. Deflated heritability results in both decreased selection response and increased inbreeding.