Definition of the Subject. The term Animal Breeding refers to the human-guided genetic improvement of phenotypic traits in domestic

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2 Page Number: 0 Date:25/4/11 Time:23:49:09 1 A 2 Animal Breeding, Foundations of 3 GUILHERME J. M. ROSA 4 University of Wisconsin 5 Madison, WI, USA 6 Article Outline 7 Glossary 8 Definition of the Subject 9 Introduction 10 Principles of Selection 11 Mixed Model Methodology 12 Marker-Assisted Selection 13 Future Directions 14 Bibliography 15 Glossary 16 Bayesian Inference Statistical inference approach 17 based on the combination of prior information 18 and evidence (i.e., observations) for estimation or 19 hypothesis testing. In Bayesian analysis the prior 20 information is updated with the experimental data 21 to generate the posterior distribution of unknowns, 22 such as model parameters. The name Bayesian 23 comes from the use of the Bayes theorem in the 24 updating process. 25 Breeding Value A measure of the genetic merit of an 26 individual for breeding purposes. 27 Genetic Correlation The correlation between traits 28 that is caused by genetic as opposed to environ- 29 mental factors. Genetic correlations can be caused 30 by pleiotropy (genes that affect multiple traits 31 simultaneously) or by linkage disequilibrium 32 between genes affecting the different traits. 33 Genomic Selection Genomic selection is a form of 34 marker-assisted selection in which genetic markers 35 covering the whole genome are used such that all quantitative trait loci (QTL) are in linkage disequilibrium with at least one marker. Heritability (Narrow Sense) The fraction of the phenotypic variance that is due to additive genetic effects. Infinitesimal Genetic Model A genetic model that assumes that a trait is influenced by a very large (effectively infinite) number of loci, each with infinitesimal effect. Linkage Disequilibrium Nonrandom association of alleles at two or more loci, leading to combinations of alleles (haplotypes) that are more or less frequent in a population than would be expected from a random formation of haplotypes from alleles based on their frequencies. Mixed Models A mixed-effects model (or simply mixed model) is a statistical model containing both fixed and random effects. Such models are useful in a wide variety of disciplines in the physical, biological, and social sciences, especially for the analysis of data with repeated measurements on each statistical unit or with measurements taken on clusters of related statistical units. Population Genetics The study of allele frequency distribution and change under the influence of the four main evolutionary processes: selection, genetic drift, mutation, and migration. Quantitative Genetics The study of complex traits (e.g., production and reproductive traits, disease resistance) and their underlying genetic mechanisms. It is effectively an extension of simple Mendelian inheritance in that the combined effect of the many underlying genes results in a continuous distribution of phenotypic values or of some underlying scale or liability thereof. Definition of the Subject The term Animal Breeding refers to the human-guided genetic improvement of phenotypic traits in domestic Robert A. Meyers (ed.), Encyclopedia of Sustainability Science and Technology, DOI / , # Springer Science+Business Media, LLC 2011

3 Page Number: 0 Date:25/4/11 Time:23:49:09 2 A Animal Breeding, Foundations of 74 animals such as livestock and companion species [1]. 75 Animal breeding is based on principles of Quantitative 76 Genetics [2 4] and aims to increase the frequency of 77 favorable alleles and allelic combinations in the popu- 78 lation, which is achieved through selection of superior 79 individuals and specific mating systems strategies. 80 Selection methods and mating strategies are developed 81 by combining principles of quantitative and popula- 82 tion genetics with sophisticated statistical methods and 83 computational algorithms for integrating phenotypic, 84 pedigree, and genomic information, along with the 85 utilization of reproductive technologies that allow for 86 larger progeny cohorts from superior animals as well as 87 shorter generation intervals. 88 Through selection and mating of superior animals 89 the frequency of favorable alleles is increased, so the 90 overall additive genetic merit of a population is 91 increased through successive generations [5]. Selection 92 can be regarded as the most important tool for the 93 improvement of lines or breeds within a specific species 94 in terms of additive genetic effects. Such lines or breeds 95 can be then intermated such that nonadditive genetic 96 effects such as dominance and epistasis can be 97 exploited through specific inter- and intralocus allelic 98 combinations [1 4]. 99 The theoretical foundations of population and 100 quantitative genetics can be traced back to the work 101 of R. A. Fisher, J. B. S. Haldane, and S. Wright. The 102 rational animal breeding has its origins in the work of 103 J. L. Lush, who made substantial contributions to ani- 104 mal genetics and biometrics research and is generally 105 referred to as the father of modern scientific animal 106 breeding [1]. 107 More recent theoretical developments in popula- 108 tion and quantitative genetics have been fostered by 109 researchers such as C. C. Cockerham, C. W. Cotterman, 110 J. F. Crow, W. J. Ewens, W. G. Hill, M. Kimura, 111 G. Malécot, T. Nagylaki, and B. S. Weir, among others. 112 A landmark in the area of animal breeding and genetics 113 is the development of mixed model methodology, first 114 proposed by C. R. Henderson, which has been used 115 extensively in many applications in the field, ranging 116 from breeding value prediction under the infinitesimal 117 assumption to gene mapping and segregation analysis. 118 Most recently, Bayesian methods, Monte Carlo, and 119 resampling techniques have been employed to fit and 120 evaluate complex models in different contexts, including nonlinear systems, generalized models, survival analysis, and situations in which the number of parameters or covariates surpasses the number of observations, such as in association analysis and whole-genome marker-assisted selection using high density panels of single nucleotide polymorphism (SNP) markers. Introduction Since domestication, artificial selection has greatly changed the shape, size, and production and reproduction performance of livestock and companion animal species. For example, there is an incredible diversity of canine breeds and between dogs and their wolf ancestors from differences in overall appearance to behavior and their ability to perform specific tasks. Although to a lesser degree, the same can be observed in many other companion animal species, such as cats and horses. With livestock species, tremendous genetic changes have been accomplished as well, markedly in the last 50 years or so. For example, Fig. 1 depicts the average growth curves of broilers from selected and control populations. These results refer to a population of birds selected for over 40 years for increased growth rate and another population kept without artificial selection, with both groups derived from the same base population, starting in 1957 [6]. In the experiment presented in Fig. 1, the two groups of birds were fed diets typical of 1957 and 2001, such that the interaction between genetics and feed, as well as the genetic contribution to the phenotypic differences observed, could be assessed. It is seen that the 2001 genetics group presented an average body weight of about 4 kg at 56 days of age, while its 1957 counterpart weighed only 800 g or so. Moreover, it is shown that 85 90% of this fivefold improvement is accounted for by genetics with the remaining 10 15% to nutrition. Similar levels of genetic improvement can also be observed in many other species, such as swine, beef and dairy cattle, and some species of fish. For example, as illustrated in Fig. 2, the average breeding value for milk yield in the US Holstein, and Red and White populations has increased over 3,500 kg in the last 50 years. Such genetic improvements have been accomplished mostly through the selection and breeding of

4 Page Number: 0 Date:25/4/11 Time:23:49:10 Animal Breeding, Foundations of A superior animals, which can be chosen using specific 167 statistical methods such as those discussed on the fol- 168 lowing sections. In this chapter, the discussion will 169 focus on methods developed for normally distributed 170 (Gaussian) traits, under the infinitesimal assumption, 171 i.e., that traits are affected by a large (virtually infinite) 172 number of genes of small effects [2 4], although this 173 assumption is somewhat alleviated in marker- assisted 174 selection, which is discussed later. 175 Principles of Selection 176 Basic Genetic Model for Quantitative Traits 177 The basic genetic model can be expressed as [2, 3, 7]: y i m þ g i þ e i ð1þ 178 where y i is the phenotypic value of animal i (i.e., the 179 animal s performance for a specific trait); m is the 180 population mean (average performance of the ani- 181 mals); g i is the genotypic value of the animal, expressed 182 as a deviation from the mean; and e i is a term 183 representing environmental factors affecting the ani- 184 mal s performance, also expressed as a deviation from 185 the mean. Hence, it is assumed that E½g i Š0 and 186 E½e i Š0, such that E½y i Šm, where E½:Š represents 187 the expectation function. Moreover, the variance of y i 188 is given by Var½ y i Šs 2 y s2 g þ s2 e, where s2 g Var½g iš 189 and s 2 e Var½e iš are the genetic and environmental 190 variances, respectively. Normally distributed traits, 191 i.e., phenotypic traits with a bell-shaped distribution, 192 are generally represented as y i N m; s 2 y. Such dis- 193 tribution has a probability density function that can be 194 described as [2, 4]: ( ) f ðy i Þqffiffiffiffiffiffiffiffiffiffi 1 exp 1 2ps 2 y 2s 2 y ðy i mþ for 1< y i < 1, 1< m < 1, and s 2 y > 0, 196 which can be represented as in Fig. 3. To simplify the 197 notation used throughout the text, it is noted that 198 either random variables or their realizations will be 199 represented with lower case letters. However, the con- 200 text should make it clear to the reader when a letter 201 represents one or the other. 202 The genetic component g i of Model (1) can be 203 partitioned into additive ða i Þ and nonadditive ðc i Þ 204 genetic effects, i.e., g i a i þ c i, where a i is also called ; breeding value, and c i refers to the gene combination value, which encompasses interaction effects between alleles within each locus (i.e., dominance effects) or between alleles in different loci (i.e., epistatic effects). Hence, Model (1) can be expressed as: y i m þ a i þ c i þ e i ð2þ where a i Nð0; s 2 a Þ, c i Nð0; s 2 c Þ, and e i Nð0; s 2 e Þ, 211 with all these terms assumed independent from each 212 other. The phenotypic variance can be then expressed 213 as Var½y i Šs 2 y s2 a þ s2 c þ s2 e, from which two 214 important definitions are derived. The first one is called 215 broad sense heritability, given by H 2 s 2 g =s2 y, where 216 s 2 g s2 a þ s2 c, which gives the proportion of the phe- 217 notypic variance that is due to genetic effects. The 218 second, called narrow sense heritability, refers to 219 the specific contribution of additive genetic effects 220 to the phenotypic variance, i.e., h 2 s 2 a =s2 y. These 221 two quantities, particularly narrow sense heritability, 222 will be further discussed and used in the next sections. 223 The breeding value of an individual ða i Þ is equal to 224 the sum of additive effects of individual alleles within 225 and across loci, and it is sometimes called additive 226 genetic deviation or additive genetic effect. Because 227 individual alleles, and therefore independent allele 228 effects, are passed from parent to offspring, the breed- 229 ing value of an individual is important for predicting 230 its progeny s performance and so it is central to selec- 231 tion of superior animals [1, 3]. The gene combination 232 value ðc i Þ is the difference between the genetic merit 233 ðg i Þ of an animal and its breeding value, i.e., 234 c i g i a i, so it is often called nonadditive genetic 235 deviation. Because the component c i involves interac- 236 tions between alleles (both within and between loci), 237 and only a single allele (as opposed to a pair of alleles) 238 in each locus is transmitted from parents to offspring, 239 nonadditive effects are not transmitted in a predictable 240 manner. Hence, while average breeding value in a pop- 241 ulation can be changed through selection of superior 242 animals, the gene combination value should be 243 explored through specific mating systems. Here, the 244 discussion will focus on selection approaches and the 245 genetic improvement of a population in terms of addi- 246 tive genetic effects only. For a discussion on mating 247 systems, such as inbreeding and outbreeding strategies, 248 see for example, [1, 7, 8]. Additional discussion on

5 Page Number: 0 Date:25/4/11 Time:23:49:10 4 A Animal Breeding, Foundations of 250 inbreeding depression and heterosis (or hybrid vigor) 251 can be found in [3, 4]. 252 As discussed previously, the breeding value of an 253 individual is equal to the sum of its independent allele 254 effects. Because a parent passes a random sample of half 255 of its alleles to its progeny, an animal s breeding value is 256 twice what is often called transmitting ability or 257 expected progeny difference [1, 5]. The expected 258 breeding value of an offspring ða o Þ is then equal to 259 the average of its parents breeding values (the same 260 as the sum of its parents transmitting abilities), i.e., 261 E½a o ja s ; a d Š a sþa d 2, where a s and a d represent the 262 (realized) breeding values of the offspring s sire and 263 dam, respectively. However, there will be variability in 264 terms of breeding values within a full-sib family 265 because of the random sampling of parents alleles 266 that each offspring receives, the so-called Mendelian 267 sampling [4]. 268 The breeding value of an individual can be 269 expressed as a function of its parents breeding values 270 as a o 0:5a s þ 0:5a d þ d, where d refers to the Men- 271 delian sampling component. It is interesting to notice 272 that the variance of breeding values in a specific gener- 273 ation is equal to Var½a o Š0:25Var½a s Šþ0:25Var½a d Š 274 þvar½dš. Assuming the same additive genetic vari- 275 ance across generations and for both sexes (i.e., 276 Var½a o ŠVar½a s Š Var½a d Šs 2 a ), it is shown that 277 the Mendelian sampling variance is equal to half the 278 additive genetic variance, i.e., Var½dŠ s 2 a = Phenotypic Selection 280 The most traditional approach of genetic improvement 281 of livestock (and more generally any domestic animal 282 or plant species) is based on selection of animals with 283 the best performance, or phenotypic selection [1 4]. 284 Accordingly, given a group of animals supposedly 285 reared in similar environmental conditions, only 286 those with the highest performance are allowed to 287 breed to produce the next generation. As discussed 288 previously (Model 2), the performance of each animal 289 is a combination of its breeding value and all other 290 nonadditive genetic effects and environmental factors, 291 such that a superior performance does not always rep- 292 resent superior breeding value. Nonetheless, whenever 293 s 2 a > 0, there will be a positive correlation between 294 performance and breeding value, and the effectiveness of phenotypic selection (i.e., selection response) will increase with such correlation. To illustrate this concept consider Fig. 4, in which a scatter plot of breeding values and phenotypes (centered on zero, i.e., y i m) for a few fictitious animals is presented. As indicated before, in this chapter the discussion will be focused on selection approaches and the genetic improvement of a population in terms of additive genetic effects only, such that Model (2) can be conveniently reexpressed as: y i m þ a i þ e i ð3þ value y i, assumed e i Nð0; s 2 e Þ. where e i c i þ e i represents all nonadditive genetic 305 and environmental effects affecting the phenotypic Assuming that each effect in Model (3) is indepen- 308 dent from each other, the covariance between pheno- 309 type and breeding value is given by: 310 Cov½y i ; a i ŠCov½m þ a i þ e i ; a i ŠVar½a i Šs 2 a ; such that the correlation between phenotype and 311 breeding value is: 312 r yi ;a i Cov½y i; a i Š pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 a s p a ffiffiffiffi h Var½y i ŠVar½a i Š s y s a s 2 ; y i.e., the square root of the (narrow sense) heritability. As the breeding values of animals are unknown in practice, what phenotypic selection does is to predict (or estimate) the animals breeding values based on their own performance. The prediction is based on the regression of breeding values on phenotypes, and the regression coefficient (slope) is given by: b ai y i Cov½y i; a i Š Var½y i Š s2 a s 2 h 2 : y This means that an animal s estimated breeding value (EBV) based solely on its performance (and with a single measurement only) can be expressed as: ^a i h 2 ðy i mþ: The correlation between such EBV (which is a linear transformation of y i ) and the true breeding value ða i Þ is r^ai ;a i Cov½^a i;a i Š pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 s pffiffiffiffiffiffiffiffiffiffi 2 a h, which Var½^a i ŠVar½a i Š h 4 s 2 y s2 a is generally referred to as prediction accuracy in the animal breeding literature [5]. The square of the

6 Page Number: 0 Date:25/4/11 Time:23:49:10 Animal Breeding, Foundations of A accuracy in this case is equal to the heritability of the 329 trait and is often called prediction reliability. The 330 prediction accuracy (and consequently the reliability) 331 can be increased by using additional sources of infor- 332 mation on an animal (such as repeated measurements 333 of the trait or performance of progeny and other rela- 334 tives) when estimating its breeding value. An example 335 is with selection indexes and mixed model methodol- 336 ogy, which will be discussed later in this chapter. 337 As indicated in Fig. 4, the selected animals (i.e., the 338 best performing animals) will have an average pheno- 339 typic value equal to S and an average breeding value 340 equal to R. The expected average breeding value (and 341 also the expected phenotypic performance) of the 342 progeny of the selected animals is also R, as illustrated 343 in Fig. 5, and the ratio R/S is equal to the heritability 344 ðh 2 Þ of the trait under selection. The genetic progress 345 after one generation of selection is then given by: R h 2 S; 346 where R m P m and S m S m, with m P, m S, and m 347 representing the average phenotypic performance of 348 the progeny (generation 1), of the selected animals, 349 and of the selection candidate (generation 0) 350 populations, respectively. 351 The selection differential (S) can also be expressed 352 as S is y, where i m S m s y is called selection inten- 353 sity, and represents the selection differential in terms 354 of phenotypic standard deviations. In addition, as R 355 represents the genetic progress expected in a single 356 generation of selection, the genetic improvement per 357 unit of time is then given by R R=L, where L is the 358 generation interval. Hence, the expected genetic pro- 359 gress when phenotypic selection on a single trait is 360 employed is [1, 3]: R h2 is y L ; 361 which, given that s y s a =h, can be expressed also as: R his a L : 362 This equation is a special form of the so-called 363 breeder s equation (or key equation ), for the case 364 of phenotypic selection. In its general form, the 365 breeder s equation is expressed as [5]: R Accuracy Intensity Variation ; Generation interval meaning that the genetic progress per unit of time is proportional to the accuracy of breeding values prediction, to the selection intensity, and to the genetic variation, and inversely proportional to the generation interval. Hence, to increase the genetic progress in a population (e.g., breed or line) through selection, animal breeders (and similarly plant breeders) work to improve the four components of the equation above. As the genetic variability is a natural characteristic of a population and cannot be easily changed, genetic progress is generally incremented by improving prediction accuracy (e.g., by using specific statistical techniques to combine different sources of information regarding the animals genetic merit), by increasing the selection intensity, and by shortening the generation interval, which can be accomplished using molecular genetics techniques (e.g., the use of marker-assisted selection) and biotechnology approaches (e.g., artificial insemination). It is important to mention that the breeder s equation discussed here can be extended for more complex scenarios, such as when males and females contribute differently for some components of the equation [5]. For example, prediction accuracies and selection intensity are generally higher for males if artificial insemination is used. Another important issue to mention here is that selection not only shifts the mean of the breeding values in a population but also changes the genetic variance (and heritability). A primary cause of the change in genetic variance is due to the fact that selected parents represent one tail of the phenotypic distribution, therefore their phenotypic variance is smaller than that of the whole candidates-for-selection population. This leads to a reduction in both the phenotypic and additive genetic variances in the progeny population, which is known as the Bulmer effect [2]. In addition, as selection modifies allele frequencies toward the fixation of favorable alleles, selection in one direction over many generations is also expected to reduce the genetic variation. Additional discussion on effects of selection on variance and other short-term and long-term consequences of artificial selection can be found, for example, in [2, 3]

7 Page Number: 0 Date:25/4/11 Time:23:49:10 6 A Animal 410 In the remainder of this chapter specific statistical 411 techniques (such as the selection index, BLUP, and 412 genomic selection) for the improvement of accuracy, 413 intensity, and generation interval, and consequently 414 the increase of genetic progress from artificial selection, 415 will be discussed. 416 Correlated Response and Indirect Selection 417 If two traits x and y are genetically correlated, direct 418 selection on one of the traits (say y) will also cause 419 a genetic change in the other trait (trait x), which is 420 called correlated response [3]. Correlated response to 421 selection ðr xy Þ, that is, genetic change in trait x as 422 a consequence of direct selection on trait y, can be 423 predicted by: R xy b xy R y ; 424 where R y is the genetic progress of trait y through direct 425 selection on itself, and b xy is the genetic regression 426 coefficient, given by: b xy Covða x; a y Þ s 2 a y ; 427 where Covða x ; a y Þ is the genetic covariance between 428 traits x and y. 429 The genetic correlation between two traits x and y is 430 given by: r ax ;a y Covða x; a y Þ s ax s ay ; 431 such that Covða x ; a y Þr ax ;a y s ax s ay and the genetic 432 regression can be expressed as: b xy r a x ;a y s ax s ay s 2 a y r ax ;a y s ax s ay : 433 Using this term, and recalling the selection response 434 formula discussed before, given by R y h y i y s ay, the 435 correlated response can then be expressed as: R xy r ax ;a y s ax s ay h y i y s ay r ax ;a y s ax h y i y ; 436 or, given that s ax h x s yx, it can be finally written as: R xy r ax ;a y h x h y i y s yx : Breeding, Foundations of 437 Such an equation can be used either to monitor poten- 438 tial genetic changes in correlated traits when performing direct selection on a specific trait of economic importance or, alternatively, to explore indirect selection strategies using indicator traits [5]. The latter use may be of interest when a trait of economic importance (e.g., trait x) is difficult or expensive to measure, or it is expressed later in an animal s life, so it may be advantageous to select on a correlated trait (e.g., trait y), which would be the indicator trait. To assess the effectiveness of indirect selection relative to direct selection, one may look at the ratio of expected genetic progress per unit of time in each scenario, i.e., R xy R x r a x ;a y s ax h y i y =L y h x i x s ax =L x r ax ;a y h y h x i y i x L x L y : r a x ;a y h y i y L x h x i x L y So, it can be seen that this ratio can be higher than 1 (meaning that the indirect selection is more effective than the direct selection) depending on the genetic correlation between the economic and the indicator traits, the ratios of their heritabilities, and their potential selection intensities and generation intervals. Selection Index In section Phenotypic Selection, selection based on a single measurement on each animal was discussed. However, it is not always possible to observe the phenotype for all animals, such as traits that are expressed in only one sex or that require the sacrifice of animals to be measured, etc. In addition, even when it is possible to measure the phenotypic trait in each animal, information from relatives can be used to obtain earlier or more reliable predictions of breeding values. In this section, the prediction of breeding values using different sources of information (e.g., multiple measurements of the trait in each animal and progeny performance) will be discussed and a methodology (the selection index) that combines multiple sources of information into a single prediction for each animal will be presented. When multiple measurements of the same trait are recorded (e.g., milk yield in multiple lactations), breeding values can be predicted using the average of observations ðy i Þ from each animal as ^a i b ai y i ðy i mþ. However, to derive the genetic regression of breeding value on average phenotypic value, Model (3) must be

8 Page Number: 0 Date:25/4/11 Time:23:49:11 Animal Breeding, Foundations of A expanded to include an additional term, which is 480 discussed next. 481 It can be shown empirically that the covariance (or 482 resemblance) between repeated measurements on the 483 same animal is larger than s 2 a, which is what would be 484 expected under the assumptions of Model (3). This 485 additional source of covariance between records for 486 the same animal refers to environmental factors that 487 affect all records similarly, the so-called permanent 488 environmental effects [1, 4]. Under these circum- 489 stances, the Model (3) can be extended to: y ij m þ a i þ p i þ e ij ð4þ 490 where y ij represents the observation j ðj 1;...; n i Þ on 491 animal i, with n i being the total number of records on 492 animal i; m and a i Nð0; s 2 aþ are as defined previously; 493 p i refers to the permanent environmental effects affect- 494 ing records on animal i, assumed p i Nð0; s 2 pþ; and 495 e ij Nð0; s 2 eþ represent residual effects (nonadditive 496 genetic and temporary environmental effects) associ- 497 ated with observation y ij. In addition, it is assumed 498 that all random terms in Model (4) are independent 499 from each other, i.e., Covða i ; p i ÞCovða i ; e ij Þ 500 Covðp i ; e ij ÞCovðe ij ; e ij 0Þ0 for any i, j, and j ðj 6 j 0 Þ. 502 Under these settings, the average phenotypic value 503 of an animal is given by y ij m þ a i þ p i þ e ij, where 504 e i 1 n i P n i j1 e ij, such that its variance is given by 505 Var½y i Šs 2 a þ s2 p þ s2 e =n i, and the covariance between 506 a i and y i is Covða i ; y i Þs 2 a. In this case, the regression 507 of breeding values on phenotypic means is given by: b ai y i Cov½a i; y i Š Var½y i Š s 2 a s 2 a þ s2 p þ s2 e =n : i Noting that r 1 s2 e, the variance of the 515 s 2 a þs2 pþs 2 e average phenotypic value of an animal can be expressed 516 as a function of the repeatability as Var½y i Š 517 ½r þð1 rþ=n i Šs 2 y, such that the genetic regression 518 becomes: 519 s 2 a n i h 2 b ai y i ½r þð1 rþ=n i Šs 2 y 1 þðn i 1Þ=r : The prediction accuracy in this case, i.e., the correlation between an animal s estimated breeding value using repeated records and its true breeding value is given by: Covðy r^ai ;a i r yi ;a i i ; a i Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðy i ÞVarða i Þ s 2 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r þð1 rþ=n i s y s a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n qffiffiffiffiffiffiffiffiffi i h b ai y 1 þðn i 1Þr i : Hence, it can be seen that compared with single record phenotypic selection, there is a gain in accuracy when predictions are based on repeated records and that the gain will depend on the values of r and n i ; higher gain in accuracy is obtained when r is low and when n i is high. Another alternative to predict breeding values is to use progeny performance, which is often employed for predicting breeding values of males for traits where records can be obtained only on females, such as milk yield. For example, let y i be the average of single records on n i progeny of sire i, and assume that the sire was mated to a random sample of females not related to him. In this case, each progeny record can be expressed as: An important definition related to repeated mea- 509 surements refers to repeatability ðrþ, which is given by 510 the intraclass correlation, i.e., the ratio of the within- 511 individual (or between repeated measurements) to the 512 phenotypic variances [1, 4]: r s2 a þ s2 p s 2 y s2 a þ s2 p s 2 a þ s2 p þ ; s2 e 513 and measures the correlation between records on the 514 same animal. y ij m þ 1 2 a i þ 1 2 d ij þ d ij þ e ij ; where a i is the breeding value of a specific sire i; d ij is the breeding value of dam j ðj 1;...; n i Þ mated with sire i; and d ij and e ij refer to the Mendelian sampling and residual (environmental) components associated with the observation y ij. Using this notation, the following model can be used to describe the progeny average of sire i:

9 Page Number: 0 Date:25/4/11 Time:23:49:11 8 A Animal Breeding, Foundations of y i m þ 1 2 a i þ 1 2 d i þ d i þ e i where y i 1 n i P n i e i 1 n i P n i i1 i1 y ij, di 1 n i P n i i1 d ij, di 1 n i P n i i1 ð5þ d ij, and e ij. Given that E½ d i Š0 and E½ d i Š0, the 548 breeding value of sire i can be then predicted by 549 ^a i b ai y i ðy i mþ, where b ai y i Cov½a i ; y i Š=Var½y i Š. 550 It is shown that: 551 and Covða i ; y i ÞCovða i ; a i =2Þ s 2 a =2 Var½y i ŠVar 1 2 a i þ 1 d 2 i þ d i þ e i 1 4 s2 a þ 1 s 2 a þ s2 a þ s2 e 4 n i 2n i n i ðn i þ 3Þs 2 a þ 4s2 e 4n i ðn i þ 3Þh 2 þ 4ð1 h 2 Þ 4n i k þ 1 k s 2 y n ; i 552 where k h 2 =4 is the intraclass correlation between 553 half-sibs, such that the genetic regression coefficient is 554 given by: b ai y i s 2 a =2 ½k þð1 kþ=n i Šs 2 y h 2 s 2 y =2 ½h 2 =4 þð1 h 2 =4Þ=n i Šs 2 y 2n i h 2 4 þðn i 1Þh 2 ; 555 and the prediction accuracy by: r ai ;y i Cov½a i;y i Š h 2 s 2 y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =2 Var½a i ŠVar½y i Š h 2 s 2 y ½k þð1 kþ=n išs 2 y sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n i h 2 =4 1 þðn i 1Þk sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n i h 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þðn i 1Þh 2 b ai y i =2; 556 which approaches unity (one) as the number of prog- 557 eny records increases. s 2 y Up to this point, it has been discussed how breeding values can be predicted using different sources of information, such as an animal s own performance (either a single record or multiple measurements) or progeny performance. Other sources of information that could also be used are the performance of parents, sibling, or other kinds of relatives. However, what generally happens is that multiple sources of information are available simultaneously, so the question becomes how to best combine all the information in order to improve prediction accuracy. Here, a classical approach will be discussed, the selection index, and later on in this chapter a more general and modern alternative, based on mixed model methodology, will be presented. Consider, for example, that there are three sources of information available on animal i (represented here as y i1, y i2, and y i3, and expressed as deviations from their means), so the goal is to predict the animal s breeding value with a linear combination of such information, i.e., ^a i b i1 y i1 þ b i2 y i2 þ b i3 y i3 ; such that the prediction accuracy (i.e., correlation between predicted and true breeding value) is maximized. Maximization of r^ai ;a i is equivalent to the maximization of logðr^ai ;a i Þ, which is generally easier to accomplish. The log correlation can be expressed as (here, to simplify the notation, the index i indicating the animal is dropped): " # s^a;a logðr^a;a Þlog pffiffiffiffiffiffiffiffiffi s 2^a logðs^a;a Þ 1 s2 2 s2^a 1 2 s2 a ; a where the covariance between ^a and a, and the variance of ^a are given respectively by: and s^a;a b 1 s y1 ;a þ b 2 s y2 ;a þ b 3 s y3 ;a s 2^a b2 1 s2 y 1 þ 2b 1 b 2 s y1 ;y 2 þ 2b 1 b 3 s y1 ;y 3 þ b 2 2 s2 y 2 þ 2b 2 b 3 s y2 ;y 3 þ b 2 3 s2 y 3 : Substituting these expressions into logðr^a;a Þ, taking the partial derivatives of logðr^a;a Þ with respect to each of

10 Page Number: 0 Date:25/4/11 Time:23:49:11 Animal Breeding, Foundations of A the regression coefficients b j ðj 1; 2; 3Þ, and setting 594 them to zero, gives the following set of equations: logðr^a;a Þ s y 1 ;a b 1s 2 y 1 þ b 2 s y1 ;y 2 þ b 3 s y1 ;y 1 s^a;a s >< logðr^a;a Þ s y 2 ;a b 1s y1 ;y 2 þ b 2 s 2 y 2 þ b 3 s y2 ;y 2 s^a;a s logðr^a;a Þ s y 3 ;a b 1s y1 ;y 3 þ b 2 s y2 ;y 3 þ b 3 s 2 y >: 3 s^a;a s 2^a 595 which can be rearranged as: 8 >< b 1 s 2 y 1 þ b 2 s y1 ;y 2 þ b 3 s y1 ;y 3 ks y1 ;a b 1 s y1 ;y 2 þ b 2 s 2 y 2 þ b 3 s y2 ;y 3 ks y2 ;a >: b 1 s y1 ;y 3 þ b 2 s y2 ;y 3 þ b 3 s 2 y 3 ks y3 ;a 596 where k s2^a =s^a;a. 597 Extending the system for any number m of compo- 598 nents (i.e., sources of information), these equations can 599 be expressed in matrix notation as: Pb kc; 2 3 s 2 y 1 s y1 ;y 2 s y1 ;y m s y1 ;y 2 s 2 y 2 s y2 ;y m where P is the covari s y1 ;y m s y2 ;y m s 2 y m 601 ance matrix of the vector y ½y 1 ; y 2 ;...; y m Š 0, 602 b ½b 1 ; b 2 ;...; b m Š 0 is the vector of regression coeffi- 603 cients (weights) of each source of information, and 604 c ½s y1 ;a; s y2 ;a;...; s ym ;aš 0 is the vector of covariances 605 between each piece of information and the breeding 606 value of the animal, such that the weights b of the index 607 ^a b 0 y are given by b kp 1 c. 608 It should be noted that the constant k does not 609 change the relative size of the regression coefficients b 610 or the value of r^a;a, so it can be set to 1. In fact, if instead 611 of maximizing r^a;a, the average square prediction error 612 E½^a aš 2 is minimized, then s 2^a s^a;a and the system 613 (usually called selection index equations) becomes: b P 1 c: 614 The correlation between the index and the true pffiffiffiffiffiffiffiffiffi 615 breeding value is given by r^a;a s^a;a = s 2^a s2 a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s^a;a =s 2 1 a m b j s yj ;a: 616 s 2 a j1 Multiple-Trait Selection Usually more than one trait is considered in a selection program, as multiple traits may be economically important in a production system (e.g., [9]). There are many strategies for multi-trait selection, including the tandem approach (which selects rotationally one trait at a time) and the independent culling levels strategy (which sets minimum performance levels for each of the traits of interest), but they are generally suboptimal. Here, the selection of a combination of multiple traits evaluated in economic terms will be discussed. Such a combination of traits is generally called aggregate breeding value or breeding objective, and can be expressed as [3]: T w 0 a w 1 a 1 þ w 2 a 2 þ...þ w k a k ; where w ðw 1 ; w 2 ;...; w k Þ 0 is the vector of economic weights (expressed as net economic value per unit of trait) for k traits of linear economic value, and a ða 1 ; a 2 ;...; a k Þ 0 is a vector of breeding values relative to the k traits defining T. Here again, to simplify the notation, a subscript indexing the animal is suppressed. Suppose records are available for m traits, which may or may not be included in the k traits describing the breeding objective. The goal then is to predict T based on the m traits observed, using the so-called economic selection index. The theory of selection index was introduced in the previous subsection as a means of combining multiple sources of information to predict breeding values for a specific trait. Here, similar methodology will be considered, but will be used to combine information from multiple traits to predict an overall economic merit for each animal, i.e., ^T I v 0 y v 1 y 1 þ v 2 y 2 þ...þ v m y m ; where ^T is the predicted overall economic merit of an animal, v ðv 1 ; v 2 ;...; v m Þ 0 is the vector of weighting factors, and y ðy 1 ; y 2 ;...; y m Þ 0 is the vector of phenotypic measurements. An alternative for determining the weights v ðv 1 ; v 2 ;...; v m Þ 0 is to first predict separately the breeding values a j, j 1; 2;...; k, for each trait involved in the breeding objective, using information from all the traits with measurements,

11 Page Number: 0 Date:25/4/11 Time:23:49:11 10 A Animal 659 y ðy 1 ; y 2 ;...; y m Þ 0. Afterward, such predictions are 660 substituted for the true breeding values in the breeding 661 objective equation, and then coefficients are grouped 662 accordingly. 663 The breeding values a j for each trait can be 664 predicted by ^a j b j1 y 1 þ b j2 y 2 þ...þ b jm y m, in 665 which the weights are obtained as usual, to maximize 666 r^aj ;a j or minimize E½^a j a j Š 2. The equations which 667 define the weights for the prediction of a j are then 668 given by: 8 b j1 s 2 y 1 þ b j2 s y1 ;y 2 þ...þ b jm s y1 ;y m s y1 ;a j >< b j1 s y1 ;y 2 þ b j2 s 2 y 2 þ...þ b jm s y2 ;y m s y2 ;a j..... >: b j1 s y1 ;y m þ b j2 s y2 ;y m þ...þ b jm s 2 y m s ym ;a j 669 This procedure is repeated for all k traits in 670 the breeding objective, and the predictions 671 ^a ð^a 1 ; ^a 2 ;...; ^a k Þ 0 are then substituted for the true 672 values a ða 1 ; a 2 ;...; a k Þ 0 in the aggregate breeding 673 value, i.e., ^T w 1^a 1 þ w 2^a 2 þ...þ w k^a k : Breeding, Foundations of 674 This overall index estimating T can be rewritten as 675 I v 1 y 1 þ v 2 y 2 þ...þ v m y m, by using appropriate 676 multiplications and grouping of coefficients. It is 677 shown that each coefficient v i is given by 678 v i w 1 b 1i þ w 2 b 2i þ...þ w k b ki, with i 1; 2;...; m. 679 Another way of deriving the weights 680 v ðv 1 ; v 2 ;...; v m Þ 0 defining the economic selection 681 index I v 0 y is to maximize the correlation r T;I, 682 which will generate the following equations: 8 v 1 s 2 y 1 þ v 2 s y1 ;y 2 þ...þ v m s y1 ;y m s y1 ;T >< v 1 s y1 ;y 2 þ v 2 s 2 y 2 þ...þ v m s y2 ;y m s y2 ;T >: v 1 s y1 ;y m þ v 2 s y2 ;y m þ...þ v m s 2 y m s ym ;T 683 where s yi ;T is the covariance between each measured 684 trait i ði 1; 2;...; mþ and the linear function 685 T w 0 a, i.e., the aggregate breeding value. It can be 686 shown that both approaches for determining the 687 weights v ðv 1 ; v 2 ;...; v m Þ 0 are equivalent. Mixed Model Methodology Introduction Many statistical methods for analysis of genetic data are specific (or more appropriate) for phenotypic measurements obtained from planned experimental designs with balanced data sets. While such situations may be possible within laboratory or greenhouse experimental settings, data from natural populations and agricultural species are generally highly unbalanced and fragmented by numerous kinds of relationships. Culling of data to accommodate conventional statistical techniques (such as those discussed to this point) may introduce bias and/or lead to a substantial loss of information. The mixed model methodology, on the other hand, allows efficient estimation of genetic parameters (such as variance components and heritability) and breeding values while accommodating extended pedigrees, unequal family sizes, overlapping generations, sex-limited traits, assortative mating, and natural or artificial selection. The single trait prediction methods discussed in the previous section use only a single source of information or, when multiple sources of information are available, they require them to be split into independent subgroups, i.e., specific groups of relatives such as halfsibs, full-sibs, progeny, etc. However, in practice the data may be extremely complex due to the intricate pedigree structure commonly found in livestock species, e.g., beef and dairy cattle populations. Other drawbacks of the selection index include an inability to account for genetic trend over time and that the phenotypes must be pre-adjusted for environmental effects, which can be done, for example, using the average of contemporary groups of animals. However, contemporary group effects can be inferred only under the unrealistic assumption that they are genetically equal. Hence, a selection index can be reliably applied only to individual animals within same herd and born in same year. In view of such limitations, linear mixed models (models including both fixed and random effects) and best linear unbiased predictions (BLUP) of breeding values were developed [10 12]. The BLUP methodology uses performance information from all known relatives to estimate breeding values, can be applied to whole herds or large populations using data from

12 Page Number: 0 Date:25/4/11 Time:23:49:12 Animal Breeding, Foundations of A many years, and can also accommodate genetic differ- 735 ences between contemporary groups. Presently, mixed 736 models are widely used in many fields of science as 737 a flexible tool for the analysis of data where responses 738 are clustered around some random effects, such that 739 there is a natural dependence between observations in 740 the same cluster. Examples of applications of mixed 741 models in genetics and genomics include gene mapping 742 and association analysis (e.g., [13, 14]), and gene 743 expression assays using microarrays [15, 16] or 744 RT-PCR [17], to name a few. 745 In some applications of mixed models the central 746 objective is the estimation and hypothesis testing 747 regarding fixed effects (e.g., treatment effects in an 748 experimental study), in which case the random effects 749 (e.g., block effects) are nuisance effects. In animal 750 breeding, however, the main goal is the prediction of 751 realized values of random effects (breeding values of 752 animals), and the fixed effects are generally environ- 753 mental factors that should be taken into account to 754 adjust the observed phenotypic values. A third appli- 755 cation or goal of mixed models is the estimation of 756 variance components, such as genetic and environmen- 757 tal variances, or functions of them, such as heritability 758 and repeatability. 759 In this section some basics regarding mixed models 760 are briefly reviewed, with some emphasis toward the 761 prediction of random effects, and subsequently some 762 specific applications of the mixed model methodology 763 in animal breeding and genetics are presented. 764 A linear mixed-effects model is defined as: y MVN u Xb 0 ; V GZ 0 ZG ; G where V ZGZ 0 þ S. From the properties of multivariate normal distributions, E½ujyŠ is given by: E½ujyŠ E½uŠþCov½u; y 0 ŠVar 1 ½yŠðy E½yŠÞ; such that in this case: E½ujyŠ GZ 0 V 1 ðy XbÞ GZ 0 ðzgz 0 þ SÞ 1 ðy XbÞ: This expression, however, depends on the fixed effects values b, which also need to be inferred from the data. The fixed effects are then typically replaced by their estimates, such that predictions are made based on the following expression: ^u GZ 0 V 1 ðy X^bÞ: To estimate the fixed effects b, all random effects in Model (6) can be combined into a single vector, j Zu þ «, such that the following fixed effects model is obtained: y Xb þ j. It is shown that the expectation of the x term is E½jŠ E½Zu þ «Š ZE½uŠ þe½«š 0, and that its variance is Var½jŠ Var½Zu þ «Š ZVar½uŠZ 0 þ Var½«Š ZGZ 0 þ S V. Under these settings, the distribution of y is multivariate normal with mean vector Xb and covariance matrix V, i.e., y MVNðXb; VÞ, and the maximum likelihood estimator of b can be shown to be: y Xb þ Zu þ «ð6þ ^b ðx 0 V 1 XÞ 1 X 0 V 1 y; 765 where y is the vector of responses (observations), b is 766 a vector of fixed effects, u is a vector of random effects, 767 X and Z are known incidence matrices relate y to the 768 vectors b and u, respectively, and e is a vector of 769 residual terms. Generally, it is assumed that u and e 770 are independent from each other and normally distrib- 771 uted with zero-mean vectors and variance covariance 772 matrices G and S, respectively. 773 As mentioned before, in animal breeding a central 774 goal refers to the prediction of random effects (breed- 775 ing values). In linear (Gaussian) models as in (6) such 776 predictions are given by the conditional expectation 777 of u given the data, i.e., E½ujyŠ. Given the model spec- 778 ifications above, the joint distribution of y and u is: which is distributed as ^b MVNðb; ðx 0 V 1 XÞ 1 Þ.If the design matrix X is not full column rank, a generalized inverse of X 0 V 1 X must be used to obtain a solution b 0 ðx 0 V 1 XÞ X 0 V 1 y of the system, from which estimable functions u Lb are estimated as ^u Lb 0. The solutions ^b and ^u discussed before require V 1. As V can be of huge dimensions, especially in animal breeding applications, its inverse is generally computationally demanding if not unfeasible. However, Henderson [18] presented the mixed model equations (MME) to estimate b and u simultaneously, without the need for computing V 1. The MME were derived

13 Page Number: 0 Date:25/4/11 Time:23:49:12 12 A Animal Breeding, Foundations of 812 by maximizing (for b and u) the joint density of y and 813 u, expressed as: pðy; uþ /jsj 1=2 jgj 1=2 exp 1 ðy Xb ZuÞ0 2 S 1 ðy Xb ZuÞ 1 2 u0 G 1 u : 814 The logarithm of this function is: log½pðy;uþš / jsjþjgjþðy Xb ZuÞ 0 S 1 ðy Xb ZuÞþu 0 G 1 u jsjþjgjþy 0 S 1 y 2y 0 S 1 Xb 2y 0 S 1 Zu þ b 0 X 0 S 1 Xb þ 2b 0 X 0 S 1 Zu þ u 0 Z 0 S 1 Zu þ u 0 G 1 u: 815 The derivatives regarding b and u are: 2 2 X 6 0 S 1 y X 0 S 1 X ^b X 0 S 1 3 Z^u 5 4 Z 0 S 1 y Z 0 S 1 ^b Z 0 S 1 Z^u G 1^u 816 Equating them to zero gives the following system: X 0 S 1 X ^b þ X 0 S 1 Z^u Z 0 S 1 X ^b þ Z 0 S 1 Z^u þ G 1^u X 0 S 1 y Z 0 S 1 ; y 817 which can be expressed as: X 0 S 1 X X 0 S 1 Z Z 0 S 1 X Z 0 S 1 Z þ G 1 ^b ^u X 0 S 1 y Z 0 S 1 ; y 818 known as the mixed model equations (MME). 819 Using the second part of the MME, 820 such that Z 0 S 1 X ^b þðz 0 S 1 Z þ G 1 Þ^u Z 0 S 1 y; ^u ðz 0 S 1 Z þ G 1 Þ 1 Z 0 S 1 ðy X ^bþ: 821 It can be shown that this expression is equivalent to 822 ^u GZ 0 ðzgz 0 þ SÞ 1 ðy X ^bþ and, more impor- 823 tantly, that ^u is the best linear unbiased predictor 824 (BLUP) of u. Using this result into the first part of the 825 MME, X 0 S 1 X ^b þ X 0 S 1 Z^u X 0 S 1 y X 0 S 1 X ^b þ X 0 S 1 ZðZ 0 S 1 Z þ G 1 Þ 1 Z 0 S 1 ðy X ^bþ X 0 S 1 y ^b fx 0 ½S 1 S 1 ZðZ 0 S 1 Z þ G 1 Þ 1 Z 0 S 1 ŠXg 1 : X 0 ½S 1 S 1 ZðZ 0 S 1 Z þ G 1 Þ 1 Z 0 S 1 Šy Similarly, it is shown that this expression is equivalent to ^b ðx 0 V 1 XÞ 1 X 0 V 1 y, which is the best linear unbiased estimator (BLUE) of b. It is important to note that ^b and ^u require knowledge of G and S, or at least some function of them. As these matrices are rarely known, the practical approach is to replace G and S by some sort of point estimates ^G and ^S into the MME. Many methods have been proposed to estimate variance components in mixed-effects models. The simplest is the analysis of variance (ANOVA) method, which works well for simple models (such as a one-way structure) or balanced data (such as data from designed experiments with no missing data), but they are not indicated for more complex models and data structures such as those generally found in the animal breeding context. Alternative methods proposed for estimating variance components in more complex scenarios include the expected mean squares approach of Henderson [19] and the minimum norm quadratic unbiased estimation [20]. However, maximum likelihood-based methods are currently the most popular (see, for example, [21]), especially the restricted (or residual) maximum likelihood (REML) approach [22], which attempts to correct for the well-known bias in the classical maximum likelihood (ML) estimation of variance components. Additional literature on variance component estimation and mixed model methodology can be found, for example, in [23 27]. The Animal Model The advent of mixed-effects models has undoubtedly revolutionized the animal breeding field, and today they are widely used in the genetic improvement of many livestock and companion animal species. In this subsection some of the applications of mixed models for the genetic evaluation of populations using phenotypic and pedigree information will be presented. In the following section applications incorporating molecular marker information will be discussed as well. As a first application of mixed models in animal breeding, the so-called animal model is considered