Use of random regression model as an alternative for multibreed relationship matrix

Introduction

Two models of breeding value and variance component estimation for multibreed data have been presented in Lo et al. (1993), Cantet & Fernando (1995), and García-Cortés & Toro (2006). Lo et al. (1993), and Cantet & Fernando (1995) build a single covariance matrix to describe relationships between breeding values. Thus, there is one covariance matrix having breed proportion information and variance components, and one breeding value for each animal. In contrast, García-Cortés & Toro (2006) splits the single breeding value to its breed and breed segregation specific components. In this split breeding value model, each breeding value component is a random effect with a specific relationship matrix, and the breeding value components are uncorrelated in the variance structure.

The original multibreed model can be solved by standard linear mixed model equations (Henderson 1984). However, the covariance structure for the breeding values contains the variance components making its use in practice numerically expensive when variance components are estimated. When the covariance structure is built externally, the split breeding value model allows using standard variance component estimation software in analysis of multibreed data where the variance components multiply the correlation structure of breeding values. However, neither of these multibreed models can use genomic marker information. Objective of this study was to derive random regression models that are approximations to the multibreed model. The approximation allows an easy approach to use multibreed data and also allows genomic data models.

Additive multibreed linear mixed models

Multibreed models need information on observations, fixed effects, pedigree information etc. but also information on the breed composition of animals. Each animal in the pedigree must have breed proportion values that indicate proportion of genes in the animal from the different breeds in the analysis. Let be proportion of genes in animal i from breed p. The breed proportions of animal i sum to one. For a purebred animal, only one of the breed proportion values is one, and the others are zero.

Linear mixed effect model by Lo et al. (1993) is as follows:

(1)

where y is vector of observations, b is vector of fixed effects, X is design matrix linking fixed effects and observations, a has total breeding values, Z is incidence matrix linking breeding values with observations and e is random residual vector. It is assumed that the breeding values and residuals are normally distributed: , and . Thus, and . Matrix G describes variances and covariances in a multibreed population where individual is allowed to have unique breed composition and each breed has individual genetic variance. Lo et al. (1993) describe rules for obtaining the relationship matrix G using pedigree, breed proportion and variance component information. Note that the additive covariance matrix G cannot be expressed as the numerator relationship matrix times additive genetic variance but contains the necessary variances in the matrix.

García-Cortés & Toro (2006) split the G matrix to its components:

(2)

where n_b is number of breeds, A_p and A_pp′ are partial numerator relationship matrices for breed p and segregation between breeds p and p′, is the additive genetic variance of breed p and is the genetic segregation variance between breeds p and p′. The authors gave algorithms to calculate the required partial numerator relationship matrices. Splitting the covariance matrix G to its breeding value components allows using model

(3)

with assumptions and .

The mixed model equations by model (3) is much larger than by model (1) but has the advantage of allowing separation of the variance terms from the covariance matrix as a multiplier. In practice, this allows implementation of computationally more efficient breeding value and variance component estimation approaches. Model (3) can be used in standard variance component estimation software such as ASREML (Gilmour et al. 2009) when the partial relationship matrices or their inverses are available as external covariance structures stored in files. However, special software has to be written to analyse large data sets. In addition, the model does not allow use of genomic marker information. In the following, we derive a random regression approximation to model (3) that can be easily extended to use genomic marker information.

Random regression approximation to the multibreed model

Consider breeding value variance of individual i in (2):

where subscript ii refers to diagonal element associated with animal i in corresponding partial relationship matrix. The diagonal elements of partial relationship matrices A_p and A_pp′ are (García-Cortés & Toro 2006) as follows:

and

respectively, where is the proportion of genes of the animal i coming from breed p, s and d denote sire and dam of individual i, respectively, and . These formulas have the same structure as diagonal of standard numerator relationship matrix A where diagonal of animal i is . When all animals are 100% of, say, breed one, then is always one, are zero and are zero giving A₁ = A and A_p>1 = 0.

Diagonal elements of partial relationship matrices A_p and A_pp′ are proportional to and , respectively, when the partial relationship coefficients and , respectively, are zero. We will assume this proportionality to hold for the terms and as well for all individuals. Thus, the approximation assumes that all diagonal elements have form and where is diagonal element in A for animal i, and asterisk (*) in superscript is used to denote approximation of and . In our approximate model, breeding value variance of individual i is given as follows:

In the model (3) of García-Cortés & Toro (2006), diagonal elements of partial relationship matrices A_p and A_pp′ are functions of relationship coefficient between parents that depend on the breed proportions of the parents. In our approximation, we assume that the diagonal elements are functions of the breed proportions of individual itself.

The formulas to calculate off-diagonal elements of the partial relationship matrices A_p and A_pp′ are and , respectively. Corresponding formula in the standard numerator relationship matrix A has the same form . The approximation can be extended to the off-diagonal elements where we calculate and .

Denote approximate partial relationship matrices by and where F_p is diagonal matrix having square roots of breed proportions of individuals, and diagonal matrix H_pp′ has square roots of values. Note that the diagonal matrices F_p and H_pp′ contain square roots similarly as in the Cholesky decomposition of the partial relationship matrix given in García-Cortés & Toro (2006). The full approximate G matrix can be written as follows:

The G* matrix could be used as an approximation to G in the mixed model equations in model (1) but with the same inherent problem that the variance components are part of the covariance structure and cannot be factored as multipliers.

The split breeding value approach by García-Cortés & Toro (2006) can be used for the approximation as well. Linear mixed effects model for the approximation has the same appearance as model (3) but it is now assumed that and . Mixed model equations using these assumptions will lead to separate breeding value–specific partial covariance matrices as for model (3) where a_p and a_pp′ have been replaced by and , respectively.

The need to build covariance matrices for and using breed proportion information F_p and H_pp′ can be relaxed by using equivalent random regression model. The matrices F_p and H_pp′ can be factored out from the covariance structure of and , respectively, to be part of the model equations. Then, the model is given as follows:

(4)

where Z_p = ZF_p and Z_pp′ = ZH_pp′ model (4) assumes that and where diagonal matrices T_p and T_pp′ are incidence matrices having ones for animals with nonzero values in F_p and H_pp′, respectively. Thus, T_pAT_p matrix has the same elements as ordinary A matrix but row and column values are zero for purebred animals of breeds other than p.

Total breeding value is directly estimated in model (1). Total breeding value by the equivalent split breeding value model (3) is sum of the component breeding values: . Similarly, total breeding value by the approximate random regression model (4) is .

Multibreed random regression model and genomic data

Genomic data models like G-BLUP have the same form as model (1), for example, VanRaden (2008). However, the G matrix given in formula (2) is replaced by genetic variance times genomic relationship matrix G_M formed by using genomic SNP marker information. Often, the matrix has form G_M = Z_MZ′_M. For example, VanRaden (2008) gave two commonly used genomic relationship matrices, which can be expressed in this form. Genomic relationship matrix G_M and pedigree-based relationship matrix A describe relationships in population without trait specific variance parameter, which is included in the G matrix.

Model (4) can be applied for the genomic data model by using genomic relationship matrix G_M instead of the numerator relationship matrix A. When all genotyped animals are breed crosses, that is, there are no genotyped purebred animals; matrices T_p and T_pp′ are identity matrices. Consequently, the mixed model equations simplify to using the same genomic relationship matrix for all breeding value components of the random regression model. The above-presented approximate model with genomic relationship matrix was used in Makgahlela et al. (2013) to analyse multibreed genomic data except that terms u_pp′ were unaccounted.

G-BLUP used for estimating genomic breeding values has an alternative equivalent method (Strandén & Garrick, 2009). Genomic information can be used to estimate marker effects, and, then, these can be used to calculate genomic breeding values, that is, . Thus, similarly as for the random regression model, we can rewrite the multibreed G-BLUP model as a genomic marker model

where it is assumed that the marker effects have and .

Example data by García-Cortés and Toro

Pedigree data and variance components presented in García-Cortés & Toro (2006) are used to illustrate the random regression model (Table 1). A two-breed model by Cantet & Fernando (1995) is used: y = Xb + Za₁ + Za₂ + Za₁₂ + e where Z is identity matrix because all animals have an observation. There are 11 animals of which animals 1, 2 and 5 are purebred animals from breed 1, and animals 3, 4 and 7 are purebred animals from breed 2. The fixed effects are herd effect and breed proportion regression effects. The animals are in two herds. Individual animal breed proportions and segregation terms were calculated using the given pedigree.

Applying (4) the random regression model is give as follows:

and for animal i, it is give as follows:

where t_i is herd of animal i and c_p is fixed regression effect of breed p. Mixed model equations for the random regression model are given as follows:

where the variance ratios are α₁ = 4, α₂ = 2 and α₁₂ = 8. The fixed effects vector b has both the herd effects t and the two breed regression effects c.

Because all animals have an observation, the Z matrices are diagonal. Diagonal elements of matrices Z₁, Z₂ and Z₁₂ are [1 1 0 0 1 0.707 0 0.866 0.5 0.707 0.935], [0 0 1 1 0 0.707 1 0.5 0.866 0.707 0.354] and [0 0 0 0 0 0 0 0.707 0.707 0.866 0.612], respectively.

Upper triangle of the ordinary numerator relationship matrix A is given below:

Approximate partial relationship matrix for the random regression effect of breed one is where diagonal matrix F₁ has square roots of breed proportions f₁ on the diagonal (Table 1). Thus, has zeros in rows and columns of the ordinary relationship matrix A for the purebred animals of breed 2, that is, animals 3, 4 and 7. Likewise, matrix has zero rows and columns for purebred animals of breed 1, that is, 1, 2 and 5. The only nonzero elements in matrix are in the lower right hand corner from animal 8 onwards. These animals have nonzero h₁₂(Table 1). Hence, matrices , and are based on the standard numerator relationship matrix A but some columns and rows have been zeroed according to zero breed proportion or segregation value, and other elements have been weighted by breed proportion information. Mixed model equations have matrices T₁AT₁, T₂AT₂ and T₁₂AT₁₂ that have the same structure of zero rows and columns as the approximate partial relationship matrices. Consequently, many rows and columns of the mixed model equations will have zeros. Corresponding solutions in the mixed model equations were set to zero. In addition, herd effect one was restricted to be zero for the coefficient matrix to have full rank.

Let . Upper triangle of the total covariance matrix is given as follows:

Table 2 has the diagonal elements of total genetic covariance matrix G as given in García-Cortés & Toro (2006), and the random regression model covariance matrix G*. The diagonal elements are the same for animals 1 to 7. There are small differences in the diagonal elements of animals 8 to 11. Some differences exist in the off-diagonal elements as well. Values in the G matrix by García-Cortés & Toro (2006) and our approximate G* are equal in the block of animals from one to five, and animal 7. The largest differences in the off-diagonal elements are those between animal 3 and 6, and between animal 3 and animals 8 to 11.

We illustrate reasons for differences between the G and approximate G* matrix by an example. Animal 6 is progeny of purebred animals 2 and 3 of breed 1 and 2, respectively. Thus, animal 6 has breed proportion of 0.5 for breeds one and two, and the segregation term is zero (Table 1). Covariance between animals 3 and 6 is one in G but 0.71 in G*. In matrix G, this value is due to the term . Element (3,6) in A₂ is and . Element (3,6) is zero in A₁ and A₁₂ because animal 3 is 100% of breed 2. Similarly, element (3,6) in G* is due to term where element (3,6) in is . The fact that animal 6 is an F₁ cross between two purebred animals leads our approximation to decrease the relationship coefficient too much because this coefficient should not be influenced by breed proportion of individual but only its parents. Thus, in general, the more different are the breed proportions between parents of individual, the more different are the corresponding covariance terms of individual in G* from those in G.

Estimated breeding values (EBV) by the methods were similar (Table 2). Largest difference was for animal 3, which had the largest differences between off-diagonal elements of G and G*. The base population had four animals numbered from 1 to 4. EBVs for the purebred animals 1 and 4 were similar by both methods because these animals had only purebred progeny. Purebred animals 2 and 3 had crossbred progeny that leads to larger differences in off-diagonal elements of G and G*, and EBVs by the two methods. Hence, the approximation is likely to work well when the population is highly admixed, that is, only a small fraction are purebred or first generation F₁ crossbred animals. When the population consists of lines of purebred animals that are used to make F₁ crossbred animals, the approximation is likely to work poorest. The small example (Table 1) had quite many purebred and F₁ animals (63%), which should illustrate a poor case scenario for the approximation. Still, however, correlation of EBV between García-Cortés & Toro (2006) and our approximation showed a remarkably high correlation of 0.987.

Cross breeding often attempts to maximize the differences in breed composition of parent animals to attain positive heterosis. Our approximation is perhaps not well suited to such breeding structure. Our model was developed for analyses of admixed population such as the Nordic red dairy cattle (RDC). RDC population can still be roughly divided by country to subpopulations in terms of breed composition defined by breed proportions, although nowadays the population has common breeding pool of bulls. Makgahlela et al. (2013) applied subset of our approximate model in genomic model context to the RDC population. In their study, validation reliabilities by the approximate model were always at least as good as those by the model that did not account breed heterogeneity. This suggests that the approximation worked reasonably well, which can be expected because their study had only very small fraction of purebred bulls.

Conclusions

We presented a random regression model for the analysis of multibreed data. The approximate model approach presented allows a simple way to incorporate breed proportion information and different genetic variances of origin breeds into multibreed analysis and also models including genomic data.