## Abstract

## Key words

## Introduction

**RRM**) for national genetic evaluation of dairy cattle was adopted at the onset of the millennium (

**TD**) observations were modeled based on Danish RRM variance component estimates (

**RR**) covariance function (

**CF**) for the additive genetic animal effect of the meta-model. The motivation for developing a meta-model was to achieve a model as good as or better than the one already implemented in each respective country. Since the first joint model, additional efforts have been made to harmonize the data from the participating countries as well as the models for these data. In 2008, Swedish 305-d yield observations were replaced by TD observations.

**HOL**), Nordic Red Cattle (

**RDC**), and Jersey (

**JER**) were officially adopted in February 2012.

## Materials and Methods

### Data

#### Breeds

**FIC**); HOL and RDC are the main dairy breeds, of which HOL cows are predominant in Denmark and Sweden, and RDC cows in Finland (Table 1). Herds with JER cows are only found in Denmark and in the south of Sweden, and indigenous FIC cows only in Finland. Crossbreeding is used in all the main breeds, both between different breed strains and between breeds. The latter is especially the case for the Danish RDC population, which is a synthetic breed of old Red Danish Cattle, Swedish Red Breed, Brown Swiss, and Red Holstein. The Finnish and Swedish Red Cattle populations also have their own separate histories, making RDC a genetically very heterogeneous breed. Connectedness of the populations across the 3 countries was originally established mainly by the use of common Nordic or international sires.

Breed | Holstein | Nordic Red Cattle | Jersey | Finncattle | |||||
---|---|---|---|---|---|---|---|---|---|

D | F | S | D | F | S | D | S | F | |

Holstein Friesian | 96.2 | 84.2 | 91.7 | 20.6 | 0.5 | 1.3 | 0.1 | 1.0 | 3.5 |

European Black and White | 3.7 | 10.8 | 6.6 | 0.4 | 0.3 | 0.8 | 0.4 | 1.7 | |

Finnish Ayrshire | 3.5 | 0.4 | 11.0 | 59.3 | 28.5 | 0.3 | 8.1 | ||

Swedish Red Breed | 0.5 | 0.9 | 22.3 | 22.2 | 43.2 | 0.9 | 1.7 | ||

Red Danish Cattle | 0.1 | 22.5 | 1.5 | 6.6 | 0.1 | ||||

Norwegian Red Cattle | 0.7 | 0.2 | 2.1 | 5.7 | 7.4 | 0.1 | 0.6 | ||

Canadian Ayrshire | 0.2 | 0.1 | 5.6 | 9.7 | 8.0 | 0.1 | 0.4 | ||

American Brown Swiss | 13.6 | 0.7 | 4.1 | ||||||

Danish Jersey | 0.1 | 57.9 | 58.5 | ||||||

American Jersey | 40.2 | 36.7 | |||||||

New Zealand Jersey | 1.7 | 1.5 | |||||||

Finncattle | 0.1 | 83.9 | |||||||

Other breeds | 0.1 | 1.8 | 0.1 | 0.1 | 0.1 | 0.5 |

#### Test-Day Records

Breed | Denmark | Finland | Sweden | |||
---|---|---|---|---|---|---|

TD records | Cows | TD records | Cows | TD records | Cows | |

Holstein | 59,081,435 | 3,382,401 | 14,366,861 | 595,524 | 18,921,594 | 1,048,669 |

Nordic Red Cattle | 8,131,043 | 478,688 | 39,938,682 | 1,570,252 | 17,888,622 | 975,227 |

Jersey | 10,164,986 | 582,922 | 204,977 | 11,480 | ||

Finncattle | 544,330 | 23,504 |

#### Variance Components

- Lidauer M.H.
- Madsen P.
- Matilainen K.
- Mäntysaari E.A.
- Strandén I.
- Thompson R.
- Pösö J.
- Pedersen J.
- Nielsen U.S.
- Eriksson J.-Å.
- Johansson K.
- Aamand G.P.

*Interbull Bull.*2009; 40: 37-41

where

**y**is a vector of observations;

**b**is a vector of fixed effects;

**h**,

**c**,

**p**, and

**a**are vectors of random effects; and

**e**is a vector of random residuals. Vector

**b**includes the fixed effects herd × 2-yr calving period, calving age, days carried calf, regression function on DIM

*d*nested within 2-yr calving periods, and regression on breed heterozygosity. Vector

**h**contains the random herd × test-day (

**HTD**) effects. Vectors

**c**,

**p**, and

**a**are vectors of random regression coefficients for the herd lactation curve nested within herd × 2-yr calving period, nonhereditary animal effects, and additive genetic animal effects, respectively. The random regression functions for nonhereditary and additive genetic animal effects include 4 terms comprising a second-order Legendre polynomial and an exponential term exp(−0.04

*d*). When comparing the −2logL values obtained for the various functions with different orders of Legendre polynomials and exponential terms, we found that including an exponential term improved the fit for fat yield, which is consistent with

**e**were nested within 12 consecutive lactation periods from

*d*= 8 to

*d*= 365 with intervals of 3 × 2 wk, 3 × 3 wk, 3 × 7 wk, and 3 × 5 wk. The matrices

**X**,

**Z**,

**Q**

_{c}, and

**Q**are incidence and covariable matrices that relate the appropriate effects to each observation. Furthermore, it was assumed that

where

**I**is an identity matrix of size equal to number of effect levels,

**H**is a 9 × 9 covariance matrix for the HTD effect,

**C**is a 27 × 27 covariance matrix of the herd lactation curve regression coefficients,

**P**and

**G**are 36 × 36 covariance matrices of the nonhereditary and additive genetic regression coefficients, and

**A**is the numerator relationship matrix. The matrix for the residuals is in block diagonal form ${\sum}_{+}{\text{E}}_{\text{l},\omega},$ where each diagonal block is of size 3 × 3 and contains the residual covariances for an observation triplet of lactation

*l*and stage of lactation period

*ω*. Both methods for variance component estimation analysis reached the same conclusion (

- Lidauer M.H.
- Madsen P.
- Matilainen K.
- Mäntysaari E.A.
- Strandén I.
- Thompson R.
- Pösö J.
- Pedersen J.
- Nielsen U.S.
- Eriksson J.-Å.
- Johansson K.
- Aamand G.P.

*Interbull Bull.*2009; 40: 37-41

*d*= 15,

*d*= 45, …,

*d*= 285), are given in Table 3.

Trait | Parity | Holstein | Nordic Red Cattle | Jersey | |||
---|---|---|---|---|---|---|---|

D | S | D | F | S | D | ||

Milk | 1 | 0.40 | 0.39 | 0.42 | 0.39 | 0.44 | 0.45 |

2 | 0.29 | 0.29 | 0.35 | 0.34 | 0.33 | 0.30 | |

3 | 0.29 | 0.25 | 0.34 | 0.31 | 0.34 | 0.25 | |

Protein | 1 | 0.35 | 0.35 | 0.38 | 0.34 | 0.43 | 0.39 |

2 | 0.25 | 0.28 | 0.35 | 0.34 | 0.35 | 0.28 | |

3 | 0.27 | 0.26 | 0.35 | 0.32 | 0.36 | 0.22 | |

Fat | 1 | 0.38 | 0.39 | 0.39 | 0.36 | 0.43 | 0.38 |

2 | 0.30 | 0.33 | 0.35 | 0.35 | 0.34 | 0.26 | |

3 | 0.29 | 0.29 | 0.35 | 0.34 | 0.36 | 0.23 |

### Building of Covariance Functions

Lactation | Trait | ||
---|---|---|---|

Milk | Protein | Fat | |

1 | 4.0 | 0.13 | 0.18 |

2 | 5.5 | 0.18 | 0.26 |

3+ | 6.0 | 0.20 | 0.28 |

#### Additive Genetic Animal Effects

**G**for random regression coefficients describing the additive genetic animal effects could have been applied to the evaluation model without any modification. Nevertheless, the rank of

**G**was reduced to make the model more parsimonious and to decrease the number of equations per animal. This rank reduction was done using eigenvalue decomposition and only retaining the significant eigenvalues: $\text{G}={\text{V}}_{\text{g}}{\text{D}}_{\text{g}}{\text{V}}_{\text{g}}^{\text{T}}\cong {\text{V}}_{\text{g}}^{*}{\text{D}}_{\text{g}}^{*}{\text{V}}_{\text{g}}^{*\text{T}},$ where

**V**

_{g}and

**D**

_{g}are eigenfunction and eigenvalue matrices, ${\text{V}}_{\text{g}}^{*}$ and ${\text{D}}_{\text{g}}^{*}$ are corresponding matrices that include only the significant eigenfunctions, and superscript T indicates the transpose of the matrix. Thus, the additive genetic (co)variance matrix was approximated by $\text{var}\left(\text{a}\right)\text{\hspace{0.17em}}\cong \text{A}\otimes {\text{U}}_{\text{g}}{\text{U}}_{\text{g}}^{\text{T}},$ where ${\text{U}}_{\text{g}}=\text{Q}{\text{V}}_{\text{g}}^{*}{\text{D}}_{\text{g}}^{*\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$ The diagonal matrix ${\text{D}}_{\text{g}}^{*}$ included the 15 largest eigenvalues of

**D**

_{g}and explained 99.5, 99.0, and 99.9% of the variation in

**G**for HOL, RDC, and JER, respectively.

#### Nonhereditary Animal and Residual Effects

**C**,

**P**, and

**E**were used to construct an overall 63 × 63

**R**matrix by summing

where ${\Phi}_{\text{c}}={\text{I}}_{9\times 9}\otimes {\phi}_{\text{c}},$ $\Phi ={\text{I}}_{9\times 9}\otimes \phi ,$ and ${\text{C}}^{\#}$ is a block diagonal matrix of

**C**that ignores the correlations between lactations. Matrix φ consisted of 7 rows with covariables for the second-order Legendre function and the exponential term exp(−0.04

*d*) for 7 different DIM

*d*within the lactation period

*d*= 8 to

*d*= 365. Matrix φ

_{c}was the same as φ but without the intercept column. Matrix

**R**

_{e}was constructed from the estimated

**E**

*l*

_{,}

*ω*

_{3×3}submatrices that correspond to the 7 chosen DIM; namely,

*d*= {20, 50, 80, 150, 220, 280, 330}. In a first attempt, covariables for 36 different DIM that were evenly distributed within lactation were used to construct the overall

**R**. However, fitting the CF only to the 7 presented DIM resulted in better predictability of EBV for cows having an extreme observation at the beginning of lactation.

**R**was decomposed into a CF for nonhereditary animal effects $\left(\Phi {\text{K}}_{\text{p}}{\Phi}^{\text{T}}\right)$ and into a measurement error covariance matrix (

**M**) by applying a maximum likelihood algorithm (

**K**

_{p}and

**M**:

Note that the size of

**K**

_{p}is 36 × 36 and measurement errors are correlated only within lactation; that is,

**M**is a block diagonal matrix with block size 3 × 3. Fitting the CF to

**R**resulted in the same measurement error variances for all DIM within lactation, which simplifies the adjustment for heterogeneous variances, as explained later.

**K**

_{p}was reduced. Therefore, an eigenvalue decomposition was performed on each diagonal block (

**K**

_{p11},

**K**

_{p22},

**K**

_{p33}), and the 9 largest eigenvalues were retained within each block. For instance, for the first-lactation diagonal block, this resulted in ${\text{K}}_{\text{p}11}={\text{V}}_{\text{p}11}{\text{D}}_{\text{p}11}{\text{V}}_{\text{p}11}^{\text{T}}\cong {\text{V}}_{\text{p}11}^{*}{\text{D}}_{\text{p}11}^{*}{\text{V}}_{\text{p}11}^{*\text{T}}$ and a scaled eigenfunction matrix ${\text{W}}_{11}^{*}={\text{V}}_{\text{p}11}^{*}{\text{D}}_{\text{p}11}^{*\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$ The off-diagonal blocks of the rank-reduced matrix ${\text{D}}_{\text{p}}^{*}\text{\hspace{0.17em}}$ were fitted by applying generalized right and left inverses, as proposed by

where the covariable matrix for lactation

*l*is ${\text{U}}_{\text{p}l}={\text{I}}_{3\times 3}\otimes \phi {\text{W}}_{ll}^{*}.$ The rank-reduced CF ${\text{U}}_{\text{p}}{\text{D}}_{\text{p}}^{*}{\text{U}}_{\text{p}}^{\text{T}}$ has rank 27 and explained 99.8% or more of the variation described by the original CF $\left(\Phi {\text{K}}_{\text{p}}{\Phi}^{\text{T}}\right)$ for HOL, RDC, and JER.

#### Covariance Functions for Finnish Later-Lactation Observations

**K**

_{p33}into 2 matrices: ${\text{K}}_{\text{p}33}={\text{K}}_{\text{p}33}^{*}+{\text{K}}_{\text{w}},$ and by building for the third lactation submatrix of

**K**

_{p}a symmetric matrix resembling the covariances between second and third lactations. Hence,

where

**C**

_{p32}is the second to third lactation submatrix of the correlation matrix of

**K**

_{p}. Then, in matrix

**K**

_{p},

**K**

_{p33}was replaced by ${\text{K}}_{\text{p}33}^{*},$ and the CF for nonhereditary animal effects for Finnish later-lactation TD observations were determined as explained above. To construct the CF for additional nonhereditary animal effects, the 7 largest eigenfunctions of

**K**

_{w}were used: $\left[{\text{I}}_{3\times 3}\otimes \phi \right]{\text{K}}_{\text{w}}{\left[{\text{I}}_{3\times 3}\otimes \phi \right]}^{\text{T}}\text{\hspace{0.17em}}\cong {\text{U}}_{\text{w}}{\text{U}}_{\text{w}}^{\text{T}}.$ The obtained CF explained 99.8% of the variation described by

**K**

_{w}.

Trait | DIM | ||||||
---|---|---|---|---|---|---|---|

20 | 50 | 80 | 150 | 220 | 280 | 330 | |

Milk | 0.32 | 0.42 | 0.46 | 0.53 | 0.58 | 0.60 | 0.58 |

Protein | 0.31 | 0.40 | 0.45 | 0.52 | 0.59 | 0.61 | 0.60 |

Fat | 0.26 | 0.33 | 0.37 | 0.46 | 0.53 | 0.58 | 0.59 |

### Modeling of Breed Effects

#### Heterosis and Recombination Loss

Lidauer, M., E. A. Mäntysaari, I. Strandén, J. Pösö, J. Pedersen, U. S. Nielsen, K. Johansson, J.-Å. Eriksson, P. Madsen, and G. P. Aamand. 2006a. Random heterosis and recombination loss effects in a multibreed evaluation for Nordic red dairy cattle. Abstract c24–02 in Proc. 8th World Congr. Genet. Appl. Livest. Prod., Belo Horizonte, Brazil. Brazilian Society of Animal Breeding, Belo Horizonte, MG, Brazil.

#### Calving Age by Breed

^{2}were centered to zero according to mean calving age to have the calving age effect free from genetic-level differences between breeds; that is, α = (calving age in days − mean calving age)/365. Breed-specific deviations from the common mean calving age curves were modeled by including additional linear and quadratic regression coefficients, which were multiplied by the breed proportion of animal. Breed-specific calving age curves were modeled for Finnish Ayrshire, Swedish Red Breed, American Brown Swiss, HOL, and FIC.

#### Genetic Groups

### The Nordic Test-Day Model

where

*y*

_{tld:cfhijmnopqrsuvz}= observation

*z*for trait

*t*(milk yield, protein yield, fat yield) in lactation

*l*(1, 2, 3+) of DIM

*d*(8, …, 365) in parity

*p*(1, 2, 3, 4, 5+), for cow

*o*that calved at age

*n*, in country

*c*(DNK, FIN, SWE), herd

*h*, and belongs to contemporary group

*i*(primiparous, multiparous cows), 5-yr production period

*f*, production year

*j*, production month

*m*, calving year-season class

*s*, calving age class

*u*, days carried calf class

*q*, and dry period class

*r*; λ

_{tlc:hjmp}= multiplicative heterogeneous variance adjustment factor for stratum

*hjmp*;

*HY*

_{t:hji}= fixed effect of herd × year × contemporary group; $h{s}_{t:hfi}\phi {\left(d\right)}_{2}$ = fixed linear regression on DIM

*d*nested within herd × 5-yr period × contemporary group, where $\phi {\left(d\right)}_{2}$ is a linear Legendre polynomial covariable;

*YM*

_{t:cjmp}= fixed effect of production year × production month × parity class nested within country; ${\sum}_{k=1}^{5}{b}_{t:cpsuk}\phi {\left(d\right)}_{k}$ = fixed regression function on DIM

*d*nested within country × parity class × calving year-season class (Jan–Mar, Apr–June, July–Sep, Oct–Dec) × calving age class (25% youngest, 25% second youngest, 50% oldest), where $\phi \left(d\right)$ is a vector containing the covariates of a third-order Legendre polynomial (without intercept) plus exponential terms ${e}^{-0.04d}$ and ${e}^{-0.15d};$ ${\sum}_{w=1}^{3}{\sum}_{k=1}^{2}{g}_{t:cpf}{\alpha}_{opnk}{\pi}_{ow}$ = fixed regression function on calving age × breed proportion nested within country × parity class × 5-yr period, where

**α**

*opn*is a vector containing the covariates of a quadratic polynomial (without intercept) for calving age

*n*of cow

*o*in parity

*p*, and ${\pi}_{o}$ is a vector of breed proportion for cow

*o*;

*CC*

_{t:cqpf}= fixed effect of days carried calf classes (10-d classes) nested within country × parity class × 5-yr period;

*DD*

_{t:crpf}= fixed effect of days dry classes (week classes) nested within country × parity class × 5-yr period for observations from multiparous cows; $h{e}_{tl}{\xi}_{T,o}$ = fixed linear regression on total (

*T*) heterosis of cow

*o*across countries; $r{e}_{tl}{\rho}_{T,o}$ = fixed linear regression on total (

*T*) recombination loss of cow

*o*across countries;

*htd*

_{t:hjmi}= random effect of herd × test-day × contemporary group; ${\sum}_{k=1}^{5}h{e}_{tl:ck}{\xi}_{ok}$ = random regressions for heterosis nested within country, where ${\xi}_{o}$ is a vector of heterozygosity covariates for specific breed-crosses; ${\sum}_{k=1}^{5}r{e}_{tl:ck}\text{\hspace{0.17em}}{\rho}_{ok}$ = random regressions for recombination loss nested within country, where

**ρ**

*o*is a vector of recombination loss covariates for specific breed-crosses; ${\sum}_{k=1}^{9}{p}_{l:ok}\text{\hspace{0.17em}}{U}_{p}{\left(d\right)}_{tl:ck}$ = random regressions for nonhereditary animal effects for milk, protein, and fat yields among stage of lactation nested within lactation, where ${U}_{p}{\left(d\right)}_{tl:c}$ is a vector of trait- and lactation-specific CF covariates for DIM

*d*; ${\sum}_{k=1}^{7}{w}_{xok}\text{\hspace{0.17em}}{U}_{w}{\left(d\right)}_{t:ck}$ = random regressions for nonhereditary animal effects for milk, protein, and fat yields among stage of lactation nested within later lactation

*x*(3, …, 10) of Finnish cows, where ${U}_{w}{\left(d\right)}_{t:c}$ is a vector of trait-specific CF covariates for DIM

*d*; ${\sum}_{k=1}^{15}{a}_{ok}\text{\hspace{0.17em}}{U}_{g}{\left(d\right)}_{tl:k}$ = random regressions for additive genetic animal effects for all 9 traits and among stage of lactation, where ${U}_{g}{\left(d\right)}_{tl}$ is a vector of trait- and lactation-specific CF covariates for DIM

*d*; and

*e*

_{tld:cfhijmnopqrsuvxz}= random residual.

*htd**h*) =

**H**, var(

*hetl:c*) = 1.0, var(

*retl:c*) = 1.0, var(

*p**o*) = ${\text{D}}_{\text{p}}^{*},$ var(

*w**zo*) =

**I**

_{7×7}, var(

*a**o*) =

**I**

_{15×15}, and var(

**) =**

*e***I**⊗

**M**

^{*}. The residual (co)variance matrix

**M**

^{*}was built from

**M**by reducing the estimated residual correlations between traits within lactation by 33%. The reduction in residual correlations was found important to increase the predictability of EBV for cows with extreme observations.

### Adjustment for Heterogeneous Variance

**HV**) in across-country evaluation. The approach involves alternate solving of a variance model and a mean model; that is, the evaluation model. Solutions from the variance model are used to adjust each TD observation based on its stratum variance, and the mean model is then solved using the adjusted TD observations. This allows us to retain the heterogeneity of variance explained by the evaluation model, which is important because the same contemporary comparison group may include cows of different breeds. The HV adjustment approach uses the residuals of the evaluation model. A log-linear model, the variance model, is fitted to the observations of heterogeneity to obtain the multiplicative adjustment factors for TD observations.

#### Variance Model

*t*, contemporary group

*i*, and country

*c*combination, the following single-trait log-linear model was applied:

where

*s*

_{tlc:hjmp}= heterogeneity observation for stratum

*hjmp*; ${\beta}_{{1}_{tc:jmp}}$ = fixed production year × month × parity class effect; ${\beta}_{{2}_{tic:hj}}$ = random effect of herd × production year; and ${\u03f5}_{tic:hjmp}\text{\hspace{0.17em}}$ = random residual.

_{2}effect was modeled based on the results of a simulation study by

#### Homogeneous Genetic Variance Across Countries

**M**

^{*}used for the evaluation model. The method yields homogeneous genetic variances, given that heterogeneity of variance influences all effects in the model in proportionality and that the estimated variance components are close to the true variance components. To ensure the same genetic variance across countries, we modified the multiplicative mixed model approach by applying the evaluation model residual variances in

**M**

^{*}only for the strata of an arbitrarily chosen base country (SWE), whereas own sets of residual variances were used for the other 2 countries (DNK, FIN). Suitable residual variances for DNK and FIN were obtained by a calibration procedure during model development. Thus, the multiplicative model was solved and genetic variances were re-estimated from EBV by a full model sampling approach (

**M**

^{*}for the evaluation model.

### Ebv

*d*= 8 to DIM

*d*= 312. Thus, the yield EBV of animal

*o*for trait

*t*in lactation

*l*is

where ${\stackrel{\u02c6}{a}}_{o}$ comprises the 15 estimated random regression coefficients of the CF that describes animal

*o*’s additive genetic effects. A breeding value for persistency of production was calculated as the sum of losses or gains in daily EBV from DIM

*d*= 101 to DIM

*d*= 300 compared with the EBV for DIM

*d*= 100:

For purposes of practical breeding work, a combined index for milk yield, protein yield, and fat yield is published for each animal:

where ${\overline{EBV}}_{tl}$ is the average EBV of cows with observations born in 2008 to 2010 and

*stl*is the standard deviation of EBV of bulls born in 1997 to 1998 having an EBV reliability >0.6.

### Robustness Against Extreme Observations

**YD**) were calculated for Danish HOL cows from the solutions of model [2], which were then used as a dependent variable in breeding value estimation by the models under study:

where

*YD*

_{tld:coz}is a YD

*z*for trait

*t*(milk yield, protein yield, fat yield) in lactation

*l*(1, 2, 3) of DIM

*d*(8, …, 365) for cow

*o*in country

*c*(DNK). All of the studied models included a mean effect but differed in the way the random animal and residual effects were modeled.

#### Model A

#### Model B

**R**matrix was constructed by considering 36 DIM points within lactation;

*d*= {10, 20, ..., 360}. The rank of the obtained CF $\Phi {\text{K}}_{\text{p}}{\Phi}^{\text{T}}$ was reduced across all lactations, and eigenfunctions with the 17 largest eigenvalues were applied for modeling of non-hereditary animal effects. The obtained estimates for

**M**were used as variance components for residual effects. This model is consistent with an earlier model that has passed the official validation test run by the Interbull Centre but has never been placed into use.

#### Model C

#### Model D

*o*, where a yield index was calculated as $\text{\hspace{0.17em}}{I}_{o}=0.8\text{\hspace{0.17em}}{I}_{protein,o}+0.4\text{\hspace{0.17em}}{I}_{fat,o}-0.2\text{\hspace{0.17em}}{I}_{milk,o}.$ The robustness of the models was then assessed using the obtained yield indices.

### The Solving Algorithm

^{−7}. The solutions for the RRM (i.e., the mean model) were considered converged when the relative change between solutions of consecutive iterations was less than 10

^{−9}. Both the mean model and variance model were solved by the preconditioned conjugate gradient method using parallel computing, as described in

## Results and Discussion

### Covariance Functions

### Modeling of Breed Effects

#### Genetic Groups

#### Heterosis and Recombination Loss

Lidauer, M., E. A. Mäntysaari, I. Strandén, J. Pösö, J. Pedersen, U. S. Nielsen, K. Johansson, J.-Å. Eriksson, P. Madsen, and G. P. Aamand. 2006a. Random heterosis and recombination loss effects in a multibreed evaluation for Nordic red dairy cattle. Abstract c24–02 in Proc. 8th World Congr. Genet. Appl. Livest. Prod., Belo Horizonte, Brazil. Brazilian Society of Animal Breeding, Belo Horizonte, MG, Brazil.

Cross | Heterosis | Recombination loss | ||||
---|---|---|---|---|---|---|

First | Second | Third | First | Second | Third | |

Finnish Ayrshire | ||||||

× Swedish Red Breed | 5.67 | 5.72 | 4.64 | −3.06 | −2.19 | −1.81 |

× Canadian Ayrshire | 2.80 | 3.44 | 2.04 | 0.52 | −0.63 | −0.13 |

× Norwegian Red Cattle | 2.30 | 3.61 | 3.07 | −3.98 | −3.04 | −1.54 |

× Holstein | 3.11 | 3.02 | 2.30 | −1.25 | −1.27 | −0.87 |

Red Danish Cattle | ||||||

× Swedish Red Breed | 6.49 | 5.54 | 4.69 | −2.01 | −2.81 | −2.61 |

× Holstein | 3.24 | 3.81 | 3.37 | −3.24 | −3.17 | −3.00 |

× American Brown Swiss | 4.84 | 5.46 | 5.45 | −1.79 | −2.34 | −2.59 |

Swedish Red Breed | ||||||

× Finnish Ayrshire | 4.87 | 5.12 | 4.15 | −3.39 | −2.25 | −1.34 |

× Canadian Ayrshire | 2.84 | 4.90 | 2.13 | −1.65 | −0.85 | −0.14 |

× Norwegian Red Cattle | 1.97 | 3.26 | 1.24 | −4.57 | −3.49 | −2.99 |

Overall mean | 3.98 | 4.31 | 3.63 | −2.95 | −2.46 | −1.38 |

#### Calving Age × Breed Proportion

### Heterogeneous Variance Adjustment

### EBV

### Robustness Against Extreme Observations

Extreme observation | Model | |||
---|---|---|---|---|

A | B | C | D | |

Included | 115 | 122 | 114 | 110 |

Excluded | 110 | 110 | 110 | 109 |

### Considerations on Solving the Models

## Conclusions

## References

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