# Principal component approach in variance component estimation for international sire evaluation

Anna-Maria Tyrisevä, Karin Meyer, Freddy W. Fikse, Vincent Ducrocq, Jette Jakobsen, Martin H. Lidauer, Esa A. Mäntysaari

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Anna-Maria Tyrisevä, Karin Meyer, Freddy W. Fikse, Vincent Ducrocq, Jette Jakobsen, et al. Principal component approach in variance component estimation for international sire evaluation. Genetics Selection Evolution, 2011, 43, online (may), Non paginé. 10.1186/1297-9686-43-21. hal-01191069

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Distributed under a Creative Commons Attribution 4.0 International License
Principal component approach in variance
component estimation for international sire
evaluation

Anna-Maria Tyrisevä<fnref n="1" />*, Karin Meyer², W Freddy Fikse³, Vincent Ducrocq⁴, Jette Jakobsen⁵, Martin H Lidauer<fnref n="1" /> and
Esa A Mäntysaari<fnref n="1" />

**Abstract**

**Background:** The dairy cattle breeding industry is a highly globalized business, which needs internationally comparable and reliable breeding values of sires. The international Bull Evaluation Service, Interbull, was established in 1983 to respond to this need. Currently, Interbull performs multiple-trait across country evaluations (MACE) for several traits and breeds in dairy cattle and provides international breeding values to its member countries. Estimating parameters for MACE is challenging since the structure of datasets and conventional use of multiple-trait models easily result in over-parameterized genetic covariance matrices. The number of parameters to be estimated can be reduced by taking into account only the leading principal components of the traits considered. For MACE, this is readily implemented in a random regression model.

**Methods:** This article compares two principal component approaches to estimate variance components for MACE using real datasets. The methods tested were a REML approach that directly estimates the genetic principal components (direct PC) and the so-called bottom-up REML approach (bottom-up PC), in which traits are sequentially added to the analysis and the statistically significant genetic principal components are retained. Furthermore, this article evaluates the utility of the bottom-up PC approach to determine the appropriate rank of the (co)variance matrix.

**Results:** Our study demonstrates the usefulness of both approaches and shows that they can be applied to large multi-country models considering all concerned countries simultaneously. These strategies can thus replace the current practice of estimating the covariance components required through a series of analyses involving selected subsets of traits. Our results support the importance of using the appropriate rank in the genetic (co)variance matrix. Using too low a rank resulted in biased parameter estimates, whereas too high a rank did not result in bias, but increased standard errors of the estimates and notably the computing time.

**Conclusions:** In terms of estimation's accuracy, both principal component approaches performed equally well and permitted the use of more parsimonious models through random regression MACE. The advantage of the bottom-up PC approach is that it does not need any previous knowledge on the rank. However, with a predetermined rank, the direct PC approach needs less computing time than the bottom-up PC.

<footnote>* Correspondence: anna-maria.tyriseva@mtt.fi</footnote>

<footnote n="1">¹Biotechnology and Food Research, Biometrical Genetics, MTT Agrifood
Research Finland, 31600 Jokioinen, Finland
Full list of author information is available at the end of the article</footnote>
**Background**

Globalization of dairy cattle breeding requires accurate and comparable international breeding values for dairy bulls. The international Bull Evaluation Service, Inter-bull, has for years performed international genetic evaluations for dairy cattle for several traits, serving the cattle breeders worldwide. Due to different trait definitions and evaluation models in countries participating in the international genetic evaluation of dairy bulls, biological traits like protein yield are treated as different, but genetically correlated traits across countries [1]. Therefore, each bull will have a breeding value on the base and scale of each participating country. For protein yield in Holstein, this currently leads to 28 breeding values per bull and the number of participating countries is expected to increase. Such a model is challenging for those responsible for the evaluations and estimation of the corresponding genetic parameters. The size of the (co)variance matrix is large: for 28 traits, the genetic covariance matrix of the classical, unstructured, multiple-trait model comprises 406 distinct covariance components. Furthermore, the full rank model becomes over-parameterized due to high genetic correlations. In addition, links between populations are determined by the amount of exchange of genetic material among the populations and can vary in strength. These special characteristics have led to a situation, where variance components e.g. for protein yield in Holstein are estimated in sub-sets of countries, and are then combined to build-up a complete (co)variance matrix [2,3]. Also, country sub-setting is not problem-free since it is often necessary to apply a “bending” procedure in order to obtain a positive definite (co)variance matrix when combining estimates from the analyses of sub-sets [4]. Even if the complete data could be analyzed simultaneously, variance component estimation would remain a challenge since the usual estimation methods are very slow or unstable, when the (co)variance matrices are ill-conditioned. Mäntysaari [5] has hypothesized that with the high genetic correlations among countries, estimation of parameters for the full size (co)variance matrix may underestimate the genetic correlations and yield unexpected partial correlations. As an extreme case, this can result in a situation where the bull's daughter performance in one country can effect negatively the bull's EBV in another country. This has been illustrated by van der Beek [6].

Different solutions have been proposed to deal with
the problem of over-parameterisation. Madsen et al. [7]
have introduced a modification of the average informa-
tion (AI) algorithm that could be applied to estimate
heterogeneous residual variance, residual covariance
structure and matrices of reduced rank. Rekaya et al. [8]
have employed structural models to estimate genetic

(co)variances. They modelled genetic, management and
environmental similarities to explain the genetic (co)var-
iance structure among countries and to obtain more
accurate estimates of genetic correlations. The authors
considered the method useful, especially when there was
a lack of genetic ties between countries. However, they
noted a 15 to 20% increase in computing time compared
to the standard multivariate model. Leclerc et al. [9]
have approached the structural models in a different
way. They selected a subset of well-connected base
countries to build a multi-dimensional space. The coor-
dinates defined by these countries were used to estimate
a distance between base countries and other countries
and thus the genetic correlations between them. This
decreased the number of parameters to be estimated
compared to the unstructured variance component
matrix for the multiple-trait across country evaluation
(MACE) approach [10]. However, when they studied a
field dataset, a relatively large number of dimensions
was needed to model the genetic correlations appropri-
ately and the estimation process often led to local max-
ima, decreasing the utility of the approach.

The principal component (PC) approach has also been
investigated as a possible solution to deal with the pro-
blems of variance component estimation for the interna-
tional genetic evaluation of dairy bulls. This approach is
of special interest because it allows for a dimension
reduction. Principal components are independent, linear
functions of the original traits. PC are obtained through
an eigenvalue decomposition of a covariance or correla-
tion matrix, which yields its eigenvectors and corre-
sponding eigenvalues. Eigenvalues describe the
magnitude of the variance that the eigenvectors explain.
For highly correlated traits, the first few principal com-
ponents explain the major part of the variation in the
data and those with the smallest contribution on the
variance can be excluded without notably altering the
accuracy of the estimates, e.g. [11]. Factor analysis (FA)
is closely related to the PC approach, but it models part
of the variance to be trait-specific. Thus, generally it
does not lead to a reduction in rank (assuming all trait-
specific variances are non-zero), but benefits from the
more parsimonous structure of the (co)variance matrix.
Leclerc et al. [12] have studied both PC and FA
approaches, but instead of estimating parameters
directly from the complete data, they used a subset of
well-linked base countries, performed a dimension
reduction for the subset and estimated a contribution of
the other countries to these PC or factors.

The above studies were motivated by an attempt to
reduce the number of parameters in the variance com-
ponent estimation for MACE, but except for the study
of Rekaya et al. [8], they were based on data sub-setting.
Kirkpatrick and Meyer [13] and Mäntysaari [5] have
suggested two different PC approaches meant to use
complete datasets. Kirkpatrick and Meyer [13] have
introduced a direct PC approach that exploits only lead-
ing principal components to model the variation in a
multivariate system to improve the precision of the esti-
mation and to reduce the computational burden inher-
ent in the analysis of large and complex datasets.
However, the approach was not specifically designed for
MACE and has not been tested for such datasets. The
bottom-up PC approach, introduced by Mäntysaari [5],
is based on the random regression (RR) MACE model
that enables rank reduction. It adds traits, i.e. countries,
sequentially in the analysis and defines a correct rank in
each step, until all countries are included and the final
rank is determined. The bottom-up PC approach was
designed to estimate the genetic parameters of large,
over-parameterized datasets, for which the estimation of
the complete, full rank dataset might not be possible. So
far it has only been tested on a simulated dataset. This
article studies the value of the direct and the bottom-up
PC approaches to estimate the variance components for
MACE using real datasets and evaluates the validity of
the bottom-up PC approach to determine the appropri-
ate rank of the (co)variance matrix.

Methods

Random regression MACE

Classical MACE [10] including *t* countries is applied
using the model

$$
y_i = X_i b + Z_i u_i + \varepsilon_i \quad (1)
$$

where $y_i$ is a $n_i$ vector of national de-regressed breeding
values for bull $i$, $b$ is a vector of $t$ country effects, $u_i$ is a vec-
tor of $t$ different international breeding values for bull $i$ and
$\varepsilon_i$ is a $n_i$ vector of residuals. $X_i$ and $Z_i$ are incidence
matrices and the variance of the bull's breeding values is
$\text{Var}(u_i) = \mathbf{G}$. Differences in residual variances, $\text{var}(\varepsilon_i)$, were
taken into account by carrying out a weighted analysis. Spe-
cifically, this involved fitting residual variances at unity and
scaling the other terms in the model (1) with weights, $w_{ij} =
\text{EDC}_{ij}/g_{jj}\lambda_{j}$, where $g_{jj}$ is the sire variance of the $j$'th country,
$\lambda_j = (4 - h_j^2)/h_j^2$ with heritabilities $h_j^2$ provided by each par-
ticipating country $j$ and $\text{EDC}_{ij}$ is the bull's effective daughter
contribution in country $j$ [14]. Contrary to the official
MACE evaluations, in this study animals with unknown
parentage were not grouped into phantom parent groups.

Following [5], the genetic (co)variance matrix of the
sire effects can be rewritten as

$$
G = \text{SCS}, \tag{2}
$$

and C can be further decomposed into

$$
C = VDV^T, \tag{3}
$$

in which **S** is a diagonal matrix of genetic standard
deviations, **C** is a genetic correlation matrix, **D** is the
matrix of eigenvalues of **C** and **V** is the matrix of the
corresponding eigenvectors. This allows the classical
MACE model to be rewritten as an equivalent random
regression MACE model [5,15]:

$$
y_i = X_i b + Z_i S V v_i + \varepsilon_i, \quad (4)
$$

where $v_i$ is a vector of $t$ regression coefficients for bull $i$ with $\text{var}(v_i) = \mathbf{D}$.

Estimation of the G matrix with appropriate rank

Formulating the classical MACE model as a RR MACE
model enables a rank reduction of the genetic (co)var-
iance matrix [16]. If **G** is close to singular, then the *r*
largest eigenvalues, *r* < *t*, explain the essential part of
the variance in **G**. Thus, **G** can be replaced with

$$
G_r = SV_r D_r V_r^T S, \quad (5)
$$

where the $r \times r$ $D_r$ contains the $r$ largest eigenvalues and the $t \times r$ matrix $V_r$, the $r$ corresponding eigenvectors [17]. Consequently, the $t \times t$ matrix $G_r$ has now only $r(2t - r + 1)/2$ parameters.

Bottom-up PC approach

The bottom-up PC approach is comprised of a sequence
of REML analyses that starts with a sub-set of traits.
New traits/countries are added one by one into the ana-
lysis, and after each trait addition step the correct rank
of the model is determined. The latter can be inferred
based on the size of the smallest eigenvalues of **G** [5] or
of the correlation matrix or by using likelihood based
model selection tools such as Akaike's information cri-
terion (AIC) [18], which takes into account both the
magnitude of the likelihood and the number of para-
meters in the model, thus penalizing for overparameter-
ized models. The latter was used in this study. For given
starting values in each step, we decomposed **G** into **S**
and **D**, estimated **D** conditional on **S** and combined **S**
and **D** to update **G**. At the beginning of the analysis,
starting values provided by Interbull were used and in
the subsequent steps, estimates were obtained from the
previous steps.

The rationale behind the bottom-up algorithm is to
select in each step the highest rank, which is still justi-
fied by the AIC criteria. Each time a new country/trait,
$k + 1$, is added to the analysis, the variance of the pre-
vious traits is already completely described by the $r$
eigenvectors. The genetic variance of the new trait and
its covariance with the previous eigenvectors is esti-
mated and if it is considered to provide new information
on breeding values, the new breeding value equation
and the new rank, $r + 1$, is kept.
Implementation for MACE:

1. Initial step

(a) choose k countries as starting sub-set

(b) use starting values G₀, take EDCij and λj for
bull i to model the residual variance by applying
weights wij

(c) estimate k × k matrix $\hat{G}_r$ for the k starting countries under the full rank model, r = k

(d) calculate Akaike's information criterion value AIC<sub>r</sub> = 2 log L + 2p, where log L is the maximum log Likelihood and p = r(r + 1)/2 the number of parameters

2. Determination of the correct rank

(a) for a given rank decompose
$$
\hat{G}_r = \hat{S}_r \hat{C}_r \hat{S}_{r-1}, \quad \hat{C}_r = \hat{V}_r \hat{D}_r \hat{V}_r^T
$$

(b) derive $\hat{G}_{r-1} = \hat{S}_r \hat{C}_{r-1} \hat{S}_r$, where $\hat{C}_{r-1}$ is obtained from $\hat{C}_r$ by removing the smallest eigenvalue from $\hat{D}_r$ and the corresponding eigenvector from $\hat{V}_r$

(c) update the weights using $\hat{G}_{r-1}$, EDC$_{ij}$ and $\lambda_j$

(d) estimate a new $\hat{D}_{r-1}$ with $\hat{S}_r$ and $\hat{V}_{r-1}$ as cov-ariables by fitting model (5).

(e) calculate AIC<sub>r-1</sub>

(f) select the best model ("rank reduction" step)

• after the initial step: while AIC<sub>r-1</sub> < AIC<sub>r</sub>, set r = r-1 and repeat step 2, otherwise take $\hat{V}_r$ and $\hat{D}_r$ and proceed to step 3

• after the country addition step: if AIC<sub>r-1</sub> < AIC<sub>r</sub>, replace $\hat{V}_r$ and $\hat{D}_r$ with $\hat{V}_{r-1}$ and $\hat{D}_{r-1}$, otherwise take $\hat{V}_r$ and $\hat{D}_r$ and proceed to step 3

3. Addition of a new country/trait

(a) if $k < t$, $k = k + 1$ and $r = r + 1$

• add a new row and column of zeros to $\hat{V}_r$ and $\hat{D}_r$, and set the $k^{th}$ element of $\hat{V}_r$ to 1 and the $r^{th}$ diagonal element of $\hat{D}_r$ to twice the average genetic variance from countries $j = 1, k$. Two times the mean value was used as a starting value for estimation of the variance of a new country to improve the convergence of iteration.

(b) update the weights using $\hat{G}_r$, EDC$_{ij}$ and $\lambda_j$ ($w_{ij} = EDC_{ij}/g_{ji}\lambda_j$)

(c) estimate a new $\hat{D}_r$ and backtransform to $\hat{G}_r$
using Equation (5)

(d) calculate AIC<sub>r</sub>

4. repeat steps 2 and 3 until $k = t$

5. Final step: update the weights and re-estimate the parameters

Direct PC approach

Genetic principal components can be estimated directly from the data [13]. The genetic (co)variance matrix is decomposed into matrices of eigenvalues and eigenvectors and only the leading principal components with notable contribution to the total variance are selected to estimate the genetic parameters. The direct estimation method requires a priori knowledge of the number of principal components fitted in the model or it must be estimated.

Defining the correct rank of matrix

Meyer and Kirkpatrick [19] noticed that selecting too
low a rank in the direct PC approach can lead to pick-
ing up the wrong subset of PC, which can result in
biased estimates. Thus, it is important to select the cor-
rect rank when the direct PC approach is employed. We
followed the procedure of Meyer and Kirkpatrick [19],
to determine the appropriate rank and to test the cap-
ability of the bottom-up PC approach to define an
appropriate rank. First, the (co)variance matrix for pro-
tein yield provided by Interbull was decomposed. Then
we studied the magnitude of the eigenvalues to make an
informed guess of the correct rank. After this, we per-
formed several direct PC analyses with ranks bracketing
this value. And finally, we examined the values of Log L
and AIC, the sum of the eigenvalues, the magnitude of
the leading eigenvalues to determine the correct rank.
In addition, average quadratic deviations between p opti-
mal and sub-optimal models, √Δr, were calculated to
indicate changes in the estimates of genetic correlations
while moving away from the optimal model [11]. √Δr
was defined as

$$
\sqrt{\Delta r} = \sqrt{2 \sum_{i=1}^{t} \sum_{j=i+1}^{t} \frac{(r_{ij,m} - r_{ij,20})^2}{t \times (t-1)}}, \quad (6)
$$

where *t* is the number of traits and *r*<sub>*ij*,m</sub> is the estimated genetic correlation between traits *i* and *j* from an analysis fitting *m* PC. The genetic correlations from the sub-optimal models were contrasted with the estimates from the direct PC rank 20 model (*r*<sub>*ij*,20</sub>), which was the optimal rank selected by the bottom-up approach.

When the rank of the model is appropriately defined,
[19] AIC should be at its minimum and the magnitude
of the leading principal components and the sum of the
eigenvalues stabilized, indicating that there is no re-par-
titioning of the genetic variance into the residual var-
iance, which is the case if too few principal components
are fitted [11]. Further, the improvement of the Log
Likelihood beyond the optimal model is expected to be
negligible.
## Differences between the direct and bottom-up PC approaches

The parameterization in the bottom-up PC approach differs from the direct PC approach in the matrix that is used for the eigenvalue decomposition. In the bottom-up PC approach, the eigenvalue decomposition was done on the correlation matrix, while in the direct PC approach the parameterization was on the (co)variance matrix [13]. For both PC approaches, the heterogeneity in residual variances were taken into account using weights, as outlined above. In the bottom-up PC approach, they were updated after each REML run, implying that $h_j^2$ were fixed, whereas $h_j^2$ were estimated in the direct PC approach.

## Test application

Data of the MACE Interbull Holstein protein yield and somatic cell count (SCC) evaluations were used for testing. Deregressed breeding values [20] for protein yield came from the August 2007 evaluation, consisting of 25 countries and those for SCC from the April 2009 evaluation comprising 23 countries. Table 1 lists the countries participating in the international evaluations in 2007 for protein yield and in 2009 for SCC. The number of countries differs between biological traits since some of countries - often those who joined the international evaluation only recently - provide data only for production traits. In addition, new countries join the MACE evaluation over time, so the number of countries

<table><caption>Table 1 Structure of the datasets for protein yield and somatic cell count (SCC).</caption><thead><tr><th rowspan="3">Country</th><th rowspan="3">Code</th><th colspan="6">Protein yield</th><th colspan="6">SCC</th></tr><tr><th colspan="4">Number of bulls</th><th colspan="2">Common bulls<sup>a</sup></th><th colspan="2">Number of bulls</th><th colspan="2">Common bulls<sup>a</sup></th></tr><tr><th>Total</th><th>Foreign bulls, % <sup>c</sup></th><th>Min<sup>b</sup></th><th>Max<sup>b</sup></th><th>Mean</th><th>Total</th><th>Foreign bulls<sup>c</sup>, %</th><th>Min<sup>b</sup></th><th>Max<sup>b</sup></th><th>Mean</th></tr></thead><tbody><tr><td>Canada</td><td>CAN</td><td>7028</td><td>33</td><td>2</td><td>1044</td><td>267</td><td>7730</td><td>34</td><td>4</td><td>1191</td><td>331</td></tr><tr><td>Germany</td><td>DEU</td><td>16734</td><td>23</td><td>56</td><td>1194</td><td>370</td><td>18624</td><td>25</td><td>49</td><td>1526</td><td>469</td></tr><tr><td>Dnk-Fin-Swe<sup>d</sup></td><td>DFS</td><td>8900</td><td>13</td><td>12</td><td>590</td><td>248</td><td>9459</td><td>13</td><td>19</td><td>731</td><td>314</td></tr><tr><td>France</td><td>FRA</td><td>11127</td><td>20</td><td>3</td><td>568</td><td>220</td><td>12254</td><td>19</td><td>7</td><td>622</td><td>274</td></tr><tr><td>Italy</td><td>ITA</td><td>6322</td><td>20</td><td>8</td><td>607</td><td>253</td><td>7254</td><td>23</td><td>11</td><td>777</td><td>338</td></tr><tr><td>The Netherlands</td><td>NLD</td><td>9696</td><td>24</td><td>26</td><td>1194</td><td>346</td><td>10935</td><td>26</td><td>37</td><td>1526</td><td>481</td></tr><tr><td>USA</td><td>USA</td><td>23380</td><td>6</td><td>6</td><td>1044</td><td>410</td><td>25281</td><td>6</td><td>10</td><td>1191</td><td>507</td></tr><tr><td>Switzerland</td><td>CHE</td><td>715</td><td>37</td><td>4</td><td>209</td><td>118</td><td>946</td><td>45</td><td>9</td><td>325</td><td>182</td></tr><tr><td>Great Britain</td><td>GBR</td><td>4361</td><td>51</td><td>7</td><td>873</td><td>316</td><td>4017</td><td>55</td><td>12</td><td>855</td><td>377</td></tr><tr><td>New Zealand</td><td>NZL</td><td>4253</td><td>24</td><td>3</td><td>560</td><td>209</td><td>4886</td><td>22</td><td>6</td><td>725</td><td>255</td></tr><tr><td>Australia</td><td>AUS</td><td>4950</td><td>26</td><td>5</td><td>681</td><td>216</td><td>5404</td><td>31</td><td>12</td><td>895</td><td>325</td></tr><tr><td>Belgium</td><td>BEL</td><td>634</td><td>97</td><td>12</td><td>425</td><td>143</td><td>665</td><td>97</td><td>14</td><td>466</td><td>166</td></tr><tr><td>Ireland</td><td>IRL</td><td>1260</td><td>79</td><td>0</td><td>354</td><td>153</td><td>1337</td><td>96</td><td>3</td><td>388</td><td>183</td></tr><tr><td>Spain</td><td>ESP</td><td>1499</td><td>48</td><td>2</td><td>408</td><td>203</td><td>1720</td><td>45</td><td>3</td><td>455</td><td>246</td></tr><tr><td>Czech Republic</td><td>CZE</td><td>2036</td><td>75</td><td>12</td><td>590</td><td>202</td><td>2453</td><td>75</td><td>17</td><td>768</td><td>279</td></tr><tr><td>Slovenia</td><td>SVN</td><td>196</td><td>55</td><td>5</td><td>68</td><td>32</td><td>-<sup>e</sup></td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Estonia</td><td>EST</td><td>472</td><td>46</td><td>2</td><td>93</td><td>30</td><td>556</td><td>49</td><td>6</td><td>117</td><td>40</td></tr><tr><td>Israel</td><td>ISR</td><td>773</td><td>11</td><td>0</td><td>59</td><td>27</td><td>853</td><td>11</td><td>1</td><td>68</td><td>33</td></tr><tr><td>Swiss Red Hol<sup>f</sup></td><td>CHR</td><td>1162</td><td>45</td><td>3</td><td>256</td><td>103</td><td>1359</td><td>42</td><td>10</td><td>327</td><td>147</td></tr><tr><td>French Red Hol<sup>f</sup></td><td>FRR</td><td>145</td><td>72</td><td>0</td><td>73</td><td>9</td><td>168</td><td>71</td><td>1</td><td>84</td><td>15</td></tr><tr><td>Hungary</td><td>HUN</td><td>1898</td><td>46</td><td>2</td><td>502</td><td>192</td><td>1638</td><td>63</td><td>5</td><td>573</td><td>246</td></tr><tr><td>Poland</td><td>POL</td><td>5071</td><td>16</td><td>0</td><td>295</td><td>118</td><td>-<sup>e</sup></td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>South Africa</td><td>ZAF</td><td>920</td><td>48</td><td>1</td><td>372</td><td>148</td><td>882</td><td>54</td><td>3</td><td>402</td><td>180</td></tr><tr><td>Japan</td><td>JPN</td><td>3177</td><td>67</td><td>1</td><td>226</td><td>97</td><td>3562</td><td>63</td><td>1</td><td>272</td><td>123</td></tr><tr><td>Latvia</td><td>LVA</td><td>232</td><td>71</td><td>6</td><td>71</td><td>29</td><td>-<sup>e</sup></td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Danish Red Hol<sup>f</sup></td><td>DNR</td><td>-<sup>e</sup></td><td>-</td><td>-</td><td>-</td><td>-</td><td>232</td><td>38</td><td>1</td><td>83</td><td>16</td></tr></tbody><tfoot><tr><td colspan="2">Total number of bulls</td><td>116941</td><td></td><td></td><td></td><td></td><td>122215</td><td></td><td></td><td></td><td></td></tr></tfoot></table>

<footnote><sup>a</sup> With other countries</footnote>

<footnote><sup>b</sup> Minimum (min) and maximum (max) values</footnote>

<footnote><sup>c</sup> Bull's country of first registration is embedded in its international identity and was extracted from it</footnote>

<footnote><sup>d</sup> Denmark, Finland and Sweden</footnote>

<footnote><sup>e</sup> Country does not participate in international evaluation for this trait</footnote>

<footnote><sup>f</sup> Holstein</footnote>
involved increases gradually. We followed Interbull's practice by listing countries in all figures and tables (except Table 1 for SCC) based on their joining date for the evaluation of each biological trait.

The total number of records was 116 941 for protein yield and 122 215 for SCC. These represented 103 676 and 100 551 bulls with deregressed breeding values, respectively. The number of bulls with records in protein yield varied from 145 to 23 380 among countries, with a mean of 4 678 bulls per country.

Corresponding values for SCC were 168 to 25 281, with a mean of 5 314 bulls per country. For both biological traits, bulls were used mainly in one country; only 5% of the bulls were used in two countries and 1% in three countries. Further, only 286 bulls (i.e. 0.3%) with records for protein yield and 321 bulls (i.e. 0.3%) with records for SCC were used in more than 10 countries. Breeding policies vary notably among countries in terms of how much countries rely on their own breeding schemes or whether they import most of their breeding animals. USA is an example of a country that has a long tradition of Holstein breeding: only 6% of the bulls were imported bulls for the 2007 protein yield data (Table 1). Conversely, Belgium is an example of a country that leans heavily on import: in the same data, 97% of the Holstein bulls used in Belgium were imported (Table 1). The number of common bulls between countries varied from zero to 1 194 for protein yield, with a mean of 178, and for SCC from one to 1 526, with a mean of 240. Substantial variation existed in the number of common bulls among countries. For both biological traits, French Red Holstein shared the smallest number of common bulls with the other countries and the USA, as a popular trading partner, shared the most.

Bottom-up PC runs were performed for both traits. Direct PC runs with ranks 15, 17, 19, 20 and 25 were carried out for protein yield to evaluate the optimal rank using the methods proposed by Meyer and Kirkpatrick [19]. For SCC, however, only the rank suggested by the bottom-up PC approach was used in the direct PC analyses.

The sensitivity of the bottom-up PC approach to differ-
ent orders of country addition was tested for a sub-
set of nine countries: France, USA, Czech Republic, Lat-
via, Poland, New-Zealand, Australia, Slovenia and Ire-
land. These nine countries that were well and loosely
linked, represented different hemispheres, and different
managing systems and thus constituted a representative
sample of all countries involved in the Interbull evalua-
tion. Two different orders were tested. Order1 was the
order of introduction of the countries above and order2
was the reverse of order1. For both orders, the analysis
started with four countries.

The order of country addition should not affect the
estimates, if only non-significant eigenvalues are
excluded. To test this, we modified the bottom-up PC
approach. Instead of selecting the best model based on
the AIC (steps 2e-f, 3d), we determined a rank based on
the proportion of explained variance in the transforma-
tion step 2a. Therefore, steps 2b-d became optional,
depending on whether the rank was reduced or not. We
tested three scenarios: the modified bottom-up approach
was required to include 97, 99, or 99.5% of the total var-
iance in the transformation step. For comparison, a full
fit direct PC analysis (rank 9) and a basic bottom-up
analysis were carried out for the sub-set of nine
countries.

The WOMBAT software [21] was used for the direct PC analyses, as well as for the variance component estimation in the bottom-up PC approach. The average information REML algorithm was applied for both approaches. Bull pedigrees were based on sire and maternal grand sire information. Genetic correlations estimated by Interbull in their test runs (protein yield: test run preceding August 2007 evaluation, SCC: test run preceding April 2009 evaluation) were used for comparison.

**Results and Discussion**

Bottom-up approach - effect of the order of country
addition on the results

Table 2 shows the effects of varying the order in which
countries are added in the modified bottom-up PC
approach on estimates of genetic correlations among the
nine countries considered. Explaining 97, 99, and 99.5%
of the total variance required the inclusion of the 6, 7 or
8 largest eigenvalues, respectively. Results clearly
revealed the importance of the correct rank selection.
When 99.5% of the variance in the eigenvalues was
taken into account (rank 8), the order of the country
addition had no influence on the estimates of the
genetic correlations. Thus, relatively large number of PC
were required to explain all necessary variation in the
data. When a larger proportion of the variance in the
eigenvalues was removed (ranks 7 and 6), the order of
the countries added in the analysis affected the estimates
of the genetic correlations. Especially the genetic corre-
lations of Slovenia and Latvia with the other countries
changed notably with the change in the order. Even
though the variance explained by the 6th and 7th PC
was small, those PC were, however, essential to be
included in the analysis to ensure that all necessary PC
were picked up. This phenomenon has also been
observed in other studies [22,11]. The bottom-up PC
approach and using AIC to determine the rank resulted
in rank 8 as well, indicating that the algorithm was able
to find the correct rank.
<table><caption>Table 2 The effect of the order of country addition on the estimates of the bottom-up PC approach for protein yield</caption><thead><tr><th colspan="2">Countries<sup>a</sup></th><th rowspan="2">Genetic correlations, direct PC 9</th><th colspan="3">Differences</th></tr><tr><th>1</th><th>2</th><th>Direct PC 9 vs. Bottom-up PC rank 8</th><th>Bottom-up PC order1<sup>b</sup> vs. order2<sup>c</sup></th><th></th></tr><tr><th></th><th></th><th></th><th></th><th>rank 8</th><th>rank 7</th><th>rank 6</th></tr></thead><tbody><tr><td>FRA</td><td>USA</td><td>0.87</td><td>0</td><td>0</td><td>0</td><td>0.04</td></tr><tr><td>FRA</td><td>CZE</td><td>0.58</td><td>0</td><td>0</td><td>0</td><td>0.03</td></tr><tr><td>FRA</td><td>LVA</td><td>0.24</td><td>-0.02</td><td>0</td><td>0</td><td>0.24</td></tr><tr><td>FRA</td><td>POL</td><td>0.65</td><td>0</td><td>0</td><td>0</td><td>-0.02</td></tr><tr><td>FRA</td><td>NZL</td><td>0.68</td><td>0</td><td>0</td><td>0</td><td>-0.07</td></tr><tr><td>FRA</td><td>AUS</td><td>0.76</td><td>0</td><td>0</td><td>0</td><td>-0.01</td></tr><tr><td>FRA</td><td>SVN</td><td>0.51</td><td>-0.01</td><td>0.02</td><td>-0.14</td><td>-0.17</td></tr><tr><td>FRA</td><td>IRL</td><td>0.78</td><td>0</td><td>0</td><td>0.01</td><td>0</td></tr><tr><td>USA</td><td>CZE</td><td>0.59</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>USA</td><td>LVA</td><td>0.31</td><td>-0.01</td><td>0.01</td><td>0.02</td><td>-0.40</td></tr><tr><td>USA</td><td>POL</td><td>0.56</td><td>0</td><td>0</td><td>0</td><td>0.02</td></tr><tr><td>USA</td><td>NZL</td><td>0.54</td><td>0</td><td>0</td><td>0</td><td>-0.02</td></tr><tr><td>USA</td><td>AUS</td><td>0.65</td><td>0</td><td>0</td><td>0</td><td>0.05</td></tr><tr><td>USA</td><td>SVN</td><td>0.36</td><td>0.02</td><td>-0.03</td><td>-0.12</td><td>-0.08</td></tr><tr><td>USA</td><td>IRL</td><td>0.63</td><td>0</td><td>0</td><td>0.02</td><td>0.08</td></tr><tr><td>CZE</td><td>LVA</td><td>0.09</td><td>-0.04</td><td>0</td><td>0.03</td><td>-0.02</td></tr><tr><td>CZE</td><td>POL</td><td>0.55</td><td>0</td><td>0</td><td>0</td><td>-0.05</td></tr><tr><td>CZE</td><td>NZL</td><td>0.47</td><td>0</td><td>0.01</td><td>0.01</td><td>0</td></tr><tr><td>CZE</td><td>AUS</td><td>0.53</td><td>0</td><td>0</td><td>0</td><td>-0.06</td></tr><tr><td>CZE</td><td>SVN</td><td>0.44</td><td>0</td><td>0.04</td><td>0</td><td>-0.04</td></tr><tr><td>CZE</td><td>IRL</td><td>0.51</td><td>0.01</td><td>0</td><td>-0.02</td><td>-0.04</td></tr><tr><td>LVA</td><td>POL</td><td>0.62</td><td>-0.01</td><td>0</td><td>-0.01</td><td>-0.28</td></tr><tr><td>LVA</td><td>NZL</td><td>0.15</td><td>-0.05</td><td>0.02</td><td>-0.01</td><td>0.13</td></tr><tr><td>LVA</td><td>AUS</td><td>0.51</td><td>-0.03</td><td>0.01</td><td>-0.01</td><td>-0.08</td></tr><tr><td>LVA</td><td>SVN</td><td>0.21</td><td>0.07</td><td>-0.01</td><td>-0.12</td><td>0.16</td></tr><tr><td>LVA</td><td>IRL</td><td>0.33</td><td>0.02</td><td>0.02</td><td>-0.02</td><td>0.08</td></tr><tr><td>POL</td><td>NZL</td><td>0.49</td><td>0</td><td>0</td><td>0</td><td>0.06</td></tr><tr><td>POL</td><td>AUS</td><td>0.70</td><td>0</td><td>0</td><td>0</td><td>0.07</td></tr><tr><td>POL</td><td>SVN</td><td>0.57</td><td>0.01</td><td>0</td><td>-0.04</td><td>0.06</td></tr><tr><td>POL</td><td>IRL</td><td>0.68</td><td>0</td><td>0</td><td>0</td><td>0.04</td></tr><tr><td>NZL</td><td>AUS</td><td>0.80</td><td>0</td><td>0</td><td>0</td><td>0.01</td></tr><tr><td>NZL</td><td>SVN</td><td>0.34</td><td>-0.01</td><td>0.03</td><td>-0.14</td><td>-0.33</td></tr><tr><td>NZL</td><td>IRL</td><td>0.81</td><td>-0.01</td><td>0</td><td>0.01</td><td>-0.05</td></tr><tr><td>AUS</td><td>SVN</td><td>0.42</td><td>0.01</td><td>0.01</td><td>-0.14</td><td>-0.07</td></tr><tr><td>AUS</td><td>IRL</td><td>0.84</td><td>0</td><td>0</td><td>0.01</td><td>0.07</td></tr><tr><td>SVN</td><td>IRL</td><td>0.74</td><td>-0.03</td><td>0</td><td>-0.12</td><td>-0.13</td></tr><tr><td>Mean</td><td></td><td>0.54</td><td>-0.002</td><td>0.003</td><td>-0.021</td><td>-0.022</td></tr><tr><td>Mean_abs<sup>d</sup></td><td></td><td>0.54</td><td>0.010</td><td>0.006</td><td>0.028</td><td>0.085</td></tr><tr><td>Max</td><td></td><td>0.87</td><td>0.07</td><td>0.04</td><td>0.14</td><td>0.40</td></tr></tbody></table>

For comparison, the estimates of the genetic correlations from the direct PC full rank model and the differences in the estimates of the genetic correlations from the direct PC full rank and the bottom-up PC rank 8 models are also presented. The mean and maximum (max) values of genetic correlations from the direct PC full fit and mean and max differences from above comparisons are shown at the bottom of the table.

<footnote>a Keys of the country codes are shown in Table 1</footnote>

<footnote>b Order 1: FRA, USA, CZE, LVA, POL, NZL, AUS, SVN, IRL</footnote>

<footnote>c Order 2 is reverse to order 1</footnote>

<footnote>d Mean of the absolute differences</footnote>
### Correct rank

Information used for the model selection of the protein yield data under the direct PC approach is summarized in Table 3. AIC for the 25-trait analysis was highest for a model fitting 19 PC and log likelihood did not increase significantly beyond rank 19. The sums of eigenvalues and the leading PC were, in practice, identical between models fitting ranks 19, 20 and 25. Furthermore, the last five eigenvalues equalled zero with a precision of two decimals, thus they included basically no information. Based on the $\sqrt{\Delta r}$ values, estimates of genetic correlations from the models fitting ranks 19, 20 and 25 were almost identical. Differences in the estimates started to increase, as the rank was dropped to 17

<table><caption>Table 3 Selection of the appropriate rank for protein yield under the direct PC approach.</caption><thead><tr><th></th><th>Rank 15</th><th>Rank 17</th><th>Rank 19</th><th>Rank 20</th><th>Full fit</th></tr></thead><tbody><tr><td>-1/2 AIC<sup>a</sup></td><td>-68</td><td>-19</td><td>0</td><td>-4</td><td>-19</td></tr><tr><td>log L<sup>b</sup></td><td>-105</td><td>-36</td><td>-2</td><td>0</td><td>0</td></tr><tr><td>√Δr</td><td>0.029</td><td>0.017</td><td>0.004</td><td>0</td><td>0.001</td></tr><tr><td>No of parameters</td><td>271</td><td>290</td><td>305</td><td>311</td><td>325</td></tr><tr><td>Sum of eigenvalues</td><td>1696</td><td>1695</td><td>1695</td><td>1695</td><td>1695</td></tr><tr><td>E1<sup>d</sup></td><td>1326</td><td>1330</td><td>1331</td><td>1331</td><td>1331</td></tr><tr><td>E2</td><td>78.9</td><td>76.7</td><td>76.1</td><td>76.1</td><td>76.0</td></tr><tr><td>E3</td><td>69.8</td><td>65.0</td><td>60.3</td><td>60.1</td><td>60.1</td></tr><tr><td>E4</td><td>43.6</td><td>44.5</td><td>47.4</td><td>47.2</td><td>47.1</td></tr><tr><td>E5</td><td>36.6</td><td>35.2</td><td>33.2</td><td>33.0</td><td>33.1</td></tr><tr><td>E6</td><td>30.9</td><td>30.4</td><td>28.8</td><td>28.6</td><td>28.6</td></tr><tr><td>E7</td><td>22.3</td><td>21.3</td><td>21.4</td><td>21.3</td><td>21.3</td></tr><tr><td>E8</td><td>19.7</td><td>17.8</td><td>17.2</td><td>17.3</td><td>17.2</td></tr><tr><td>E9</td><td>15.0</td><td>15.4</td><td>16.2</td><td>15.9</td><td>16.0</td></tr><tr><td>E10</td><td>12.9</td><td>12.3</td><td>12.3</td><td>12.3</td><td>12.3</td></tr><tr><td>E11</td><td>10.6</td><td>10.5</td><td>10.6</td><td>10.6</td><td>10.6</td></tr><tr><td>E12</td><td>9.8</td><td>9.9</td><td>8.8</td><td>8.5</td><td>8.5</td></tr><tr><td>E13</td><td>9.2</td><td>8.6</td><td>8.4</td><td>8.3</td><td>8.3</td></tr><tr><td>E14</td><td>6.3</td><td>6.5</td><td>6.5</td><td>6.7</td><td>6.7</td></tr><tr><td>E15</td><td>4.3</td><td>5.2</td><td>5.2</td><td>5.2</td><td>5.2</td></tr><tr><td>E16</td><td></td><td>3.9</td><td>4.2</td><td>4.1</td><td>4.1</td></tr><tr><td>E17</td><td></td><td>2.7</td><td>3.2</td><td>3.3</td><td>3.3</td></tr><tr><td>E18</td><td></td><td></td><td>2.8</td><td>2.8</td><td>2.8</td></tr><tr><td>E19</td><td></td><td></td><td>1.1</td><td>1.3</td><td>1.3</td></tr><tr><td>E20</td><td></td><td></td><td></td><td>1.1</td><td>1.2</td></tr><tr><td>E21</td><td></td><td></td><td></td><td></td><td>0.0</td></tr><tr><td>E22</td><td></td><td></td><td></td><td></td><td>0.0</td></tr><tr><td>E23</td><td></td><td></td><td></td><td></td><td>0.0</td></tr><tr><td>E24</td><td></td><td></td><td></td><td></td><td>0.0</td></tr><tr><td>E25</td><td></td><td></td><td></td><td></td><td>0.0</td></tr></tbody></table>

and 15. Thus, results suggested that either rank 19 or 20 is the appropriate rank to describe the genetic variation in protein yield. This means a reduction from 5 to 6% in the number of parameters needed to describe the complete 25 × 25 (co)variance matrix, because the number of parameters for the direct PC is $p = r(2t - r + 1)/2$.

The bottom-up PC run terminated with rank 20 for protein yield, indicating that the approach is able to find the correct rank. Under the bottom-up PC, G is obtained by backtransforming it and only the matrix of eigenvalues is directly estimated, thus $p = r(r + 1)/2$, and only 65% of the parameters were sufficient to describe the complete (co)variance matrix for that method. Based on the bottom-up results, the appropriate rank was 15 for SCC. Thus, only 44% of the parameters under the bottom-up PC were needed to describe the 23 × 23 (co)variance matrix for SCC, whereas the corresponding number for the direct PC rank 15 analysis was 87%.

Our results on the importance of fitting an optimal rank in the principal component analysis are supported by earlier studies by Meyer [22,11] and Meyer and Kirkpatrick [19]. While studying reduced rank multivariate animal models for beef cattle, Meyer noticed that fitting too few principal components resulted in inaccurate estimates of the genetic parameters [22,11]. A more recent study of Meyer and Kirkpatrick [19] has listed three sources of bias of reduced rank estimates: spread of sample roots, constraining estimates to the parameter space and picking up the wrong subset of the genetic PC, if too few PC are fitted.

### Comparison of genetic correlations

Figures 1 and 2 summarize the genetic correlations for protein yield and SCC, respectively. Heat map type plots demonstrate the magnitude of the genetic correlations among countries from different approaches, as well as the differences in genetic correlations between approaches. Descriptive statistics of the variation in the correlations from different approaches are collected in the tables below both figures. In general, differences in the estimates obtained with different approaches were small, especially for SCC. Genetic correlations for SCC were high in magnitude for all countries, whereas those for protein yield were very low for some countries - contrary to the biologically justified expectation of on average high genetic correlations. The different approaches did not vary in this respect.

The average estimates of genetic correlations from the direct PC rank 20, direct PC full fit, bottom-up PC rank 20 and Interbull analyses for protein yield were very similar, ranging from 0.68 to 0.70 (Figure 1). Based on the first and third quantiles and the median, the distribution of the Interbull estimates was on a somewhat

<footnote><sup>a</sup> Akaike's information criterion, expressed as deviation from highest value</footnote>

<footnote><sup>b</sup> Maximum Log Likelihood, expressed as deviation from highest value</footnote>

<footnote><sup>c</sup> A square root of the average squared deviation of the estimated genetic correlations. The estimates obtained under the direct PC rank 20 model were used as the estimates of comparison</footnote>

<footnote><sup>d</sup> Eigenvalues 1,...,25 of the G matrix</footnote>
<figure><img src="image_2.png" /></figure>

higher level compared to those of the PC approaches. Nevertheless, the Interbull estimates included the lowest value for protein yield, being as low as 0.02 between New-Zealand and Latvia. The means of the SCC estimates were much higher, from 0.87 to 0.89 (Figure 2), compared to those for protein yield. In addition, the lowest values were rather high, ranging from 0.61

(Interbull) to 0.65 (bottom-up PC). The distributions of the estimates of genetic correlations from the different approaches were very similar for SCC, although those for the Interbull were on a slightly higher level. The plots of genetic correlations also showed that over-parameterization of the model for protein yield had virtually no effect on the estimates (Figure 1) since both rank 20
<figure><img src="image_3.png" /></figure>

and 25 models resulted in almost identical genetic correlations.

Figure 3 and Table 4 illustrate the challenges of the datasets used in this study. Plotting the genetic correlations with the number of common bulls between countries revealed that for protein yield, the level of the correlation estimates increased with the number of common bulls (Figure 3). This was, however, not the case

for SCC. Furthermore, the standard deviations of the genetic correlations within classes defined by the number of common bulls were notably larger for protein yield than for SCC (Figure 3). In addition, a low number of common bulls was associated with larger differences in the estimates between the different approaches, hinting that the approaches reacted differently to challenges in the datasets.
<figure><img src="image_4.png" /></figure>

On the one hand, the lack of connections between countries hinders the estimation of genetic parameters and this can explain the low genetic correlations for e.g. Slovenia and Latvia (Tables 1 and 4). On the other hand, countries like South-Africa and New-Zealand were also associated with a lower level of genetic correlations for protein yield (Table 4). However, on average, they have strong links with the other countries (Table 1). Furthermore, the standard errors of the estimates of New-Zealand and South-Africa with the other countries were relatively small, unlike those of Slovenia and Latvia with the other countries. Based on the study of Jakobsen et al. [1], different trait definitions and national genetic evaluation models, as well as genotype by environment interactions explain the low to moderate genetic correlations in international genetic evaluations of dairy bulls.

One of the main challenges in national genetic evaluation schemes incorporating foreign bulls is to adequately model systematic genetic differences by defining genetic

<table><caption>Table 4 Dissection of the estimates of France, New-Zealand South-Africa, Slovenia and Latvia: magnitude of the genetic correlations and their standard errors for protein yield</caption><thead><tr><th></th><th>FRA<sup>a</sup></th><th>NZL</th><th>ZAF</th><th>SVN</th><th>LVA</th></tr></thead><tbody><tr><td colspan="6">Genetic correlations</td></tr><tr><td>Min</td><td>0.40</td><td>0.22</td><td>0.17</td><td>0.23</td><td>0.08</td></tr><tr><td>Median</td><td>0.80</td><td>0.56</td><td>0.49</td><td>0.51</td><td>0.43</td></tr><tr><td>Mean</td><td>0.76</td><td>0.57</td><td>0.49</td><td>0.50</td><td>0.40</td></tr><tr><td>Max</td><td>0.90</td><td>0.81</td><td>0.69</td><td>0.69</td><td>0.62</td></tr><tr><td colspan="6">SEs of genetic correlations</td></tr><tr><td>Min</td><td>0.01</td><td>0.02</td><td>0.04</td><td>0.07</td><td>0.07</td></tr><tr><td>Median</td><td>0.02</td><td>0.03</td><td>0.05</td><td>0.08</td><td>0.08</td></tr><tr><td>Mean</td><td>0.03</td><td>0.05</td><td>0.06</td><td>0.09</td><td>0.09</td></tr><tr><td>Max</td><td>0.09</td><td>0.15</td><td>0.23</td><td>0.14</td><td>0.18</td></tr></tbody></table>

groups. This holds especially for countries with small populations and where information on their daughters is scarce. This might result in proofs for foreign sires which are biased. As these national proofs are the data used in variance component estimation, inadequate genetic grouping at the national level may be one of the factors contributing to low estimates of genetic correlations for protein yield in different countries. Because selection of bulls is predominantly targeting production traits, the impact of ill-defined genetic groups on proofs for other, non-production traits is expected to be smaller. In addition, imported bulls may be more representative of the population in the country of origin as, for instance, for SCC.

Altogether, using a more parsimonious covariance structure did not resolve the problem of some small genetic correlations for protein yield. Currently, Interbull post-processes genetic correlations to correspond to the level of some justified expectation. This is done by utilizing information on the new estimates, estimates from the previous run, Interbull's own expectations and for non-Holstein breeds, correlations from Holstein [3]. Until the ultimate reason for the low estimates of genetic correlations has been identified, one alternative to the current post-processing would be to apply prior expectations under the Bayesian MACE suggested by, e. g., Mark et al. [23]. With insufficient data, the prior expectations would not be overridden and the level of the final estimates would be closer to their biologically justified expectations compared to the current non-post-processed estimates or those obtained under the PC approaches. By applying the approach suggested by Mark et al. [23], we might reduce the degree of parsimony which can be attained using the PC approaches, but this may be off-set by the prior information utilized and thus reduce mean square errors.

## Performance of the PC approaches

The run time of the direct PC analysis for protein yield reached a maximum for the rank 15 model (22 days), decreased with increasing rank, being shortest for the rank 20 model (5 days) and was 17 days for the full fit model. The memory needed for the direct PC rank 20 model for protein yield was 4.1 GB, whereas it was highest i.e. 6.3 GB for the full fit model. Thus, the costs of the over-parameterization were a longer run time and a higher RAM memory requirement without any increase in the accuracy of the estimation. Furthermore, the magnitude of standard errors of the estimates increased with the number of parameters to be estimated. This is a consequence of the increased sampling variance when estimating more parameters [see [22,11]]. Interestingly, fitting too few parameters in the model prolonged the run time. This occurred also for SCC (results not

<footnote>a Keys of the country codes are shown in Table 1</footnote>
<table><caption>Table 5 Run time (d:hr:min) and number of iterates required for analyses of protein yield</caption><thead><tr><th rowspan="2"></th><th colspan="3">Country addition step</th><th colspan="3">Rank reduction step</th><th rowspan="2">Total time</th></tr><tr><th>Countries</th><th>Iterates</th><th>Time</th><th>Rank</th><th>Iterates</th><th>Time</th></tr></thead><tbody><tr><td rowspan="15">Bottom-up PC</td><td>7</td><td>5</td><td>0:00:46</td><td>7</td><td>4</td><td>0:00:26</td><td>0:01:12</td></tr><tr><td>8</td><td>9</td><td>0:01:48</td><td>8</td><td>4</td><td>0:00:41</td><td>0:02:29</td></tr><tr><td>9</td><td>8</td><td>0:02:21</td><td>9</td><td>5</td><td>0:01:12</td><td>0:03:33</td></tr><tr><td>10</td><td>8</td><td>0:03:24</td><td>10</td><td>6</td><td>0:02:02</td><td>0:05:26</td></tr><tr><td>11</td><td>11</td><td>0:06:05</td><td>11</td><td>5</td><td>0:02:25</td><td>0:08:30</td></tr><tr><td>12</td><td>14</td><td>0:10:24</td><td>11</td><td>6</td><td>0:03:49</td><td>0:14:13</td></tr><tr><td>13</td><td>13</td><td>0:10:53</td><td>12</td><td>6</td><td>0:03:50</td><td>0:14:43</td></tr><tr><td>14</td><td>13</td><td>0:14:09</td><td>13</td><td>6</td><td>0:05:00</td><td>0:19:09</td></tr><tr><td>15</td><td>12</td><td>0:16:34</td><td>14</td><td>5</td><td>0:05:28</td><td>0:22:02</td></tr><tr><td>16</td><td>77</td><td>6:03:56</td><td>15</td><td>8</td><td>0:11:04</td><td>6:15:00</td></tr><tr><td>17</td><td>12</td><td>1:06:04</td><td>16</td><td>6</td><td>0:10:40</td><td>1:16:44</td></tr><tr><td>18</td><td>17</td><td>2:10:31</td><td>16</td><td>13</td><td>1:03:47</td><td>3:14:18</td></tr><tr><td>19</td><td>12</td><td>1:13:49</td><td>17</td><td>6</td><td>0:13:00</td><td>2:02:49</td></tr><tr><td>20</td><td>21</td><td>3:14:19</td><td>17</td><td>12</td><td>1:07:15</td><td>4:21:34</td></tr><tr><td>21</td><td>14</td><td>1:22:37</td><td>18</td><td>5</td><td>0:13:08</td><td>2:11:45</td></tr><tr><td>22</td><td>28</td><td>5:11:05</td><td>19</td><td>7</td><td>0:22:04</td><td>6:09:09</td></tr><tr><td>23</td><td>15</td><td>3:14:23</td><td>19</td><td>11</td><td>1:15:25</td><td>5:04:48</td></tr><tr><td>24</td><td>15</td><td>3:17:11</td><td>20</td><td>6</td><td>1:24:00</td><td>4:17:35</td></tr><tr><td>25</td><td>14</td><td>4:05:09</td><td>20</td><td>12</td><td>2:04:03</td><td>6:09:12</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>46:23:11</td></tr><tr><td>Direct PC</td><td>25</td><td></td><td></td><td>20</td><td>24</td><td>5:13:27</td><td></td></tr></tbody></table>

shown) and for the factor analytic models (manuscript in preparation). If the rank of the model is reduced too much, the number of available parameters is not sufficient to describe the (co)variance structure of the model, which in turn, detrimentally affects the convergence rate of the REML analysis.

The effects of the possible problems in the datasets accumulated as the bottom-up PC approach was used, which the protein yield data clearly demonstrated (Table 5). The first 15 countries introduced in the analysis were mostly well-linked countries that test many AI-bulls. They contributed 88% of the total data, but the computing time was less than 9% of the total time used. In the bottom-up approach, each time a new country/ trait is added, the (co)variance matrix must be reestimated. Furthermore, the estimation process is carried out twice because two possible models are compared to test if the country addition requires an increase of rank or not. Thus, once difficulties in the iteration process have started, they will, at least to some extent, continue to the very end of the sequential country addition-rank reduction-process. On the other hand, when no larger problems are embedded in the data, the difference in the total estimation time between direct and bottom-up

<table><caption>Table 6 Run time (d:hr:min) and number of iterates required for analyses of SCC</caption><thead><tr><th rowspan="2"></th><th colspan="3">Country addition step</th><th colspan="3">Rank reduction step</th><th rowspan="2">Total time</th></tr><tr><th>Countries</th><th>Iterates</th><th>Time</th><th>Rank</th><th>Iterates</th><th>Time</th></tr></thead><tbody><tr><td rowspan="15">Bottom-up PC</td><td>7</td><td>25</td><td>0:02:45</td><td>5</td><td>3+2+4<sup>a</sup></td><td>0:00:28</td><td>0:03:13</td></tr><tr><td>8</td><td>14</td><td>0:00:56</td><td>6</td><td>3</td><td>0:00:09</td><td>0:01:05</td></tr><tr><td>9</td><td>13</td><td>0:01:29</td><td>7</td><td>5</td><td>0:00:22</td><td>0:01:51</td></tr><tr><td>10</td><td>6</td><td>0:01:19</td><td>8</td><td>5</td><td>0:00:37</td><td>0:01:56</td></tr><tr><td>11</td><td>6</td><td>0:01:58</td><td>9</td><td>5</td><td>0:00:56</td><td>0:02:54</td></tr><tr><td>12</td><td>3</td><td>0:01:50</td><td>9</td><td>21</td><td>0:05:05</td><td>0:06:55</td></tr><tr><td>13</td><td>11</td><td>0:03:59</td><td>10</td><td>5</td><td>0:01:21</td><td>0:05:20</td></tr><tr><td>14</td><td>26</td><td>0:12:18</td><td>11</td><td>8</td><td>0:02:49</td><td>0:15:07</td></tr><tr><td>15</td><td>22</td><td>0:13:43</td><td>11</td><td>6</td><td>0:03:11</td><td>0:16:54</td></tr><tr><td>16</td><td>8</td><td>0:05:26</td><td>11</td><td>4</td><td>0:02:21</td><td>0:07:47</td></tr><tr><td>17</td><td>9</td><td>0:06:01</td><td>11</td><td>4</td><td>0:02:17</td><td>0:08:18</td></tr><tr><td>18</td><td>10</td><td>0:06:31</td><td>12</td><td>8</td><td>0:03:58</td><td>0:10:29</td></tr><tr><td>19</td><td>12</td><td>0:10:09</td><td>12</td><td>6</td><td>0:04:04</td><td>0:14:13</td></tr><tr><td>20</td><td>11</td><td>0:11:20</td><td>13</td><td>5</td><td>0:03:51</td><td>0:15:11</td></tr><tr><td>21</td><td>13</td><td>0:14:34</td><td>14</td><td>7</td><td>0:06:25</td><td>0:20:59</td></tr><tr><td>22</td><td>15</td><td>1:01:54</td><td>14</td><td>6</td><td>0:07:39</td><td>1:09:33</td></tr><tr><td>23</td><td>9</td><td>0:13:13</td><td>15</td><td>7</td><td>0:07:59</td><td>0:21:12</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>7:18:57</td></tr><tr><td>Direct PC</td><td>23</td><td></td><td></td><td>15</td><td>86</td><td>7:00:02</td><td></td></tr></tbody></table>

PC approaches is rather small, as demonstrated for SCC (Table 6).

Overall, both approaches tested in this study performed very well and estimates of genetic correlations were similar to the Interbull estimates. Both PC approaches were applied to complete datasets unlike those suggested in the earlier studies [7,8,12,9] and the current Interbull procedure [2,3]. One advantage of the direct over the bottom-up PC approach are the potentially much reduced computational requirements. This applies in particular when an analysis is started with a small subset of countries and countries with a problematic data structure are added in early on. Such problems were encountered for protein yield, resulting in the computational requirements shown in table 5. The current test version of the bottom-up PC approach has not been streamlined yet by any means. The analysis of the performance of the bottom-up approach (Tables 5 and 6) as well as preliminary tests give evidence that the computation time can be reduced by starting from a higher number of countries. Furthermore, when a new country is added, zero starting covariances between new and old countries could be replaced with covariances calculated from the mean correlation of countries already in the dataset and from the variances of those

<footnote><sup>a</sup> Three rank reduction steps were needed before the appropriate rank was found.</footnote>
countries. The advantage of the bottom-up approach is
that we estimate *r* elements of *r × r D*<sub>*r*</sub>-matrix in the
parameter estimation step and therefore, there is no
danger of picking up the wrong subset of principal
components.

**Conclusions**

This study shows that both the direct and the bottom-
up principal component approaches and the use of
models with optimal rank are useful in the variance
component estimation for MACE. Furthermore, both
approaches can be applied to large datasets and data
sub-setting is not needed. Based on the results, we
emphasize the importance of the selection of the appro-
priate rank of the (co)variance matrix to obtain good
estimates. The bottom-up PC approach is capable of
determining the appropriate rank for highly over-para-
meterized models and thus leads to a more parsimonous
variance structure. However, with a predetermined rank,
the direct PC approach needs less computing time than
the bottom-up PC. The third approach that is consid-
ered for variance component estimation for MACE is
the direct factor analytic approach that will be presented
in an upcoming paper.

**Acknowledgements**

The study was part of the cooperation project of Interbull Centre and MTT
Agrifood Research Finland.

**Author details**

¹Biotechnology and Food Research, Biometrical Genetics, MTT Agrifood Research Finland, 31600 Jokioinen, Finland. ²Animal Genetics and Breeding Unit, University of New England, Armidale NSW 2351, Australia. ³Department of Animal Breeding and Genetics, SLU, Box 7023, S-75007 Uppsala, Sweden.
⁴UMR 1313 INRA, Génétique Animale et Biologie Intégrative, 78352 Jouy-en-Josas Cedex, France. ⁵Interbull Centre, Department of Animal Breeding and Genetics, SLU, Box 7023, S-75007 Uppsala, Sweden.

**Authors' contributions**

AMT performed the statistical analyses, modified the bottom-up PC approach and wrote the first draft of the manuscript. EAM developed the bottom-up PC approach and supervised the study. KM developed the direct PC approach, modified WOMBAT for the needs of this study and supervised the study. JJ and WFF provided the datasets. MHL, VD, WFF and JJ also supervised the study. All authors contributed to the writing of the manuscript.

**Competing interests**

The authors declare that they have no competing interests.

Received: 1 November 2010 Accepted: 24 May 2011

Published: 24 May 2011

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doi:10.1186/1297-9686-43-21

Cite this article as: *Tyrisevä *et al.**: Principal component approach in variance component estimation for international sire evaluation.
*Genetics Selection Evolution* 2011 **43**:21.