 Research
 Open Access
 Published:
Unified method to integrate and blend several, potentially related, sources of information for genetic evaluation
Genetics Selection Evolution volume 46, Article number: 59 (2014)
Abstract
Background
A condition to predict unbiased estimated breeding values by best linear unbiased prediction is to use simultaneously all available data. However, this condition is not often fully met. For example, in dairy cattle, internal (i.e. local) populations lead to evaluations based only on internal records while widely used foreign sires have been selected using internally unavailable external records. In such cases, internal genetic evaluations may be less accurate and biased. Because external records are unavailable, methods were developed to combine external information that summarizes these records, i.e. external estimated breeding values and associated reliabilities, with internal records to improve accuracy of internal genetic evaluations. Two issues of these methods concern doublecounting of contributions due to relationships and due to records. These issues could be worse if external information came from several evaluations, at least partially based on the same records, and combined into a single internal evaluation. Based on a Bayesian approach, the aim of this research was to develop a unified method to integrate and blend simultaneously several sources of information into an internal genetic evaluation by avoiding doublecounting of contributions due to relationships and due to records.
Results
This research resulted in equations that integrate and blend simultaneously several sources of information and avoid doublecounting of contributions due to relationships and due to records. The performance of the developed equations was evaluated using simulated and real datasets. The results showed that the developed equations integrated and blended several sources of information well into a genetic evaluation. The developed equations also avoided doublecounting of contributions due to relationships and due to records. Furthermore, because all available external sources of information were correctly propagated, relatives of external animals benefited from the integrated information and, therefore, more reliable estimated breeding values were obtained.
Conclusions
The proposed unified method integrated and blended several sources of information well into a genetic evaluation by avoiding doublecounting of contributions due to relationships and due to records. The unified method can also be extended to other types of situations such as singlestep genomic or multitrait evaluations, combining information across different traits.
Background
Simultaneous use of all available data by best linear unbiased prediction (BLUP) is a condition to predict unbiased estimated breeding values (EBV) [1]. However, this condition is not often fully met. For example, in dairy cattle, while foreign bulls are often widely used, e.g. through artificial insemination, evaluating populations based only on internal phenotypic data (i.e. internal records) will lead to potentially biased and less accurate evaluations [2]. The reason is that external phenotypic data used to select these foreign bulls are not available at the internal level. Multiple across country evaluation (MACE), performed at an international level by International Bull Service (Interbull, Uppsala, Sweden), allows EBV, for each population scale, to be aggregated into a single ranking for international dairy sires. However, this has no influence on internal evaluations. These issues are also relevant in the setting of current developments of genomic multistep or singlestep prediction methods (e.g., [3][5]).
Because external phenotypic data are not available at the internal level, methods were developed to combine external information, i.e. external EBV and associated reliabilities (REL), with internal data to improve accuracy of internal genetic evaluations. A first type of approaches is based on performing, a posteriori, an additional step after the genetic evaluation at the internal level. These approaches combine external and internal EBV based on selection index theory (e.g., [6]), based on mixed model theory (e.g., [7]) or based on bivariate evaluations (e.g., [8]). One of the problems of a posteriori approaches is that external information used for selection will not contribute to the estimation of fixed effects at the internal level, which can create potential biases. A second type of approaches combines external information simultaneously with internal phenotypic data in genetic evaluations at the internal level. Simultaneous combination of external information and internal phenotypic data can be carried out using different methods. However, to our knowledge, the following two approaches are the most used. First, external information can be directly included by converting this information into pseudorecords for fictive daughters of external animals (e.g., [2]). Similar approaches were proposed to include external information into internal singlestep genomic evaluations (e.g., [5],[9]). Second, external information can be directly included by changing both the mean and (co)variance of the prior distributions of genetic effects in a Bayesian approach, as mentioned, for example, by Gianola and Fernando [10]. Quaas and Zhang [11],[12] and Legarra et al. [13] proposed two Bayesian derivations to integrate external information into internal genetic evaluations in the context of multibreed genetic evaluations for beef cattle. These two derivations consider external information as priors of internal genetic effects. Vandenplas and Gengler [14] compared these two derivations and proposed some improvements that concerned mainly doublecounting of contributions due to relationships among external animals. Indeed, an EBV of an animal combines information from its own records (i.e., contributions due to own records) and from records of all relatives through its parents and its progeny (i.e., contributions due to relationships) [6],[15]. Therefore, integration of EBV for relatives can cause the same contributions that are due to relationships to be counted several times, which can bias genetic evaluations at the internal level.
Both types of approaches i.e. that combine available information a posteriori or simultaneously, raise another issue if the external information results from an evaluation that combines external and internal records, which is that some contributions due to records will be considered several times when external information is combined with internal records. Although this is a major issue for common sources of external information (e.g., MACE information), to our knowledge, only a few studies have proposed solutions to the doublecounting of contributions due to records (e.g., [5],[16],[17]). The proposed solutions were developed as an additional preprocessing step before integration of external information. Furthermore, in many situations, integration of several sources of external information into genetic evaluations at the internal level may be needed but this has not been studied to our knowledge. In such cases, doublecounting of contributions due to records could be worse if external information from several evaluations were, at least partially, based on the same internal records, and/or on the same external records, and integrated into the same genetic evaluation.
Thus, the aim of this research was to develop a unified method to integrate and blend simultaneously several, potentially related, external sources of information into an internal genetic evaluation based on a Bayesian approach. In order to achieve this aim, methods were developed to avoid doublecounting of contributions due to relationships and due to records generated by the integration of several sources of information. This resulted in modified mixed model equations (MME) that integrate and blend simultaneously several sources of information and avoid doublecounting of contributions due to relationships and due to records. The performance of the developed equations was evaluated using simulated and real datasets.
Methods
Integration of several sources of external information
Assume an internal genetic evaluation (referred to with the subscript E_{0}) based on internal data (i.e. a set of phenotypic records: {\mathbf{y}}_{{\mathrm{E}}_{0}}) that provides internal information (i.e. EBV and associated REL obtained from the evaluation E_{0}). Also, assume an i^{th} external genetic evaluation (i = 1, 2, ¿, N, referred to with the subscript E_{i}) that is based on the i^{th} source of external data (i.e. the i^{th} set of phenotypic records not used by evaluation E_{0} and free of internal data: {\mathbf{y}}_{{\mathrm{E}}_{\mathrm{i}}}) and that provides the i^{th} source of external information, i.e., all available external EBV (EBV_{Ei}) and associated REL (e.g., EBV and associated REL obtained from evaluation E_{1} based only on external data E_{1,} and EBV and associated REL obtained from evaluation E_{2} based only on external data E_{2}). In addition to be free of internal data, it is also assumed that each i^{th} source of external data was free of the other N1 sources of external data. These assumptions lead to each i^{th} source of external information to be free of internal data and information, as well as of the N1 other external data and information.
Two groups of animals, hereafter called external and internal animals, are defined according to the i^{th} source of external information. Therefore, for each i^{th} source of external information, external animals (subscript A_{i} with i = 1, 2,¿, N) are defined as animals that are associated with this i^{th} source of external information and for which internal data and/or information is available or that have relationships with animals involved in the internal evaluation E_{0}. All animals that are not defined as external animals for the i^{th} source of external information are defined as internal animals (subscript {\mathrm{A}}_{\mathrm{i}}^{0}). Internal animals are then defined as animals associated with only internal information when considering the i^{th} source of external information. It is noted that external animals may be associated with different sources of external information and that an animal may be considered as external for the i^{th} source of external information and internal for the N1 other sources of external information because the definitions of external and internal animals depend only on the source of external information considered. Those definitions are summarized in Table 1. In addition, because pedigree information for animals can be easily integrated into a genetic evaluation, it is assumed that the same complete pedigree information could be used for all animals for each genetic evaluation. Concerning the notation of matrices in the following sections (e.g., {\mathbf{X}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{l}}\right)}), the subscript E_{i} refers to the i^{th} source of external information and the subscript within brackets (A_{l}) refers to the l^{th} group of animals.
The N sources of external information must be integrated into the internal evaluation E_{0}. For external animals associated with the i^{th} source of external information, all EBV_{Ei} are summarized by the vector of external EBV, {\widehat{\mathbf{u}}}_{{\mathbf{E}}_{\mathbf{i}}\left({\mathbf{A}}_{\mathbf{i}}\right)}, and by the prediction error (co)variance matrix, {\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}. Because {\widehat{\mathbf{u}}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)} could be estimated with an equivalent external genetic evaluation that includes the internal animals in the pedigree through a genetic (co)variance matrix extended to all animals for the i^{th} source of external information, {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}}=\left[\begin{array}{cc}\hfill {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}^{0}{\mathrm{A}}_{\mathrm{i}}^{0}\right)}\hfill & \hfill {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}{\mathrm{A}}_{\mathrm{i}}^{0}\right)}\hfill \\ \hfill {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}^{0}{\mathrm{A}}_{\mathrm{i}}\right)}\hfill & \hfill {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}{\mathrm{A}}_{\mathrm{i}}\right)}\hfill \end{array}\right], the vector of external EBV for all internal and external animals for the i^{th} source of external information is estimated as:
A modified set of multitrait mixed model equations that integrate N sources of external information, each summarized by {\widehat{\mathbf{u}}}_{{\mathrm{E}}_{\mathrm{i}}} and its associated prediction error (co)variance matrix {\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}} for the i^{th} source of external information, can be written as [See Additional file 1 for the derivation of the equations]:
where {\mathbf{X}}_{{\mathrm{E}}_{0}} and {\mathbf{Z}}_{{\mathrm{E}}_{0}} are incidence matrices relating records in {\mathbf{y}}_{{\mathrm{E}}_{0}} to the vector of fixed effects {\widehat{\mathbf{\beta}}}_{{\mathrm{E}}_{0}} and the vector of random additive genetic effects {\widehat{\mathbf{u}}}_{{\mathrm{E}}_{0}}, respectively, {\mathbf{G}}_{{\mathrm{E}}_{0}}^{1} is the inverse of the internal additive genetic (co)variance matrix associated with the internal genetic evaluation E_{0} that includes all internal and external animals and {\mathbf{R}}_{{\mathrm{E}}_{0}}^{1} is the inverse of the residual (co)variance matrix.
For the approximation of {\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}, it can be shown that [See Additional file 1]: {\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}={\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}+{\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}}^{\mathbf{\prime}}{\mathbf{R}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}{\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}}, where {\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}} is the incidence matrix relating records of i^{th} external data to internal and external animals and {\mathbf{R}}_{{\mathrm{E}}_{\mathrm{i}}}^{1} is the residual (co)variance matrix for the i^{th} source of external information. Thereby, {\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}}^{1} is approximated by {\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}={\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}+{\mathbf{\Lambda}}_{{\mathrm{E}}_{\mathrm{i}}}, where {\mathbf{\Lambda}}_{{\mathrm{E}}_{\mathrm{i}}} is a block diagonal variance matrix with one block per animal [12],[14] and {\mathbf{\Lambda}}_{{\mathrm{E}}_{\mathrm{i}}}\approx {\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}}^{\mathbf{\prime}}{\mathbf{R}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}{\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}}. Each diagonal block of {\mathbf{\Lambda}}_{{\mathrm{E}}_{\mathrm{i}}} is equal to {\mathbf{\Lambda}}_{{\mathrm{E}}_{\mathrm{i}}\left(\mathrm{j}\right)}{\mathbf{R}}_{0}^{1}{\mathbf{\Delta}}_{{\mathrm{E}}_{\mathrm{i}}\left(\mathrm{j}\right)} for j = 1, 2, ¿, J animals, where the matrix R_{0} is a matrix of residual (co)variance among traits and the j^{th} matrix {\mathbf{\Delta}}_{{\mathrm{E}}_{\mathrm{i}}\left(\mathrm{j}\right)} is a diagonal matrix with elements \sqrt{{\mathrm{RE}}_{\mathrm{ijk}}} where k = 1,2,¿, K traits. Element RE_{ijk} is the effective number of records, i.e. record equivalents, for the j^{th} animal for the k^{th} trait associated with the i^{th} source [14],[15]. Record equivalents express the quantity of contributions due to relationships and/or due to records considered for the evaluation of an animal. For internal animals, RE_{ijk} is equal to 0 because all contributions are only due to the relationships among external and internal animals. For external animals, if doublecounting of contributions due to relationships among them is not taken into account, {\mathrm{RE}}_{\mathrm{ijk}}=\frac{1{\mathrm{h}}_{\mathrm{k}}^{2}}{{\mathrm{h}}_{\mathrm{k}}^{2}}*\frac{{\mathrm{REL}}_{\mathrm{ijk}}}{1{\mathrm{REL}}_{\mathrm{ijk}}} for the j^{th} animal for the k^{th} trait associated with the i^{th} source, where {\mathrm{h}}_{\mathrm{k}}^{2} is the heritability of the k^{th} trait [15],[18]. If doublecounting of contributions due to relationships among external animals is taken into account, RE_{ijk} only expresses the amount of contributions due to records and can be estimated through a twostep algorithm (TSA) [14]. The first step of this TSA determines external animals associated with external information that includes only contributions due to relationships. The second step estimates the amount of contributions due to records (expressed as RE) for external animals associated with information that combines both contributions due to relationships and own records. Note that the proposed approximation of {\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}}^{\mathbf{\prime}}{\mathbf{R}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}{\mathbf{Z}}_{{\mathrm{E}}_{\mathrm{i}}} differs from the approximation proposed by Quaas and Zhang [12]. Indeed, they proposed to approximate each diagonal block of {\mathbf{\Lambda}}_{{\mathrm{E}}_{\mathrm{i}}} by {\mathbf{\Delta}}_{\mathrm{Qi}\left(\mathrm{j}\right)}{\mathbf{G}}_{0}^{1}{\mathbf{\Delta}}_{\mathrm{Qi}\left(\mathrm{j}\right)}, where the matrix G_{0} is a matrix of genetic (co)variance among traits and Δ_{Qi(j)} is a diagonal matrix with elements:
Also, the multitrait MME (1) that integrate N sources of external information differ from the usual multitrait MME only by the terms {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}{\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}\right)} and {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\mathbf{D}}_{{\mathrm{E}}_{\mathrm{i}}}^{1}{\widehat{\mathbf{u}}}_{{\mathrm{E}}_{\mathrm{i}}}\right)}:
Furthermore, it was previously assumed that the whole pedigree is available for all genetic evaluations. The additive genetic (co)variance matrices that include all internal and external animals are then equal for all genetic evaluations (i.e., {\mathbf{G}}_{{\mathrm{E}}_{0}}={\mathbf{G}}_{{\mathrm{E}}_{1}}={\mathbf{G}}_{{\mathrm{E}}_{2}}=\xbf={\mathbf{G}}_{{\mathrm{E}}_{\mathrm{N}}}). Nevertheless, each internal or external genetic evaluation could be performed as a singlestep genomic evaluation (e.g., [3],[4]) without modifications to the Bayesian derivation [See Additional file 1] because assumptions on the different matrices {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}} were not limiting. Such cases would lead to {\mathbf{G}}_{{\mathrm{E}}_{0}}\ne {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}}. For example, integration of external information provided by the usual MME into a singlestep genomic evaluation would lead to {\mathbf{G}}_{{\mathrm{E}}_{0}}\ne {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}} because {\mathbf{G}}_{{\mathrm{E}}_{0}} would include genomic information [3],[4], unlike {\mathbf{G}}_{{\mathrm{E}}_{\mathrm{i}}}.
Integration of several sources of external information by avoiding doublecounting of contributions due to records
Assumptions stated in the previous section led to each source of external information to be obtained from an external evaluation that was based only on external data and free of internal data and information, as well as of the N1 other external data and information. In practice, this assumption is not necessarily valid because a source of external information may be obtained from an external evaluation based on external data and/or information and also on internal data and/or information (e.g., EBV and associated REL obtained in country E_{1} based on external data E_{1} and on internal data E_{0}). Thus, doublecounting of contributions due to records between internal and external information must be taken into account, as detailed below.
For the i^{th} source of external information, internal information included into external information (subscript I_{i}) associated with the external animals can be summarized as {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}, i.e. the vector of internal EBV associated with external animals for which external information included both external and internal information, and by {\mathbf{D}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}, the prediction error (co)variance matrix associated with {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}.
A modified set of multitrait mixed model equations that integrate several sources of external information and take doublecounting of contributions due to records between external and internal information into account, can be written as follows [See Additional file 2]:
where {\mathbf{G}}_{{\mathrm{I}}_{\mathrm{i}}} is a genetic (co)variance matrix for all animals for the internal information included into the i^{th} source of external information, {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}}=\left[\begin{array}{c}\hfill {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}^{0}\right)}\hfill \\ \hfill {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\mathbf{G}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}^{0}{\mathrm{A}}_{\mathrm{i}}\right)}{\mathbf{G}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}{\mathrm{A}}_{\mathrm{i}}\right)}^{1}{\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}\hfill \\ \hfill {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)}\hfill \end{array}\right] is the vector of internal EBV associated with the i^{th} source of external information that includes internal information and {\mathbf{D}}_{{\mathrm{I}}_{\mathrm{i}}}^{1} is the inverse of the prediciton error (co)variance matrix associated with {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}} and approximated as detailed in the previous section.
If the i^{th} source of external information does not include internal information for external animals, the vector {\widehat{\mathbf{u}}}_{{\mathrm{I}}_{\mathrm{i}}} is undetermined and the matrix {\mathbf{D}}_{{\mathrm{I}}_{\mathrm{i}}}^{1} is equal to {\mathbf{G}}_{{\mathrm{I}}_{\mathrm{i}}}^{1}. This leads to the system of equations (1).
Blending several sources of external information by avoiding doublecounting of contributions due to records
Equations to blend several sources of external information by avoiding doublecounting of contributions due to records among internal and external data/information can be derived from the system of equations (3) by assuming that {\mathbf{y}}_{{\mathrm{E}}_{0}} has no records (i.e. that {\mathbf{y}}_{{\mathrm{E}}_{0}} is an empty vector). Then, the equation can be written as follows:
Simulated example
The system of equations (3) was tested using data simulated with the software package GNU Octave [19]. The context of the simulation was a country that imports sires from another country to generate the next generation of production animals and potential sires. Populations of the importing country (hereafter called the internal population) and of the exporting country (hereafter called the external population) were assumed to belong to the same breed. Each population included about 1000 animals distributed over five generations and was simulated from 120 female and 30 male founders. For both populations, milk yield in the first lactation was simulated for each female with progeny, following Van Vleck [20]. A herd effect nested withinpopulation was randomly assigned to each phenotypic record. To obtain enough observations per level for the herd effect, each herd included at least 40 females. Phenotypic variance and heritability were assumed to be 3.24*10^{6} kg^{2} and 0.25, respectively.
To simulate the internal and external populations, the following rules were applied to generate each new generation. First, from the second generation, both females and males older than one year old were considered as mature for breeding and a male could be mated during at most two breeding years. Second, 95% of the available females and 75% of the available males with the highest true breeding values were selected for breeding. Third, all selected females were randomly mated with the selected males. The maximum number of males mated to produce the next generation was set to 25. Furthermore, a mating could be performed only if the additive relationship coefficient between male and female was less than 0.5 and if the female had less than three progeny.
The external population was simulated first and additional rules were applied to this population. For this population, males that were selected for mating only originated from the external population and 60% of the external male offspring with the lowest true breeding values were culled in each generation. Then, the internal population was simulated. For this population, males were selected among all available internal males and a subset of selected external sires. This subset of external sires included the first 50 sires with the highest true breeding values in the external population. Also, 99% of internal male offspring with the lowest true breeding values were culled in each generation. No female offspring was culled in either population.
Using the simulated data, three genetic evaluations were performed (Table 2):

(a)
A joint evaluation (EVAL_{J}) was performed as a BLUP evaluation using the system of equations (2) and based on external and internal pedigree and data. This evaluation was assumed to be the reference.

(b)
An internal evaluation (EVAL_{I}) was performed as a BLUP evaluation using the system of equations (2) and based on internal pedigree and data.

(c)
An external evaluation (EVAL_{E}) was performed as a BLUP evaluation using the system of equations (2) and based on external pedigree and data.
Three Bayesian evaluations that integrated information provided by EVAL_{E} or by EVAL_{J} for the 50 external sires into EVAL_{I} were also performed. Because the external sires were related, doublecounting of contributions due to relationships existed and this was taken into account for the three Bayesian evaluations through the TSA [14]. Doublecounting of contributions due to records could also exist with the integration of information provided by EVAL_{J} into EVAL_{I} because EVAL_{J} and EVAL_{I} were partially based on the same data (i.e., internal data). The following three Bayesian evaluations were performed:

(d)
A Bayesian evaluation using the system of equations (1) and using EBV and PEV obtained from EVAL_{E} associated with the 50 external sires that were used inside the internal population as external information (EVAL_{BE}).

(e)
A Bayesian evaluation using the system of equations (1) and EBV and PEV obtained from EVAL_{J} associated with the 50 external sires as external information (hereafter called joint information) (EVAL_{BJ}). Although EVAL_{J} was based on external and internal data, doublecounting of contributions due to records between joint and internal information was not taken into account.

(f)
A Bayesian evaluation integrating joint information by using the system of equations (3) and taking into account doublecounting of contributions due records among internal and joint information (EVAL_{BJI}). Doublecounting of contributions due to records among internal and joint information was taken into account by using EBV and PEV obtained from EVAL_{I} associated with the 50 external sires.
The simulation was replicated 100 times. Comparisons between EVAL_{J} and EVAL_{I}, EVAL_{BE}, EVAL_{BJ}, or EVAL_{BJI} were performed separately for the 50 external sires and for the internal animals. Comparisons were based on:

(1)
Spearman¿s rank correlation coefficients (r) of EBV obtained from EVAL_{J} (EBV_{J}) with EBV obtained from EVAL_{I} (EBV_{I}), EVAL_{BE} (EBV_{BE}), EVAL_{BJ} (EBV_{BJ}), and EVAL_{BJI} (EBV_{BJI}),

(2)
regression coefficients (a) of EBV_{J} on EBV_{I}, EBV_{BE}, EBV_{BJ}, and EBV_{BJI}, and

(3)
coefficients of determination (R^{2}) associated with the regressions,

(4)
the total amount of RE (RE_{tot}) associated with external information, joint information and joint information corrected for the included internal information, and

(5)
mean squared errors (MSE) of EBV_{I}, EBV_{BE}, EBV_{BJ}, and EBV_{BJI}, expressed as a percentage of MSE obtained for EBV_{I}. For each replicate, the MSE obtained for EBV_{I} was reported to a relative value of 100 before the different computations of MSE.
Because the TSA was applied before all three Bayesian evaluations, RE_{tot} were free of contributions due to relationships estimated by the Bayesian evaluations. For an easier understanding of the results and discussion, RE can be transformed into daughter equivalents (DE) through {\mathrm{DE}}_{\mathrm{ijk}}=\frac{4{\mathrm{h}}_{\mathrm{k}}^{2}}{1{\mathrm{h}}_{\mathrm{k}}^{2}}*{\mathrm{RE}}_{\mathrm{ijk}}[18]. All results were the average of the 100 replicates.
Walloon example
Even if MACE allows the aggregation of EBV for dairy sires, internal genetic evaluations for animals not associated with MACE information (e.g., cows, calves, young sires) are not influenced by external information considered by the MACE for dairy sires and may be still biased. Therefore, integration of MACE information into internal evaluations, as well as blending of MACE and internal information, could benefit those animals. The performance of equation (4) that blends MACE and internal information was evaluated in the context of the official Walloon genetic evaluation for Holstein cattle.
The Walloon example used information for milk, fat and protein yields for Holstein cattle provided by the official Walloon genetic evaluation [21],[22]. The genetic variances were those used for the official Walloon genetic evaluation [21] and were equal to 280 425 kg^{2} for milk yield, to 522.6 kg^{2} for fat yield and to 261.5 kg^{2} for protein yield. The respective heritabilities were equal to 0.38, 0.43 and 0.41. The pedigree file was extracted from the database used for the official Walloon genetic evaluation (EVAL_{W}) and covered up to six known ancestral generations. The extraction was performed for a randomly selected group of 1909 animals (potentially genotyped) born after 1998. The selected group included sires, cows and calves that were used or were not at the internal level. After extraction, the pedigree file contained 16 234 animals.
Internal information included EBV and associated REL estimated from data provided by the Walloon Breeding Association (EBV_{W}, REL_{W}) for the EVAL_{W} for milk production of April 2013 [21],[22]. A total of 12 046 animals were associated with an available EBV_{W}. External information included EBV and REL for 1981 sires provided with the official release for the April 2013 MACE performed by Interbull (EVAL_{MACE}, EBV_{MACE}, REL_{MACE}) [23]. It should be noted that the Walloon region in Belgium participated in the April 2013 MACE. Internal and external information were harmonized between the Walloon and MACE evaluations by adjusting scales and mean differences towards the original expression of the trait in the Walloon genetic evaluations. External information was then considered to be the same trait as the internal phenotype trait.
Unlike the simulated example, no joint evaluation based on Walloon and external records was available for both external and internal animals. Because EVAL_{MACE} aggregated EBV from several national genetic evaluations for sires, it was considered as the reference for the evaluated sires. Walloon and MACE information were blended by using equation (4) for the following four cases: with or without consideration of doublecounting of contributions due to relationships and with or without consideration of doublecounting of contributions due to records (Table 3). Doublecounting of contributions due to relationships was possible because all animals associated with Walloon and/or MACE information were related. Doublecounting of contributions due to records was also possible because MACE information associated with the 1981 sires included contributions provided by EVAL_{W}. Thus, to test the importance of both doublecounting issues, the following four cases were evaluated:

(a)
Walloon and MACE information were blended without considering doublecounting of contributions due to records and due to relationships (EVAL_{BLNN}, EBV_{BLNN}, REL_{BLNN}).

(b)
Walloon and MACE information were blended by considering only doublecounting of contributions due to records (EVAL_{BLRE}, EBV_{BLRE}, REL_{BLRE}). To achieve this goal, the contribution of Walloon information into MACE information was determined based on the domestic effective daughter equivalents (EDC) associated with EBV_{MACE} and REL_{MACE} and provided with the official release for the 2013 April MACE by Interbull. MACE information free of Walloon information was reported by a domestic EDC equal to 0. A total of 601 sires were associated with an EDC greater than 0. For these 601 sires, EBV and associated REL estimated from Walloon data and contributing to the April 2013 MACE routinerun (EBV_{Wc}, REL_{Wc}) were considered by EVAL_{BLRE} to take doublecounting of contributions due to records into account. Doublecounting of contributions due to relationships was not taken into account for either Walloon or MACE information.

(c)
Walloon and MACE information were blended by only considering doublecounting of contributions due to relationships among all animals (EVAL_{BLR}, EBV_{BLR}, REL_{BLR}). The TSA was therefore applied for Walloon and MACE information. Doublecounting of contributions due to records was not considered.

(d)
Walloon and MACE information were blended by considering both doublecounting of contributions due to records and due to relationships (EVAL_{BL}, EBV_{BL}, REL_{BL}). Reliabilities for EBV_{BLNN}, EBV_{BLRE}, EBV_{BLR} and EBV_{BL} were computed using the equation \mathit{REL}=1\mathit{PEV}/{\sigma}_{g}^{2}, where {\sigma}_{g}^{2} is the genetic variance for the corresponding trait and PEV is the prediction error variance obtained from the diagonal element of the inverted lefthandside of the equation (4).
As explained previously, EVAL_{MACE} was considered as the reference for sires evaluated through EVAL_{MACE}. Comparisons between EVAL_{MACE} and EVAL_{W}, EVAL_{BLNN}, EVAL_{BLRE}, EVAL_{BLR} or EVAL_{BL} were performed based on:

(1)
Spearman¿s rank correlation coefficients (r) of EBV_{MACE} with EBV_{W}, EBV_{BLNN}, EBV_{BLRE}, EBV_{BLR} and EBV_{BL},

(2)
MSE of EBV_{W}, EBV_{BLNN}, EBV_{BLRE}, EBV_{BLR}, and EVAL_{BL} (i.e. mean squared errors expressed as a percentage of average MSE of EBV_{W}),

(3)
regression coefficients (a) and,

(4)
R^{2} of the regressions of EVAL_{MACE} on the five other evaluations (i.e., EVAL_{W}, EVAL_{BLNN}, EVAL_{BLRE}, EVAL_{BLR} and EVAL_{BL}),

(5)
RE_{tot} and (6) average REL.
Comparisons concerned two groups of sires. A first group of sires included 1212 sires that were associated with both Walloon and MACE information and had daughters with records in the Walloon region dataset (hereafter called ¿internally used sires¿). A second group of sires included 631 sires that were associated with both Walloon and MACE information but had no daughters with records in the Walloon region dataset (i.e. they had only foreign, or external, daughters; hereafter called ¿internally unused sires¿). The RE_{tot} were free of contributions due to relationships that were estimated by the Bayesian evaluations but could include contributions due to relationships that resulted from the previous genetic evaluation if the TSA was not applied.
The effect of blending MACE and Walloon information was also studied for internal animals that were not associated with MACE information and that were sired by internally used sires by considering (1) r between EVAL_{BL} and EVAL_{W}, EVAL_{BLNN}, EVAL_{BLRE} or EVAL_{BLR}, (2) RE_{tot} and (3) average REL. Three groups of internal animals were defined depending on their REL_{W}. The first group included internal animals that were associated with a REL_{W} lower than 0.50, the second group included internal animals that were associated with a REL_{W} between 0.50 and 0.75, and the third group included internal animals with a REL_{W} equal or higher than 0.75.
All blending evaluations were performed using a version of the BLUPF90 program [24] modified to implement the equations (1), (3) and (4).
Results and discussion
Simulated example
On average, each of the 100 simulated internal and external populations included 1048 animals. Results for r, MSE, a and R^{2} for prediction of EBV_{J} are in Table 4 for the 50 external sires and for the internal animals.
Compared to the rankings of EVAL_{I}, integration of external or joint information for the 50 external sires led to rankings of EVAL_{BE}, EVAL_{BJ} or EVAL_{BJI} that were more similar to those of EVAL_{J}. Rank correlations r increased from 0.57 for EVAL_{I} to at least 0.95 for EVAL_{BJ} for the 50 external sires and from 0.93 for EVAL_{I} to at least 0.98 for EVAL_{BJ} for internal animals (Table 4). Furthermore, MSE, a and R^{2} also showed that the integration of external or joint information for the 50 external animals with EVAL_{BE}, EVAL_{BJ} or EVAL_{BJI} led to better predictions of EBV_{J} for both external and internal animals (Table 4). Therefore, the observations that internals animals related to the 50 external sires were also better predicted by EVAL_{BE}, EVAL_{BJ} and EVAL_{BJI}, compared to EVAL_{I}, revealed that the external information propagated from the 50 external sires to relatives.
The RE_{tot} associated with EVAL_{BE} was equal to 76.3 (which also corresponded to 381.6 DE), while the RE_{tot} associated with EVAL_{BJ} was equal to 141.5 (DE = 707.7, Table 4). The higher RE_{tot} associated with EVAL_{BJ} showed that doublecounting of contributions due to records was present when joint information was integrated. Indeed, joint information contained both external and internal information. The RE_{tot} associated with EVAL_{BJI} was equal to 78.7 (DE = 393.3, Table 4). While this latter RE_{tot} is slightly higher (i.e. 3.1% on average) than the RE_{tot} associated with EVAL_{BE}, it showed that doublecounting was almost avoided when internal information was considered for the 50 external sires. A total of 96.4% of contributions due to records of internal information on average was removed from the joint information (Table 4). The remaining 3.6% of contributions due to records of internal information was doublecounted by the Bayesian evaluations and may result from the estimation of contributions due to relationships and/or from the estimation of contributions due to records among joint and internal information.
Because doublecounting of contributions due to records between joint and internal information was almost avoided, breeding values that were estimated by EVAL_{BJI} for all animals led to better predictions of EBV_{J} than EVAL_{BJ}, based on r, MSE, a and R^{2} (Table 4). Rank correlations of EBV_{J} with EBV_{BJ} and EBV_{BJI} increased from 0.979 for EVAL_{BJ} to 0.996 for EVAL_{BJI} for the internal animals and from 0.956 for EVAL_{BJ} to 0.996 for EVAL_{BJI} for the 50 external animals. The MSE decreased on average from 34.3% for EVAL_{BJ} to 6.8% for EVAL_{BJI} for the internal animals and from 17.2% for EVAL_{BJ} to 0.6% for EVAL_{BJI} for the external animals. These results again showed that integration of external/joint information for the 50 external sires influenced the prediction of internal relatives through the propagation of information from the external sires to relatives. These results show that the doublecounting of contributions due to records also affected predictions of internal animals. Furthermore, as expected, EVAL_{BE} predicted EBV_{J} slightly better than EVAL_{BJI} for both external sires and internal animals, based on the corresponding r, MSE, a and R^{2} (Table 4). The low difference in accuracy of prediction between EVAL_{BE} and EVAL_{BJI} could be attributed to the estimation of contributions due to relationships and due to records.
Based on these results, doublecounting of contributions due to records was almost avoided. Thus, the integration of information into a genetic evaluation by avoiding both contributions due to relationships and due to records performed well for external animals. Internal animals also benefited of the integration of information thanks to their relationships with external animals.
Walloon example
Of the 12 046 animals associated with available Walloon information for the three traits, 6232 animals for milk yield, 6209 animals for fat yield, and 6212 animals for protein yield were associated with information that was based only on contributions due to relationships, as estimated by the TSA. In terms of RE, contributions due to relationships represented from 14.9% for fat yield to 16.3% for milk yield of the contributions associated with Walloon information (Figure 1). Among the 1981 sires associated with MACE information, two sires were associated with information that includes only contributions due to relationships for the three traits. Both these sires had several sons among all the sires associated with an EBV_{MACE}, which explains that the contributions were considered as only due to relationships. In terms of RE, all contributions due to relationships represented on average 5.1% of the contributions associated with MACE information for the three traits. Of the 601 sires with an EBV_{Wc}, all sires were associated with information that included both contributions due to relationships and due to records. This latter observation for the 601 sires was expected because these 601 sires must have at least 10 daughters with records within 10 herds in the Walloon region to participate in the MACE evaluation.
Internally used sires
Of the internally used sires, 1212 had Walloon and MACE information and had both internal and external daughters with records. On average, each sire had 143.1 internal daughters with records. The average REL_{W} ranged from 0.74 to 0.76 (Table 5) and the average REL_{MACE} was equal to 0.88 for the three traits. Results for r, MSE, a and R^{2} for prediction of EBV_{MACE} by EVAL_{BL} are in Table 6 for the 1212 sires for milk, fat and protein yields. For the three traits, blending of Walloon and MACE information by taking doublecounting of contributions due to records and due to relationships into account (i.e. EVAL_{BL}) led to a ranking that was more similar to the MACE ranking than to the internal ranking (i.e. EVAL_{W}), although these internally used sires sired a large number of cows with records in the Walloon region. Rank correlations increased by 0.104 points for milk yield to 0.125 points for fat yield to achieve a rank correlation between EBV_{MACE} and EBV_{BL} that ranged from 0.987 to 0.990 (Table 6). The MSE, a and R^{2} showed that accuracy of predictions of EBV_{MACE} by EBV_{W} or by EBV_{BL} increased when external information was integrated. Integration of MACE information also increased the average REL by 0.14 points for fat yield to 0.16 points for milk yield (Table 5). This increase of average REL corresponded to an increase of 57.5, 51.4, and 50.9 DE per sire on average for milk, fat and protein yields, respectively. Also, the average REL_{BL} for the 1212 sires was 0.02 points higher than the average REL_{MACE} (Table 6). This difference in average REL, as well as the differences between EBV_{MACE} and EBV_{BL} based on MSE, a and R^{2} (Table 6), can be explained by the fact that MACE did not include all information available for animals in the Walloon region. Indeed, EBV_{W} of a sire was included into MACE if it had at least10 daughters with records within 10 herds at the internal level. Therefore, EBV_{W} for sires that did not fulfill this requirement were not considered by MACE, but were taken into account by the four Bayesian evaluations, which provided additional information compared to MACE information. Approximations based on estimation of contributions due to relationships and theoretical assumptions of the model may also explain some of the differences between EBV_{MACE} and EBV_{BL}. For example, MACE was considered as a national genetic evaluation. These results indicate that EVAL_{BL}, i.e. a Bayesian evaluation that blended internal information and external information and avoided most doublecounting of contributions due to records and due to relationships, was successful in integrating MACE information for internally used sires.
Doublecounting of contributions due to records and due to relationships were also not considered (i.e. EVAL_{BLNN}) or were considered separately (i.e. EVAL_{BLRE} and EVAL_{BLR}) to study their influences on prediction of EVAL_{MACE} for internally used sires. Parameters r, a and R^{2} associated with EVAL_{BLNN}, EVAL_{BLRE} and EVAL_{BLR} for the 1212 sires were similar to the r, a and R^{2} of EVAL_{BL}, although a slight advantage was observed for EVAL_{BL}. Therefore, the four blending evaluations led to similar rankings as MACE for the 1212 internally used sires (i.e., rank correlations equal to 0.99 on average; Table 6).
However, doublecounting can be observed based on MSE, RE_{tot} and REL (Table 6). With regard to doublecounting of contributions due to relationships for the 1212 internally used sires, RE that were free of contributions due to relationships (i.e. RE that included only contributions due to records) for EBV_{MACE} were equal to 30 378 (DE = 176 578) for milk yield, 23 927 (DE = 150 772) for fat yield, and 26 338 (DE = 160 416) for protein yield. These amounts of RE free of contributions due to relationships represented 96.1% of the RE that contributed to MACE information. Considering the Walloon information for the 1212 sires, RE that included only contributions due to records represented from 93.6% of all Walloon contributions for milk yield to 94.2% for fat yield. For both Walloon and MACE information associated with the internally used sires and for the three traits (i.e. for milk, fat and protein yields), less than 6.4% of all contributions were attributed to relationships (Figure 1). Such low percentages of contributions due to relationships are in agreement with selection index theory [25]. While doublecounting of contributions due to relationships was present for EVAL_{BLRE} (i.e. the blending evaluation that considered only doublecounting of contributions due to records), the contributions due to relationships were small and their doublecounting had little effect on the prediction of EBV_{MACE} for the internally used sires, compared to EVAL_{BL}, based on parameters r and MSE. However, as expected, an average increase of 1% in REL_{BLRE} was observed, compared to REL_{BL}. Thus, the REL_{BLRE} were, on average, slightly overestimated.
With regard to doublecounting of contributions due to records, based on RE, Walloon information represented from 64.3% of the total information free of contributions due to relationships associated with EVAL_{BL} for milk yield to 67.6% for fat yield (Table 6). Thus, integrated information free of contributions due to relationships and due to records (i.e. MACE information from which Walloon information was subtracted) represented 32.5% of the total information associated with EVAL_{BL} for fat yield to 35.8% for milk yield. If doublecounting of contributions due to relationships was considered only, RE_{tot} associated with EVAL_{BLR} ranged from 43 944 RE for fat yield to 52 313 RE for milk yield, while RE_{tot} associated with EVAL_{BL} ranged from 29 631 RE for fat yield to 34 141 RE for milk yield. Thus, between 14 313 and 18 172 RE were considered twice by EVAL_{BLR}. However, doublecounting of contributions due to records affected the prediction of EBV_{MACE} for internally used sires only slightly according to all parameters evaluated (Table 5). The REL_{BLR} were overestimated by 1% on average for the internally used sires, compared to REL_{BL}. Furthermore, no preference was observed between EVAL_{BLRE} and EVAL_{BLR} based on r, MSE, a and R^{2} for the three traits. Indeed, r and R^{2} were similar for these two evaluations, while EVAL_{BLRE} was more reliable based on MSE, but parameter a indicated that EVAL_{BLR} was more reliable. However, EVAL_{BLRE} had the greatest under and overestimation of true breeding values based on parameter a. Based on these results, it can be stated that doublecounting of contributions due to relationships and due to records had little effect on EBV for internally used sires.
Internally unused sires
Of the internally unused sires (i.e. that had only external daughters with records), 631 sires were associated with Walloon and MACE information. Their average REL_{W} ranged from 0.22 to 0.23 for the three traits (Table 7) and the average REL_{MACE} was equal to 0.77. Because they had only external daughters, Walloon contributions only included contributions due to relationships and no contributions due to records. Based on RE_{tot} (Table 7), Walloon contributions due to records for all 631 sires were in general well estimated by the TSA, ranging from 0.79% of the Walloon total contributions for milk yield to 0.80% for protein yield (Figure 1). The small nonzero percentage could be attributed to approximations involved in estimating the contributions due to relationships and due to records by the TSA, such as the consideration of an unknown fixed effect [14]. The nearly correct estimation of contributions due to relationships led to similar average REL_{MACE} and average REL_{BL} for the three traits (Table 7). Integration of MACE information also increased the average REL_{W} by at least 0.54 points, resulting in an average REL_{BL} equal to 0.77 for the three traits. These results for the 631 internally unused sires confirmed that MACE information already contained the main contributions due to relationships that were expressed in the Walloon information and that doublecounting of contributions due to relationships was mostly avoided. Not considering contributions due to relationships (i.e. EVAL_{BLNN} and EVAL_{BLRE}) led to overestimation of average REL by at least 3% (Table 7).
Results for r, MSE, a and R^{2} for the prediction of EBV_{MACE} by the four blending evaluations are in Table 7 for the 631 internally unused sires for the three traits. Blending of Walloon and MACE information led to similar rankings of the 631 sires for the four blending evaluations. Rank correlations between EBV_{MACE} and EBV for the four blending evaluations increased from 0.73 to 0.99 for milk yield, from 0.57 to 0.99 for fat yield and from 0.72 to 0.99 for protein yield. These rank correlations indicated that the blending method was also successful for sires with only external information for all three traits. These results were confirmed by a decrease of MSE by at least 96.9% and by regression coefficients close to 1.0, with an R^{2} equal to 0.99 for all three traits (Table 7). Because doublecounting can be only attributed to contributions due to relationships for the 631 internally unused sires, EVAL_{BLNN} and EVAL_{BLRE} led to similar parameters. This was also observed for EVAL_{BL} and EVAL_{BLR} (Table 7). Differences between these two groups of evaluations were only observed based on MSE and a (Table 7). These two parameters showed that EBV_{MACE} for the 631 sires were slightly better predicted when contributions due to relationships were considered. However, all these results showed that contributions due to relationships had little effect on the prediction of EBV_{MACE}.
VanRaden and Tooker [17] found similar correlations between EBV_{MACE} and combined EBV for sires with only external daughters (i.e. between 0.991 and 0.994 for yield traits). Their strategy consisted of computing external deregressed proofs (DRP) from EBV_{MACE} and including one extra record based on these DRP, weighted by the associated DE for the sire. Internal contributions in MACE information for sires with internal and external daughters were considered by subtracting the number of internal DE from the total and by using internal EBV instead of parent averages from EBV_{MACE} to compute external DRP. Based on Legarra et al. [13], Gengler and Vanderick [16] integrated MACE information into the official Walloon genetic evaluation for milk production. External EBV were estimated by selection index theory and internal contributions were considered as in VanRaden and Tooker [17]. Thus, while these two latter approaches and the approach proposed in this study consider internal contributions to MACE information in a similar manner [See Additional file 2], the main advantage of the proposed approach is to avoid a preprocessing deregression step or computation of external EBV.
Internal animals
The effect of the integration of MACE information on predictions was also studied for internal animals that were not associated with MACE information and that were sired by internally used sires. A total of 3331 internal animals was considered. If doublecounting of contributions due to relationships and due to records were avoided (i.e. EVAL_{BL}), integration of MACE information led to an increase of the REL from 0.32 to 0.42 for milk yield and from 0.31 to 0.42 for fat and protein yields for internal animals that had a REL_{W} less than 0.50 (Table 8). These increases were equivalent to 2.4 DE for milk yield, 2.3 DE for fat yield and 2.4 DE for protein yield. On average, no increase in REL was observed for internal animals with REL_{W} greater than 0.50 (Tables 9 and 10; Figure 2). Therefore, integration of MACE information was mostly relevant for external animals that were associated with this information and for internal animals with a low REL_{W} sired by external animals.
The effect of doublecounting was also studied in comparison to EVAL_{BL} for the 3331 internal animals that were only associated with Walloon information and that were sired by internally used sires. Own contributions due to relationships for internal animals with REL_{W} less than 0.50 represented from 85.2% of the total contributions for milk yield to 91.8% for fat yield (Table 8). These percentages ranged from 55.1% for protein yield to 57.7% for fat yield for internal animals with REL_{W} between 0.50 and 0.75, and from 15.4% for protein yield to 16.7% for fat yield for internal animals with REL_{W} greater than 0.75 (Tables 9 and 10). As stated before, these observations were as expected based on selection index theory [25], and doublecounting of own contributions due to relationships was mostly present for internal animals with low REL_{W}. However, internal animals were also affected by doublecounting of contributions due to relationships and due to records that originated from their sires (and relatives) through the contributions due to relationships. Doublecounting that originated from their own contributions and from their sires (and relatives) could be observed based on a comparison of REL_{BLRE}, REL_{BLR} and REL_{BL} and of r between EBV_{BL} and EBV_{BLRE} or EBV_{BLR} (Tables 8, 9 and 10). Doublecounting of contributions due to records that originated from sires of internal animals had minor effects on the average REL_{BLR} associated with internal animals (at most 1%) and rankings of internal animals (r ≥ 0.999; Tables 8, 9 and 10). However, doublecounting of contributions due to relationships led to an increase of average REL by at least 0.14 points for internal animals with REL_{W} less than 0.50 and by at least 0.11 points for internal animals with REL_{W} ranging from 0.50 to 0.74. The increase of average REL was lower for internal animals with REL_{W} greater than 0.75 (>0.02 points; Tables 8, 9 and 10). Although the average REL_{BLR} and REL_{BLRE} were (slightly) overestimated for both evaluations, doublecounting of contributions due to records and due to relationships had little effect on the ranking of internal animals compared to the ranking of EVAL_{BL}, regardless of the group of internal animals or trait considered. Indeed, rank correlations between EVAL_{BL} and EVAL_{BLR} or EVAL_{BLRE} were greater than 0.99 (Tables 8, 9 and 10). All these results show that doublecounting of contributions due to relationships and due to records can be ignored for the prediction of EBV for internal animals that are sired by external animals. However, all doublecounting must be taken into account to estimate REL accurately.
On the implementation
Considering all groups of animals, i.e. internally used and unused sires, as well as internal animals sired by internally used sires, our results for the Walloon example suggest that contributions due to relationships can be ignored. Indeed, the different rank correlations for EVAL_{BLRE} (i.e. the Bayesian evaluation that took only doublecounting of contributions due to records into account) were similar to the rank correlations of EVAL_{BL}. Furthermore, in practice, the TSA could be difficult to apply if a high number of animals is associated with external information because it requires the inversion of a, potentially, dense matrix for each iteration. However, effects of doublecounting of contributions due to relationships should be tested before ignoring it. For example, overestimation of REL could occur especially for traits for which contributions due to relationships would be at least as significant as contributions due to records (e.g., if the phenotypes are expensive to obtain). Furthermore, REL associated with the modified MME were estimated based on the inverted LHS. Although this was feasible for the simulated and Walloon data, this may not be feasible in most cases, and approaches that estimate REL (e.g., [15],[18]) could be modified to take into account RE (or DE) associated with external information.
The Walloon example was considered as an evaluation that blends MACE and Walloon (internal) information in the context of official Walloon genetic evaluations for Holstein cattle. However, the Walloon example can also be considered as a particular case of an internal evaluation that has no internal data and blends only sources of external information, i.e. MACE and Walloon information, that are partially based on the same information, i.e. the Walloon information. This case can be extended to more general cases for which internal data may exist and external animals are associated with at least two sources of information (e.g., E_{1} and E_{2}) that are partially based on the same external records or information. Doublecounting of external information that is shared by the sources of external information, e.g. E_{1} and E_{2}, can be avoided by the proposed approach thanks to the knowledge and availability of EBV and associated REL that are based only on external information that is shared by the sources of external information. Nevertheless, although taking external information that is shared by different sources of external information into consideration seems to be possible with the proposed approach, this may be difficult in practice because it requires that EBV and associated REL based on shared external information are known and available.
Conclusions
The proposed unified method integrated and blended several sources of information into an internal genetic evaluation in an appropriate manner. The results also showed that the proposed method was able to avoid doublecounting of contributions due to records and due to relationships. Furthermore, because all available external sources of information were correctly propagated, relatives of external animals benefited from integrated information and, therefore, received more reliable EBV. The unified method could also be used in the context of singlestep genomic evaluations to integrate external information to indirectly recover a large amount of external phenotypic information [26]. While the simulated and Walloon examples were univariate, the unified method was developed for multitrait models that, e.g., allow evaluation of only internally available traits (e.g., methane emissions, fine milk composition traits, such as fatty acids, milk proteins and other minor components), using additional external information from correlated traits (e.g., traits evaluated by Interbull).
Additional files
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Acknowledgments
J Vandenplas, as a research fellow, and N Gengler, as a former senior research associate, acknowledge the support of the National Fund for Scientific Research (Brussels, Belgium) for these positions. Additional financial support was provided by the Ministry of Agriculture of the Walloon region of Belgium (Service Public de Wallonie, Direction Générale opérationnelle ¿Agriculture, Ressources Naturelles et Environnement¿ ¿ DGARNE) through research projects D311207, D311224/S1, D311274, D311304 and D311308. Financial support for scientific visits was also provided by Wallonie Brussels International. J Vandenplas acknowledges I Misztal for hosting him at the Animal and Dairy Sciences Department of University of Georgia, G Gorjanc for hosting him at the Animal Science Department of University of Ljubljana, P Faux for helpful discussions and editing help, S Tsuruta and I Aguilar for their help concerning the BLUPF90 programs. Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the National Fund for Scientific Research (Brussels, Belgium) under Grant No. 2.5020.11. The authors thank the two anonymous reviewers for the useful comments.
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Authors’ contributions
JV developed the algorithms and the equations, conceived the experimental design, ran the tests and wrote the first draft. FC prepared data for the Walloon example. NG initiated and directed the research. All authors participated in writing the manuscript. All authors read and approved the final manuscript.
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Additional file 1:Integration of two sources of external information into a genetic evaluation. This file describes a derivation that integrates two sources of external information into a genetic evaluation, based on a Bayesian view of the mixed models [27] and similar to the Bayesian derivation of Legarra et al. [13] that integrates one source of external information into a genetic evaluation. (PDF 63 KB)
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Additional file 2:Doublecounting between internal and external information. This file describes the development to avoid doublecounting of contributions due to records between internal and external information. (PDF 25 KB)
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Vandenplas, J., Colinet, F.G. & Gengler, N. Unified method to integrate and blend several, potentially related, sources of information for genetic evaluation. Genet Sel Evol 46, 59 (2014). https://doi.org/10.1186/s1271101400593
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DOI: https://doi.org/10.1186/s1271101400593
Keywords
 Mean Square Error
 Milk Yield
 External Information
 Genetic Evaluation
 Estimate Breeding Value