# Improving accuracy of breeding values by incorporating genomic information in spatial-competition mixed models

Eduardo P Cappa, Yousry A. El-Kassaby, Facundo Munoz, Martin N. Garcia, Pamela V. Villalba, Jaroslav Klápště, Susana N. Marcucci Poltri

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## ► To cite this version:

Eduardo P Cappa, Yousry A. El-Kassaby, Facundo Munoz, Martin N. Garcia, Pamela V. Villalba, et al.. Improving accuracy of breeding values by incorporating genomic information in spatial-competition mixed models. Molecular Breeding, 2017, 37 (10), pp.1-13. 10.1007/s11032-017-0725-6 hal-02627351

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Submitted on 30 Jan 2026

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Improving accuracy of breeding values by incorporating genomic information in spatial-competition mixed models
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**Abstract**

Climate change and the increasing demand for sustainable energy resources require urgent strategies to increase the accuracy of selection in tree breeding (associated with higher gain). We investigated the combined pedigree and genomic-based relationship approach and its impact on the accuracy of predicted breeding values using data from 5-year-old *Eucalyptus grandis* progeny trial. The number of trees that can be genotyped in a tree breeding population is limited; therefore, the combined approach can be a feasible and efficient strategy to increase the genetic gain and provide more accurate predicted breeding values. We calculated the accuracy of predicted breeding values for two growth traits, diameter at breast height and total height, using two evaluation approaches: the combined approach and the classical pedigree-based approach.

We also investigated the influence of two different trait heritabilities as well as the inclusion of competition genetic effects or environmental heterogeneity in an individual-tree mixed model on the estimated variance components and accuracy of breeding values. The genomic information of genotyped trees is automatically propagated to all trees with the combined approach, including the non-genotyped mothers. This increased the accuracy of overall breeding values, except for the non-genotyped trees from the competition model. The increase in the accuracy was higher for the total height, the trait with low heritability. The combined approach is a simple, fast, and accurate genomic selection method for genetic evaluation of growth traits in *Eucalyptus grandis* and tree species in general. It is simple to implement in a traditional individual-tree mixed model and provides an easy extension to individual-tree mixed models with competition effects and/or environmental heterogeneity.

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**Keywords:** Genomic selection, *Eucalyptus grandis*, individual-tree mixed model, genetic and environmental competition effects, environmental heterogeneity, breedR.
## 58 Introduction

The recent availability of affordable high-density genomic marker panels (e.g., Diversity Array Technology (DArT) and/or Single Nucleotide Polymorphisms (SNP)) created an opportunity, whereby the expected pedigree-based relationship (Wright 1922) used for estimating genetic parameters of complex traits following Fisher's (1918) infinitesimal model is replaced by molecular marker-based realized pairwise genetic similarity between individuals (VanRaden 2008). Research in domesticated crop plants and animals has demonstrated that the marker-based realized kinship matrix (G matrix) is superior to the common pedigree-based average numerator relationship matrix (A matrix; Daetwyler et al. 2012; El-Kassaby et al. 2012; Klápště et al. 2014; Cappa et al. 2016a; Gamal et al. 2016). This superiority is attributable to the marker-based realized kinship ability to account for (1) Mendelian segregation among family members (also known as Mendelian sampling term) as their genetic similarity is estimated precisely instead of using a "universal" relationship estimate such as 0.25 and 0.5 given to half- and full-sib family members, respectively; and (2) historical pedigree that cannot be ascertained by the known contemporary pedigree.

The G matrix can be used instead of the A matrix in the individual-tree mixed model to predict breeding values by Best Linear Unbiased Prediction (BLUP). Fitting G matrix in the mixed model equations is the genomic selection method known as genomic BLUP or GBLUP (VanRaden 2008). This method is equivalent to estimating individual marker effects using ridge regression and assuming such effects to be mutually independent (Meuwissen et al. 2001). GBLUP is a very promising approach for tree breeding (El-Kassaby et al. 2012; Klápště et al. 2014; Zapata-Valenzuela et al. 2013; Muñoz et al. 2014). However, forest tree breeding populations are usually large, with thousands of progenies from many mother trees. Thus, genotyping costs and logistical issues hinder the successful large-scale implementation of GBLUP approach in tree species. Moreover, in an early program of open-pollinated tests, genotyping of mother trees from the native stand is often not possible. Therefore, a more realistic scenario would include only a sub-set of genotyped trees to obtain the G matrix (Isik 2014).

The combined pedigree-genomic relationship matrix approach (H matrix) was recently proposed by Misztal et al. (2009), Legarra et al. (2009), and Christensen and Lund (2010), where the traditional A matrix is concurrently utilized with the G matrix. Therefore, this blended relationship H matrix combines two types of genetic information: (1) pedigree information (A matrix) of the many non-genotyped individuals; and (2)
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marker-based relationship (G matrix) of a sub-set of genotyped individuals. The *H* matrix (or its inverse *H*⁻¹)
can be seen as a projection of genetic merit (or marker genotype) from genotyped to non-genotyped
individuals using pedigree relationships (Vitezica et al. 2011). Therefore, additional information generated by
including the genomic information in the combined approach acts as genetic relationship bridges connecting
information across individuals and parents, thus ultimately facilitating better information utilization during the
BLUP analysis. As a result, more reliable and accurate breeding values (in terms of the correlation between
the true and predicted breeding value) should be obtained, thus increasing the probability of correct ranking of
selection candidates. The combined approach has been widely applied in animal breeding with many
successful applications including pigs (Christensen et al. 2012), chickens (Legarra et al. 2011; Chen et al.
2011), dairy cattle (Aguilar et al. 2010), dairy sheep (Baloche et al. 2014), dairy goat (Carillier et al. 2014),
and beef cattle (Onogi et al. 2014). *A priori*, the combined approach should produce more accurate breeding
value predictions than the pedigree approach alone, also in forest genetic evaluations. However, to our
knowledge the only reference on the combined approach in forest genetic trials is that of Ogut (2012). Using a
real cloned *Pinus taeda* population and simulated data, Ogut (2012) obtained higher accuracies of predictions
from the combined relationship *H* matrix than from the traditional BLUP based on the *A* matrix.

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The accuracy of breeding values predictions also depends on the covariance structure of the random
effects. Therefore, the specification of the dispersion parameters should consider the negative correlation
caused by competition among individuals, and/or the positive spatial correlation due to the environmental
heterogeneity. Negative competition and positive spatial correlations are observable in forest genetic trials,
either for each component alone (e.g. Thomson and El-Kassaby 1988; Dutkowski et al. 2006; Cappa and
Cantet 2007 and 2008; Ye and Jayawickrama 2008; Costa e Silva et al. 2013) or in combination with each
other (Costa e Silva et al. 2013; Cappa et al. 2015; Cappa et al. 2016b). Forest genetic trial analysis including
competition effects and/or environmental heterogeneity often result in greater accuracy of breeding values and
greater genetic gain than different *a priori* experimental designs (Dutkowski et al. 2006; Costa e Silva and
Kerr 2013). Computationally, using an individual-tree mixed model to account for competition (Cappa and
Cantet 2008) and environmental heterogeneity (Cappa and Cantet 2007) or both effects simultaneously
(Cappa et al. 2015; Cappa et al. 2016b) with pedigree-based relationship matrix is similar to applying the
competition or spatial analyses with the combined *H* matrix. Since the combined approach uses traditional
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BLUP mixed model equations (Legarra et al. 2014), the extension to an individual-tree mixed model with competition genetic or environmental heterogeneity effects is straightforward. However, the use of the combined approach in these models has not been reported in forest tree breeding.

We assessed a *Eucalyptus grandis* (Hill ex Maiden) population consisting of 164 open-pollinated families including 2,026 trees measured for two growth attributes (diameter at breast height -DBH- and total height -TH), which are known to possess different heritabilities. A random sample of 187 trees from 132 families with up to three individuals per family was also genotyped with Diversity Arrays Technology (DArT) (7,680 markers), and subsequently the genotypes were used to compute the *G* matrix. The objectives of the study were to (1) compare the accuracies of the predicted breeding values of genotyped offspring and non-genotyped mothers and offspring from the combined approach vs. the traditional pedigree-based approach; and (2) study the effects of DBH and TH contrasting heritabilities after the inclusion of competition effects or environmental heterogeneity in an individual-tree mixed model on the estimated variance components and on the accuracy of predicted breeding values.

# Material and Methods

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## Progeny trial data

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An open-pollinated (OP) progeny trial of *Eucalyptus grandis* (Hill ex Maiden) (hereafter *E. grandis*) at Gobernador Virasoro (lat. 28° 02′ S, long. 56° 03′ W alt., 105 m), northern Corrientes province, Argentina, was used. The trial comprised 164 OP families from native-forest: 92 from New South Wales, and 56 from south-eastern Queensland, Australia, and 16 from two local land-race sources from Concordia, Entre Ríos province, Argentina. Nineteen genetic groups were formed according to provenance. A detailed description of this genetic material can be found in Marcó and White (2002). Briefly, this progeny trial was established as a randomized complete block design with 20 replications and one tree per plot at each replication (i.e., single-tree-plot). Five-year diameter at breast height (1.3 m above the ground level) over bark (DBH, cm) and total height (TH, m) were measured on all surviving trees (N= 2,026). A random sample of 187 trees originating from 131 families of the OP progeny trial was genotyped with a range of 1 to 3 trees per family. The total number of phenotyped trees with at least one genotyped half-sib was 1,650 (see Table 1 for summary).

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[Insert Table 1 about here]
143 Molecular markers

144 Total genomic DNA was extracted from young leaves in November 2009 using the CTAB method (Hoisington et al. 1994) with minor modifications to avoid oil precipitation. Instead of using isopropanol in the precipitation step, samples were diluted in twice the volume of 10 mM Tris-HCl, 1 mM EDTA buffer and precipitated with 2.5 volumes of ethanol and 300 mM sodium acetate.

148 All 187 randomly selected individuals were genotyped by Diversity Arrays Technology Pty Ltd (DArT P/L, Canberra, Australia) for 2,816 DArT molecular markers selected from an operational array with 7,680 (Sansaloni et al. 2010). The selected markers showed call rate values > 0.8, reproducibility values > 0.97 (reproducibility of scoring between replicated target assays), and minor allele frequency (MAF) > 0.05.

153 Statistical models of analysis

154 Preliminary analyses indicated that the DBH and TH showed strong competition and considerable spatial heterogeneity, respectively (see Online Resource 1). Moreover, according to the Akaike information criterion (AIC), competition models (see below model [2]) for DBH and spatial models (see below model [3]) for TH, were significantly better than standard models (see below model [1]) for the two genetic evaluation methods (pedigree-based and combined), except to the combination DBH trait and competition model for the combined approach (results no shown). Therefore, the DBH trait was analyzed as follows:

160 1) Standard mixed model (TM):

$$
161 \qquad y = X\beta + Z_r r + Z_d a_d + e \qquad [1]
$$

162 where $\mathbf{y}$ is the vector of phenotypic data; $\boldsymbol{\beta}$ is the vector of genetic groups as fixed effects; $\mathbf{r}$ is the vector of
163 random replicate effects; $\mathbf{a}$ is the vector of random additive genetic effects of individual trees (i.e., breeding
164 values); and $\mathbf{e}$ is the vector of random residuals; $\mathbf{X}$, $\mathbf{Z}_r$ and $\mathbf{Z}_d$ are incidence matrices relating the observations

165 ($\mathbf{y}$) to the model effects in $\boldsymbol{\beta}$, $\mathbf{r}$ and $\mathbf{a}$, respectively. The vector $\mathbf{e}$ is distributed as $\mathbf{e} \sim N(\mathbf{0}, I_{\sigma^2})$ where $\mathbf{I}$ is an
166 identity matrix, and $\sigma^2_\epsilon$ is the residual variance. For the pedigree-based approach, the $\mathbf{a}_d$ vector is assumed to
167 be distributed as $\mathbf{a}_d \sim N(\mathbf{0}, \mathbf{A} \sigma^2_\epsilon)$ where $\sigma^2_\epsilon$ is the additive genetic variance, and $\mathbf{A}$ is the expected numerator
168 relationship matrix from the pedigree information.
2) Competition mixed model (CM):

$$y = X\beta + Z_r r + Z_d a_d + Z_c a_c + Z_p p_c + e \quad [2]$$

where the effects $\beta$, $r$ and $e$, and matrices $X$ and $Z_r$ were specified as described above. The direct and competition breeding values for mothers and offspring are included in the random vector $a_d$ and $a_c$, linked to the phenotypic data $y$ through the incidence matrices $Z_d$ and $Z_c$, respectively. Matrix $Z_d$ has all elements equal to 0 except for a 1 in the corresponding column. Similarly, the $i^{th}$ row of matrix $Z_c$ has all elements equal to zero except in the position corresponding to the $m_i$ neighbours of the tree $i$, with values $f_{ij}, j = 1, ..., m_i$. These positive coefficients can be interpreted as the intensity of competition (IC) that each neighbour exerts over the phenotype of the $i^{th}$ tree. Following Cappa and Cantet (2008) and assuming that the intensity of competition is related to the inverse of the distance between $i$ and $j$, the IC for the competitors that lie either in the same row or column (R-C) of the tree $i$ ($f_{iR-C}$) and for competitors laying in the diagonal positions (D) with respect to tree $i$ ($f_{iD}$) are

$$f_{iR-C} = \sqrt{\frac{2}{2n_{iR-C} + n_{iD}}} \qquad f_{iD} = \frac{1}{\sqrt{2n_{iR-C} + n_{iD}}}$$

where $n_{iR-C}$ and $n_{iD}$ are the respective numbers of competitors laying in R-C and D. For the pedigree-based approach, the stacked vector of breeding values is distributed as

$$\begin{bmatrix} \mathbf{a}_d \\ \mathbf{a}_c \end{bmatrix} \sim N \left( \begin{bmatrix} \boldsymbol{\theta} \\ \boldsymbol{\theta} \end{bmatrix}, \begin{bmatrix} \sigma_{Ad}^2 & \sigma_{AdAc} \\ \sigma_{AdAc} & \sigma_{Ac}^2 \end{bmatrix} \otimes \mathbf{A} \right)$$

where $\sigma_{Ad}^2$ is the variance of the direct additive genetic effects, $\sigma_{Ac}^2$ is the variance of the competition breeding values, and $\sigma_{AdAc}^2$ is their covariance. The vector $p_c$ includes the environmental competition effects (or permanent environmental competition effects, Cappa and Cantet 2008; Cappa et al. 2015; Cappa et al. 2016b) such that $p_c \sim N(\boldsymbol{0}, I \sigma_p^2)$ where $\sigma_p^2$ is the variance of environmental competition effects. Finally, the matrix $Z_p$ is composed of the non-zero columns of $Z_c$.

The TH trait was analyzed using the standard model [1] and with the following spatial mixed model:

3) Spatial mixed model (SM):

$$y = X\beta + Z_r r + Z_d a_d + \xi + \eta \quad [3]$$
where the effects $\beta$, $r$, and $a_d$, and matrices $X$, $Z_r$, and $Z_d$ are specified as described above. In equation [3] an autoregressive spatial component was added, where the residual vector $e$ was partitioned into a spatially dependent ($\xi$) and independent ($\eta$) residuals (e.g., Costa e Silva 2001). Gilmour et al. (1997) suggested to model the $\xi$ vector as a separable Kronecker product of first-order autoregressive covariance structures (AR1)

on the rows (row) and the columns (col), i.e., $\sigma_{\xi}^2 [ \text{AR1}(\rho_{col}) \otimes \text{AR1}(\rho_{row}) ]$, where $\sigma_{\xi}^2$ is the spatially dependent residual variance, and $\rho_i$ the autocorrelation parameters. The vector $\eta$ was assumed pairwise independent, i.e., $I^{\sigma_{\eta}^2}$, where $\sigma_{\eta}^2$ is the independent residual variance.

In the combined approach, the matrix $A$ of the previous mixed models [1], [2], and [3], was substituted by the combined pedigree- and marker-based pairwise relationship $H$ matrix. Therefore, the vector $a_d$ of TM and SM is distributed $a_d \sim N(0, H \sigma_d^2)$, while the stacked vector of breeding values for the CM is distributed as

$$ \begin{bmatrix} a_d \\ a_c \end{bmatrix} \sim N \left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_{Ad}^2 & \sigma_{AdAc} \\ \sigma_{AdAc} & \sigma_{Ac}^2 \end{bmatrix} \otimes H \right) $$

The inverse of the relationship matrix that combines pedigree and genomic information ($H^{-1}$) was derived by Misztal et al. (2009), Legarra et al. (2009), Aguilar et al. (2010) and Christensen and Lund (2010) as

$$ H^{-1} = A^{-1} + \begin{bmatrix} 0 & 0 \\ 0 & \lambda (G^{-1} - A_{22}^{-1}) \end{bmatrix} $$

where $\lambda$ scales differences between genomic and pedigree-based information, $G^{-1}$ is the inverse of the genomic-based relationship matrix and $A_{22}^{-1}$ is the inverse of the pedigree-based relationship matrix for the genotyped individuals ($A_{22}$). We studied the influence of weighting factor $\lambda$ in the estimation of variance components for the TM model and for DBH and TH traits. However, only slight differences were found in both studied traits when $\lambda$ decreased from 1.0 (i.e., only contributions from genomic relationships) to 0.0 (i.e., only contributions from pedigree relationships) by 0.2 (i.e., $\lambda = 1.0, 0.8, 0.6, 0.4, 0.2, 0.0$). For instance, the additive (and residual) variance varied from 5.76 (and 12.11) for $\lambda = 0.0$ to 6.21 (and 11.76) for $\lambda = 1.0$ for DBH (see Online Resource 2). Therefore, the weighting factor $\lambda$ was set to 1.0 for both traits.
217 The genomic relationship **G** matrix from the dominant DArT markers was calculated following the formula suggested by Resende et. al. (2010):

$$G = \frac{(Z - P)(Z - P)^T}{\sum_k p_k (1 - p_k)}$$

219 where $p_k$ is the frequency of the code 1 at locus k, **Z** is a $n \times m$ matrix ($n$ = number of genotyped trees, $m$ = number of DArT markers) that specifies the genotypes expressed as 0/1 denoting the absence/presence of the DArT marker, and **P** is a matrix containing $p_k$ in the $k$th column.

223 A potential problem of the combined approach is that **G** and $A_{22}$ have to be expressed on the same scale (Meuwissen et al. 2011). However, $A_{22}$ involves relationships of genotyped individuals with reference to the base population (mothers), and **G** corresponds to relationships within the current population. Several methods have been developed to overcome this scale problem (Aguilar et al. 2010; Chen et al. 2011; Forni et al. 2011; Vitezica et al. 2011; Christensen et al. 2012). In this study, we scaled the **G** matrix following closely the work of Christensen et al. (2012; equation 4): $G_c = \beta G + \alpha$, where $G_c$ is the scaled **G** matrix, and $\beta$ and $\alpha$ are parameter values calculated by solving the following system of equations:

$$\text{Avg}(\text{diag}(G)) \beta = \text{Avg}(\text{diag}(A_{22}))$$

$$\text{Avg}(G) \beta + \alpha = \text{Avg}(A_{22})$$

232 The actual parameter values of $\alpha$ and $\beta$ were 0.0062 and 0.9938, respectively. Following Vitezica et al. (2011) other two possible correction of **G** were also studied: 1) $G_{\delta} = G + \mathbf{1} \mathbf{1}' \delta$, where $\delta = \frac{1}{n^2} \left( \sum_{i} \sum_{j} A_{22(i,j)} - \sum_{i} \sum_{j} G_{i,j} \right)$ (equation 5 in Vitezica et al. 2011); and 2) $G_{\text{Fst}} = (1 - \frac{1}{2} \delta) G + \mathbf{1} \mathbf{1}' \delta$, correction suggested by Powel et al. (2010) in Vitezica et al. (2011). However, given that $\beta$ was very close to one (0.9938), this parameter $\delta$ in Vitezica et al. (2011) and Powel et al. (2010) was equal to $\alpha$, i.e., 0.0062. Consequently, the **G** matrix corrected by equation proposed by Christensen et al. (2012) was equivalent to those proposed by Vitezica et al. (2011) and Powel et al (2010).

239 All the analyses were performed in R (www.r-project.org) with the package breedR (Muñoz and Sanchez, 2014) using Restricted Maximum Likelihood (REML, Patterson and Thompson 1971) inference.

241 The theoretical accuracy $r$ of the predicted breeding values was calculated using the following expression:
The acronym PEV stands for ‘prediction error variance’ of predicted breeding values, and are calculated as the diagonal elements of the inverse of the coefficient matrix from the mixed model equations of each model studied (Gilmour et al. 1995), and $F_i$ is the inbreeding coefficients of tree i. For the CM, the estimated additive

genetic variance $\hat{\sigma}_a^2$ was replaced with the estimated direct additive genetic variance $\hat{\sigma}_{Ad}^2$.

The gain in accuracy of a tree's breeding value when using its markers' information depends on the trait, the model and the group to which the tree belongs (i.e., genotyped offspring, non-genotyped offspring or parents). Furthermore, even within each of these combinations, the gains vary due to differences in relatedness among individuals as measured by the markers and ultimately reflecting the underlying Mendelian segregation process. We estimated the expected gain in accuracy for an individual by maximum likelihood using a linear regression on the trait, the model and the group.

The narrow-sense individual heritability ($\hat{h}^2$) was estimated as

$$ \hat{h}^2 = \frac{\hat{\sigma}_a^2}{\hat{\sigma}_a^2 + \hat{\sigma}_e^2} $$

where $\hat{\sigma}_a^2$ is the estimated additive genetic variance, and $\hat{\sigma}_e^2$ the estimated error variance. For CM, the additive variance $\hat{\sigma}_a^2$ was replaced with the estimated total additive genetic variance (Cappa and Cantet 2008). Assuming that the focus tree is unrelated to its competitors, and the competitors are unrelated among themselves, the total additive variance is $\hat{\sigma}_{Ad}^2 + \hat{\sigma}_{Ac}^2$. Additionally, for these two models, the denominator of the above expression also included the estimated variance of the environmental competition effects, $\hat{\sigma}_p^2$.

## 262 Results

### 263 Pedigree and combined relationship matrices

When the pedigree-based A matrix analysis was conducted, the studied 164 OP families were treated as unrelated; conversely, when the genomic-based G matrix of the genotyped offspring was implemented, the
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ancestral relationship among their mother-trees was projected and implied that 132 mothers, with genotyped offspring, are related. The mother trees in the combined *H* matrix had 4,084 (30.6 %) out of 13,366 (i.e., (164) × (164 - 1)/2) pair-wise relatedness coefficients higher than zero while that of the genotyped and the non-genotyped offspring, from families with genotyped offspring, had 56.3% of the pair-wise relationships higher than zero as compared to that of the *A* matrix which produced 0.51% higher than zero. The average relationship coefficient for the non-genotyped offspring from the same half-sib family varied from 0.10 to 0.38 for the combined *H* matrix (consistent with the expected value of 0.25 from the *A* matrix). Moreover, the relationships between mother and offspring varied from 0.32 to 0.75 (consistent with the expected value of 0.5 from the *A* matrix). Briefly, genotyped offspring generated additional information for the non-genotyped half-sib mothers and their respective offspring, while, as expected, offspring from mothers with non-genotyped offspring did not produce any additional information.

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## 278 *Variance components*

Generally, the TM model produced lower direct additive genetic variance for both traits, while CM resulted in higher additive variance for DBH, SM showed only slightly higher additive variance for TH, highlighting the effect of competition and environmental heterogeneity (Table 2). These differences among the studied models showed opposite effects on the residual variance terms as they were higher for the TM model that only focused on direct genetic effects or experimental design consideration (Table 2).

Using the *H* matrix generally produced slightly higher additive genetic (direct and indirect) and smaller residual variance estimates than the relationship *A* matrix (Table 2), resulting in similar heritability estimates. However, for TH the spatial variance from the combined-approach (2.71) was notably smaller than that from the pedigree approach (23.24). Finally, as expected, the replicate variance decreased after spatial effects (i.e., SM) were included (Table 2). However, this reduction was more pronounced for the SM with the *H* matrix than with the *A* matrix.

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[Insert **Table 2** about here]

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## 292 *Accuracy of predicted breeding values*
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293 Mean accuracies of breeding values from the extended models (i.e., CM or SM) of both investigated traits (i.e., DBH or TH), were higher than those from TM for both pedigree and combined analyses (from 3.0 to 12.5%) (Table 3). However, the increments in the average accuracy across non-genotyped mothers and non-genotyped and genotyped offspring were slightly higher from the pedigree than for the combined approaches: 10.9 vs. 6.6% (CM vs. TM), and 4.5 vs. 4.2% (SM vs.TM) for *A* and *H* analyses, respectively.

298 Accuracy comparisons favored the combined analyses over the pedigree for all individual-tree mixed models in both traits, except for the non-genotyped parent and offspring from the CM (Table 3; Fig 1; Online Resource 3). For the TM, the expected accuracy averaged across groups increased from 0.62 to 0.64 (+3.6%) for DBH and from 0.49 to 0.58 (+20.7%) for TH. For the extended mixed models (i.e., CM and SM), the increments were slightly negative for DBH (from 0.70 to 0.69, -1.3%) or positive for TH (from 0.51 to 0.61, +21.8%). However, the expected gain in the accuracy with respect to the classical pedigree approach varied among the non-genotyped mothers and offspring and genotyped offspring for the different models and traits. Thus, associated with lower heritability for TH, the expected gain in the accuracies of breeding values for parents and all offspring using the combined approach was higher for TH (varying from 9.0 to 43.5%) than for DBH (varying from -2.7 to 6.2%) (Table 3; Fig 1). In short, the accuracy of overall predicted breeding values from the combined approach increased with respect to the classical pedigree-based approach.

309 [Insert **Table 3** about here]

310 [Insert **Fig 1** about here]

311 As expected, when the number of offspring per mother increases, the accuracy of mother breeding values from the *A* and *H* matrices increases for both traits (Fig 2). For both traits, more complex models produced higher accuracies than the reduced model (i.e., TM), and the combined approach (*H* matrix) was superior to that of pedigree approach (*A* matrix), except for the trait-model combination DBH-CM. Therefore, the observed increase in accuracy resulting from using genomic data could potentially reduce the number of tested offspring.

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319 **Discussion**
In forest tree breeding, there is growing interest in applying GBLUP using a genomic individual-tree mixed model to increase genetic gains per unit of time. However, the application of GBLUP requires genotyping all the trees involved in the prediction. Although genotyping costs have dramatically decreased in recent years, genotyping a large number of trees may not be cost effective (Beaulieu et al. 2014). An alternative genomic selection method involves combining information from a few genotyped and many non-genotyped trees with pedigree-recorded data, into an overall relationship *H* matrix. The combined *H* matrix can be seen as a modification of the regular *A* matrix to accommodate genomic relationship (Legarra et al. 2014). We investigated the accuracy of breeding values of non-genotyped mothers and offspring and genotyped offspring in a population of *E. grandis* as well as the influence of the heritability of the trait and the inclusion of competition genetic effects or environmental heterogeneity in an individual-tree mixed model on the estimated variance components and accuracy of breeding values. This preliminary study intended to assess the feasibility and usefulness of including marker data in the genomic evaluation of trees, using the combined approach methodology.

Cappa et al. (2016) highlighted the advantages of the marker-based relatedness estimates over the expected categorical measure of relationships using a sample of 166 genotyped offspring from the same dataset used in this study. The combined relationship *H* matrix diffused the information from genomic markers to non-genotyped offspring and parent trees and leveraged this information by more accurately reconstructing family relationships of all trees: genotyped and non-genotyped and with or without phenotype. For example, based on their contemporary pedigree, several pairs of seemingly unrelated mothers in *A* appeared as related in *H*, highlighting the value of capturing historical pedigree through their DNA fingerprints resulting from relatedness among their offspring in *G* matrix. Therefore, this methodology is advantageous even when molecular information from the parents is lacking as it offers a backward projection of the relationships between genotyped offspring towards the relationship between mothers.

From a breeding perspective, the accuracy of the selection criteria is important for the realized genetic progress (Loberg et al. 2015). It is thus essential to know how relationship matrices (pedigree or combined) using different individual-tree mixed models affect the accuracy of the breeding values. Although conventional selection using pedigree information has been shown to effectively improve growth traits in *E. grandis* (Marcó and White 2002; Harrand et al. 2009), including more information (i.e., genomic) is expected
to increase the accuracy and therefore the probability of correct ranking of selection candidates. However, the increase in the accuracy of breeding values in forest tree breeding populations based on joint phenotypic, pedigree and genomic information remains uncertain. For both traits (DBH and TH), our results showed that the combined approach led to an increase in the average of theoretical accuracies for genotyped offspring, but also for non-genotyped offspring and mother trees using the TM, and SM models. These results confirmed that the incorporation of genomic information (i.e., DArT markers) increased the accuracies with respect to a pedigree-based approach. Several studies in animal breeding showed that the accuracy of the combined approach is greater than that of the pedigree approach (from 16.7 to 94.4%) for both phenotyped and non-phenotyped individuals (see Table 1 in Legarra et al. 2014). It should be pointed out that the number of genetic markers used in genotyping is essential in resolving historical relationships as indicated by the lack of accuracy improvement when a modest number of markers was used when compared to conventional pedigree approach (831 SNPs; Legarra et al. 2011). The authors (Legarra et al. 2011) argued that denser SNP marker coverage of the genome and a greater number of animals were needed to improve the accuracy of genomic prediction.

Genomic selection is promising for low heritability traits (e.g. Calus et al. 2008; Grattapaglia and Resende 2011). Thus, for these traits genomic selection should be more efficient than standard pedigree selection. Our results showed that the advantage of the combined over pedigree approach was larger under low heritability traits (TH) than in high heritability traits (DBH). However, the variation (i.e., standard deviation) in the accuracy of breeding values of genotyped and non-genotyped offspring from the combined approach increased for the low heritability trait (TH) with respect to the pedigree-based approach for the two mixed models studied (TM, and SM; Table 3). Even under this condition; however, the combined approach was more accurate than the pedigree approach in all cases. These findings confirm that genomic selection is particularly beneficial for low heritability traits, even when using the more complex mixed model with competition genetic effects or environmental heterogeneity.

Genetic and environmental competition effects and environmental heterogeneity can affect the precision of the predicted breeding value and need to be considered for obtaining accurate breeding value predictions. Accounting for competition and spatial effects has been shown to increase the accuracy of breeding values for both parents and offspring when pedigree was used (Dutkowski et al. 2006; Costa e Silva
and Kerr 2012). Although with lower increments, our results suggest that it is also true for the *E. grandis*
evaluation using the proposed combined approach. To our knowledge, this is the first study applying
competition or spatial analysis with the combined genomic selection method in forest genetic trials.
Therefore, making comparisons with other studies is not possible. In a crop breeding context, Lado et al.
(2013) used different models to adjust a spatial trend in phenotypic data of 384 wheat (*Triticum aestivum*)
genotypes and concluded that a correction of spatial variation is an essential to increasing the prediction
accuracy in genomic selection models.

The scaled **G** matrix reflects a translation from the relationships relative to the genotyped trees to relationships relative to the base population defined by pedigree (i.e., mothers). An incorrect scale of **G** matrix can bias the predicted breeding values of genotyped individuals relative to those of the non-genotyped ones (Chen et al. 2011; Forni et al. 2011; Simeone et al. 2011; Vitezica et al. 2011; Christensen et al. 2012). We scaled the genomic relationship matrix **G** following closely the work of Christensen et al. (2012). In a simulation study, Vitezica et al. (2011) observed less accurate genomic breeding values with incorrectly scaled **G** but only under strong selection (i.e., non-random) of genotyped individuals, and when genotyping was across 10 generations. However, in the present study, the genotyped trees were randomly selected (i.e., unselected). Moreover, when different scale parameters (λ) were used to test the effects of the differences between genomic and pedigree-based information on the dispersion parameters (Aguilar et al. 2010), only slight differences were found in both studied traits (see an example for DBH in Online Resource 2).

Although the small number of genotyped trees (188 out of 2,026) and relatively low number of
markers assayed (2,816 DArT markers) is still less than optimal for a powerful genomic selection study, the
combined approach used for the current generation should be applied to the subsequent generation to test the
predictive ability of these DArT markers. Cross-validation is difficult to implement due to the scarcity of
available trees with genomic information. However, preliminary results on genotyped offspring without
phenotype information showed, for the two traits analyzed, that the theoretical accuracy of breeding values
from the combined approach decreased (with respect to the offspring with phenotype) but remained always
higher than that obtained by the pedigree approach in all the mixed models studied (results not shown). For
example, under TM, the accuracy for the non-phenotyped and genotyped offspring was 0.41 for DBH and
0.33 for TH using the combined approach, while for the pedigree the respective values were 0.33 and 0.27.
Results from this preliminary study indicate that the *H* matrix also has superiority over the *A* matrix in the prediction for non-phenotyped and genotyped offspring. These predictive abilities are lower (0.54 and 0.51 for DBH and TH, respectively) than those reported by Resende et al. (2012) for two *Eucalyptus* populations genotyped with more than 3,000 DArT markers. However, these values were obtained for small effective population size (11 and 51) and high heritabilities (0.53 and 0.42 for DBH and TH, respectively). The predictive ability of the proposed genomic selection method needs to be investigated under higher number of genotyped trees and denser DArT marker coverage of the genome.

411

412 **Conclusions**

In addition to the PhD thesis contribution by Ogut (2012), to our knowledge this study represents the first introduction of the combined approach as a genomic selection methodology applied in forest trees. This approach provides a simple method to jointly evaluate genotyped and non-genotyped trees, even when phenotype has not been assessed for parental trees. Moreover, the method can be easily extended to competition or spatial mixed model analysis. This empirical study shows the importance of using all available information (i.e., phenotype, pedigree, and genomic) to improve the accuracy of breeding values for growth traits with low to moderate heritability in an *E. grandis* population. Genomic prediction using an individual-tree mixed model that takes into account competition effects or environmental heterogeneity was more accurate than a prediction based on the standard individual-tree mixed model. These results indicated that genomic selection for *E. grandis* is promising. However, using a higher number of markers and more trees would allow a more efficient evaluation of the genomic selection potential for growth traits in the *E. grandis* population.

425

426 Acknowledgements

The authors are grateful to Forestal Las Marias for providing the land and logistical support for the test.
Thanks to Martín Marcó, Leonel Harrand, Javier Oberschelp, and Juan López, who assisted with field work
and data collection. We also wish to thank Cintia Acuña, and María Carolina Martínez for their help with the
DNA extraction, and molecular data analysis; and Dario Grattapaglia, Carolina Sansaloni, Cesar Petroli, and
Danielle Paiva for generating DArTs data.
432

433 **Conflict of Interest**

434 The authors declare that they have no conflict of interest.

435

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564 **Figure Legend:**

565

566 **Fig. 1** Expected average percent increases of the accuracy of breeding values for mothers and offspring from the combined approach (*H* matrix) compared with the classical pedigree (*A* matrix) using the standard model (TM), competition model (CM), and spatial model (SM) for diameter at breast height (DBH) and total height (TH).

570

571 **Fig. 2** Estimated theoretical accuracy of breeding values for mothers for the pedigree (*A* matrix) and the combined (*H* matrix) approaches using the standard, spatial and competition individual-tree mixed model vs. the number of phenotyped offspring per mother for: a) diameter at breast height (DBH) and b) total height (TH). The trend lines are logarithmic curves, and showing the increment in the estimated accuracy with the increment in number of phenotyped offspring per mother.
<table><caption>**Table 1.** Open-pollinated families and genotyped and non-genotyped individuals' statistics for diameter (DBH) and total height (TH) in the *Eucalyptus grandis* data set.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">N° of records</th><th colspan="2">Mean (SD)</th></tr><tr><th>DBH (cm)</th><th>TH (m)</th></tr></thead><tbody><tr><td>All trees with phenotype</td><td>2026</td><td>18.85 (4.27)</td><td>18.87 (2.68)</td></tr><tr><td>Trees from mothers with genotyped offspring</td><td>1650</td><td>18.87 (4.24)</td><td>18.87 (2.65)</td></tr><tr><td>Genotyped offspring</td><td>187</td><td>20.81 (3.07)</td><td>20.57 (1.67)</td></tr><tr><td>All mothers</td><td>164</td><td>-</td><td>-</td></tr><tr><td>Total number of mothers with genotyped offspring</td><td>131</td><td>-</td><td>-</td></tr></tbody></table>

SD = Standard deviation.
<table><caption>**Table 2.** Estimation of genetic parameters with pedigree and combined approaches for diameter at breast height (DBH) and total height (TH) in *Eucalyptus grandis* using different individual-tree mixed models with spatial and competition effects. See text for models' abbreviations.</caption><thead><tr><th rowspan="3">Method of genetic evaluation</th><th colspan="2">DBH</th><th colspan="2">TH</th></tr><tr><th colspan="4">Models</th></tr><tr><th>TM</th><th>CM</th><th>TM</th><th>SM</th></tr></thead><tbody><tr><td><strong>Pedigree</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Additive (direct) variance</td><td>5.76</td><td>7.56</td><td>1.08</td><td>1.16</td></tr><tr><td>Additive indirect variance</td><td>-</td><td>1.12</td><td>-</td><td>-</td></tr><tr><td>Direct and competition correlation</td><td>-</td><td>-0.94</td><td>-</td><td>-</td></tr><tr><td>Replicate variance</td><td>0.19</td><td>0.25</td><td>0.75</td><td>0.06</td></tr><tr><td>Permanent environmental variance</td><td>-</td><td>0.16</td><td>-</td><td>-</td></tr><tr><td>Spatial variance</td><td>-</td><td>-</td><td>-</td><td>23.24</td></tr><tr><td>Residual Variance</td><td>12.11</td><td>9.30</td><td>5.38</td><td>4.96</td></tr><tr><td>Heritability</td><td>0.32</td><td>0.48</td><td>0.17</td><td>0.19</td></tr><tr><td><strong>Combined</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Additive (direct) variance</td><td>6.21</td><td>6.26</td><td>1.14</td><td>1.26</td></tr><tr><td>Additive indirect variance</td><td>-</td><td>0.43</td><td>-</td><td>-</td></tr><tr><td>Direct and competition correlation</td><td>-</td><td>-0.99</td><td>-</td><td>-</td></tr><tr><td>Replicate variance</td><td>0.19</td><td>0.25</td><td>0.75</td><td>0.02</td></tr><tr><td>Permanent environmental variance</td><td>-</td><td>0.10</td><td>-</td><td>-</td></tr><tr><td>Spatial variance</td><td>-</td><td>-</td><td>-</td><td>2.71</td></tr><tr><td>Residual Variance</td><td>11.76</td><td>8.52</td><td>5.34</td><td>4.85</td></tr><tr><td>Heritability</td><td>0.35</td><td>0.44</td><td>0.18</td><td>0.21</td></tr></tbody></table>
<table><caption>**Table 3.** Mean and standard deviations of estimated theoretical accuracies for the predicted breeding value based on the pedigree and combined approaches for diameter at breast height (DBH) and total height (TH) in *Eucalyptus grandis* using different individual-tree mixed models. See text for models' abbreviations.</caption><thead><tr><th rowspan="3">Method of genetic evaluation</th><th colspan="2">DBH</th><th colspan="2">TH</th></tr><tr><th colspan="4">Models</th></tr><tr><th>TM</th><th>CM</th><th>TM</th><th>SM</th></tr></thead><tbody><tr><td><strong>Pedigree</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Mothers</td><td>0.67 ± 0.07</td><td>0.72 ± 0.07</td><td>0.54 ± 0.07</td><td>0.57 ± 0.08</td></tr><tr><td>Genotyped offspring</td><td>0.60 ± 0.02</td><td>0.68 ± 0.02</td><td>0.46 ± 0.02</td><td>0.48 ± 0.02</td></tr><tr><td>Non-genotyped offspring</td><td>0.60 ± 0.03</td><td>0.68 ± 0.03</td><td>0.46 ± 0.02</td><td>0.48 ± 0.02</td></tr><tr><td><strong>Average</strong></td><td><strong>0.62 ± 0.04</strong></td><td><strong>0.70 ± 0.04</strong></td><td><strong>0.49 ± 0.04</strong></td><td><strong>0.51 ± 0.04</strong></td></tr><tr><td><strong>Combined</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Mothers</td><td>0.68 ± 0.07</td><td>0.71 ± 0.07</td><td>0.60 ± 0.08</td><td>0.62 ± 0.07</td></tr><tr><td>Genotyped offspring</td><td>0.64 ± 0.02</td><td>0.69 ± 0.02</td><td>0.66 ± 0.03</td><td>0.68 ± 0.04</td></tr><tr><td>Non-genotyped offspring</td><td>0.61 ± 0.03</td><td>0.67 ± 0.03</td><td>0.50 ± 0.03</td><td>0.53 ± 0.03</td></tr><tr><td><strong>Average</strong></td><td><strong>0.64 ± 0.04</strong></td><td><strong>0.69 ± 0.04</strong></td><td><strong>0.58 ± 0.05</strong></td><td><strong>0.61 ± 0.04</strong></td></tr></tbody></table>