Accuracy of GEBV of sires based on pooled allele frequency of their progeny

Abstract

Despite decreasing genotyping costs, in some cases individually genotyping animals is not economically feasible (e.g., in small ruminants). An alternative is to pool DNA, using the pooled allele frequency (PAF) to garner information on performance. Still, the use of PAF for prediction (estimation of genomic breeding values; GEBVs) has been limited. Two potential sources of error on accuracy of GEBV of sires, obtained from PAF of their progeny themselves lacking pedigree information, were tested: (i) pool construction error (unequal contribution of DNA from animals in pools), and (ii) technical error (variability when reading the array). Pooling design (random, extremes, K-means), pool size (5, 10, 25, 50, and 100 individuals), and selection scenario (random, phenotypic) also were considered. These factors were tested by simulating a sheep population. Accuracy of GEBV—the correlation between true and estimated values—was not substantially affected by pool construction or technical error, or selection scenario. A significant interaction, however, between pool size and design was found. Still, regardless of design, mean accuracy was higher for pools of 10 or less individuals. Mean accuracy of GEBV was 0.174 (SE 0.001) for random pooling, and 0.704 (SE 0.004) and 0.696 (SE 0.004) for extreme and K-means pooling, respectively. Non-random pooling resulted in moderate accuracy of GEBV. Overall, pooled genotypes can be used in conjunction with individual genotypes of sires for moderately accurate predictions of their genetic merit with little effect of pool construction or technical error.

Introduction

In recent years, genotyping costs have substantially decreased making it possible to genotype animals on a commercial scale, especially dairy and beef cattle (VanRaden et al. 2011). Still, in some cases, genotyping individual animals may be impractical or economically prohibitive (e.g., in small ruminants, in commercial animals in feedlots or processing plants), making it necessary to develop alternative strategies for genomic prediction and analyses. While DNA pooling has been extensively used as a means to reduce genotyping costs for association studies (McDaneld et al. 2014; Keele et al. 2015), its use for genomic prediction (estimation of breeding values) has only recently become more common (Reverter et al. 2016; Bell et al. 2017; Alexandre et al. 2019; Baller et al. 2020).

In the context of genomic prediction, Reverter et al. (2016) used a hybrid genomic relationship matrix (G) based on pooled allele frequencies (PAFs). A relationship matrix constructed this way provides a means to obtain estimates of genomic breeding values (GEBVs) for pools of individuals and relatives of individuals in a pool. Similarly, to predict breeding values of sires, Bell et al. (2017) used a hybrid G, which included individual genotypes as well as pooled genotypes. The hybrid G in Reverter et al. (2016) and Bell et al. (2017) differ in the way they were constructed. While GEBV can be obtained through genomic BLUP (GBLUP) using pooled genotype information, the effects of pool structure (e.g., number of animals in a pool, pool design) on the accuracy of GEBV have only recently been quantified (Alexandre et al. 2019).

Besides pool design and size, other sources of error may influence PAF estimates and thus potentially introduce bias in the resulting GEBV obtained from them. The contribution of an animal to a given pool is inversely proportional to the number of animals in the pool. In practice, however, small deviations occur (randomly) such that planned animal representation in a pool may differ from its actual representation (Kuehn et al. 2018), resulting in pool construction error. Moreover, when genotyping pooled DNA samples, alleles at a polymorphic locus may not be equally quantified by fluorescent intensities when reading the array (for heterozygous individuals), or PAF not exactly equal to 0 or 1 for pools of homozygous individuals (Peiris et al. 2011). Both of these would contribute to pool construction error. Given that economic resources are limited, not only are the choices of pool design and size important but also quantifying the effect of pool construction and technical error on accuracy of GEBV.

Pedigree, individual, and pooled genomic information can be simultaneously used for prediction through single-step methodologies (Baller et al. 2020). There is interest in obtaining breeding value (EBV) for sires that do not have phenotypes, and whose pedigree relationships with progeny is unknown. In such cases, pooling individuals may provide a means to obtain GEBV for their individually genotyped relatives (sires). However, in all cases when combining genotypes on individuals and pools into a single genomic relationship matrix, scaling issues may arise (Reverter et al. 2016) potentially leading to a decrease in prediction accuracy.

The main aim of this study was to determine the changes in accuracy of sire GEBV derived from pools constructed from their progeny, in the case where pedigree relationships are unknown. Potential sources of inaccuracies were (i) pool construction error, and (ii) technical error. Beyond (i) and (ii) we also investigated the effect of size and number of pools, pooling design, and scaling of the genomic relationship matrix. The effect of these factors was evaluated through a comprehensive simulation. Because the cost of individually genotyping animals is particularly challenging in small ruminants, the sheep genome was simulated. We emulated a trait with moderate heritability. In addition, we describe the theoretical properties of a model designed to simulate PAF accounting for pool construction and technical error.

Materials and methods

Population

To determine the effect that pool construction error, technical error, pool design, pool size (or number of pools), scaling, and selection scenario may have on the accuracy of sire GEBV a simulation study was conducted. A sheep population was simulated using AlphaSimR (Gaynor C., 2019). AlphaSimR allows simulation of both phenotypic and genotypic information based on user-specified parameters (Supplementary File S1). Haplotypes in the founding population were generated using MaCS (Chen et al. 2009). A single trait was generated assuming heritability (h²) of 0.3 and a phenotypic variance of 1.0. A founding population of 1000 individuals (half male and half female) was simulated with effective population size (N_e) of 350. This value was chosen based on Kijas et al. (2014) and it was intermediate between the Merino (N_e = 900) and Border Leicester (N_e = 242) breeds. The aim was to define a parental population that ultimately would result in approximately 10,000 progeny per generation.

After the founders were established, the population size was increased using random mating and by replacing 20% of the current dams and keeping an additional 10% of the female progeny. The flock consisted of one sire for every 20 females at any given time, with 60% of sires replaced each generation. Twenty-six generations were required to achieve the desired number of progeny. The number of dams ranged from 500 at generation 1 to 5350 at generation 26, while the number of sires ranged from 25 at generation 1 to 240 at generation 26. Total number of progeny born was 1000 at generation 1 and 10,700 at generation 26. The number of progeny per sire was on average 44, ranging from 40 at generation 1 to 44 at generation 26. After reaching the target population size, 15 additional generations of (i) random mating or (ii) phenotypic selection (for high phenotypes) were simulated such that the total number of generations was 41. During these additional generations, the number of sires and progeny was maintained at 240 and 10,700, respectively.

Genome

A set of 26 autosomal chromosomes was simulated. The length of each chromosome was based on the sheep genome (Howe et al. 2020) and ranged from 275,406,953 bp (or 386 cM) for chromosome one to 42,034,648 bp (or 59 cM) for chromosome 24. For the conversion of bp to cM it was assumed that 1.4 cM corresponded to 1 million bp (Prieur et al. 2017). The number of markers and QTL depended on chromosome length such that the largest chromosome had 6059 markers and 275 QTL, while the smallest chromosome had 925 markers and 42 QTL. Markers and QTL were non-overlapping. In total, 53,901 markers and 2449 QTL were simulated. Loci with MAF <0.01 were removed, resulting in approximately 40,000 SNP for subsequent analyses.

Stochastic representation of pool construction and technical error

To incorporate pool construction as well as technical error, PAF was stochastically simulated as described in the following sections.

Model for pooled allele frequency

Assuming that the simulated genotypes (from AlphaSimR) have a Binomial distribution (Gerard et al. 2018) then let X_ij be a random variable corresponding to the i^th animal’s genotype at the j^th locus (i.e., $X_{ij}\sim \text{Binomial}(2,p_j)$), such that $E[X_{ij}]=E[X_{1j}]=\cdots =E[X_{nj}]=2p_j$. The PAF can then be constructed (represented) as

$$\begin{array}{c}{\theta }_{jk}=\frac{1}{2}\sum _{i=1}^{n_k}{\pi }_i{X}_{ij},\\ i=1,\dots ,n_k,\hspace{0.17em}j=1,\dots ,P.\end{array}$$ (1)

where the θ_jk was the PAF for marker j and pool k, which includes the weight (contribution) π_i of animal i to pool k, n_k was the number of individuals in pool k, and P was the number of markers. Pool construction error due to random animal representation and technical error can be easily included using this formulation.

Pool construction error

To more realistically portray the effect of pool construction error (animals being unequally represented in a given pool) let $\mathbf{\pi }=[{\pi }_1,\dots ,{\pi }_{n_k}]$ be a vector corresponding the contribution of animal i’s DNA to pool k and let

$$\begin{array}{c}\mathbf{\pi }\sim \text{Dirichlet}([{\alpha }_1,\dots ,{\alpha }_{n_k}]),\hspace{0.17em}\text{such}\hspace{1em}\text{that},\\ E[{\pi }_i]=\frac{{\alpha }_i}{{\alpha }_0},\hspace{0.17em}\mathit{Var}[{\pi }_i]=\frac{{\alpha }_i({\alpha }_0-{\alpha }_i)}{{\alpha }_0^2({\alpha }_0+1)},\hspace{0.17em}\text{with}\\ {\alpha }_0=\sum _{i=1}^{n_k}{\alpha }_i,\end{array}$$ (2)

where α_i, $i=1,\dots ,n_k$, was the concentration parameter. It was further assumed that ${\alpha }_1=\cdots ={\alpha }_{n_k}$ (symmetric Dirichlet), so that ${\alpha }_0=n_k{\alpha }_1$ and $E[{\pi }_i]=1/n_k$. The variance of individual animal contributions ($\mathit{Var}[{\pi }_i]=(n_k-1)/[n_k^2(n_k{\alpha }_1+1)]$) will tend to zero as pool size increases. Thus, the parameter α₁ provides a measure of accuracy of pool construction. The theoretical coefficient of variation was then $\sqrt{(n_k-1)/(n_k{\alpha }_1+1)}$. A conservative value of ${\alpha }_1=10$ was used (Abrams et al. 2021), such that the coefficient of variation ranged from 28.0% to 31.4% depending on pool size. Variability in PAF due to pool size and allele frequency is shown in Figure 1. A symmetric Dirichlet distribution assumes that, on average, the contribution of an individual’s DNA to a pool is centered at the mean, and the mean is the same for all individuals. If contributions are homogeneous (i.e., no error in pool construction), then ${\pi }_1=\cdots ={\pi }_{n_k}=1/n_k$.

Figure 1.

Density of simulated pooled allele frequency as functions of allele frequency (p) and pool size (n_k). In all cases ${\alpha }_i=10$.

Technical error

Error due to variability in green and red intensity when reading the array was incorporated by the random variable U_jk. Let U_jk be a Normal random variable (representing technical error) and let $g(·)$ be the standard logistic function (i.e., $g(X)=e^X/(1+e^X)$). Then $W_{jk}=g(U_{jk})$ has a logit-Normal distribution (Atchison and Shen 1980; Frederic and Lad 2008). The distribution of U_jk is centered around the current PAF value such that

$$\begin{array}{c}{\mu }_{jk}=\text{log}\left(\frac{{\theta }_{jk}}{1-{\theta }_{jk}}\right),\\ U_{jk}\sim \text{Normal}({\mu }_{jk},\hspace{0.17em}{\sigma }_T^2),\end{array}$$ (3)

$$W_{jk}=\frac{\hspace{0.17em}\text{exp}(U_{jk})}{1+\text{exp}(U_{jk})},$$ (4)

where W_jk is the PAF for marker j and pool k with individual animal contributions and technical error included. Variability in the underlying logit scale is controlled by ${\sigma }_T^2$. Some properties of the Normal distribution on the logit scale are given below. Note that μ_jk is not defined when θ_jk equals 0 or 1. However, fixed markers (values of 0 or 1) were excluded, as done in practice, because they provide no information for genomic relationships.

Under the construction in (1) to (4), and in the absence of technical error, U_jk equals μ_jk such that $E[{\theta }_j]=E[W_j]=p_j$. On the other hand, when technical error is not negligible, the expected value can be approximated by (Appendix A):

$$E[W_j]\approx p_j\left[1-\left(\frac{1-3p_j+2p_j^2}{2}\right){\sigma }_T^2\right].$$

Deviations of the expected value of PAF from the base allele frequency are a function of ${\sigma }_T^2$ and p_j (Figure 2). That is, if $W_j~\text{Logit‐Normal}(p_j,{\sigma }_T^2)$ the expected value of PAF deviates from p_j (i.e., $E[W_j]\ne p_j$). These deviations may be of less concern when the interest is in prediction of breeding values, as PAF will be used to construct (and scale) the genomic relationship matrix rather than to estimate individual marker effects. Note that the above approximation is not defined for values of p_j = 0 or p_j = 1 because $\mu =\text{log}\left(\frac{p_j}{1-p_j}\right)$.

Figure 2.

Mean (red line), 0.025, 0.5, and 0.975 quantiles (dashed blue lines) for simulated PAF (10,000 replicates) when n_k = 25 and ${\alpha }_1=10$, at three levels of technical error (σ_T). Black line shows slope = 1.

As with the mean, closed form expressions for the variance of the constructed PAF are not available but can be approximated through numerical integration or by (shown in Appendix A)

$$\mathit{Var}[W_j]\approx \left[\frac{2p_j}{{(1+p_j)}^3}\right]{\sigma }_T^2.$$

Finally, it was assumed that the effect of ${\sigma }_T^2$ is the same for every locus (i.e., array variability was constant across loci). This was done to maintain a more conservative scenario in our simulations; that is assuming more impact due to technical error than likely in practice. Nevertheless, the properties of the technical error outlined above still hold in cases where such is not the case.

Closed expressions for the variances of PAF (in the presence of construction and array-based error) are not easily obtained. Still, the proportion of total variability in PAF due to pool construction error can be assessed by simulating replicates of PAF as in (1) without error ${\pi }_i=1/n_k$ for all $i=1,\dots ,n_k$, and as in (2) with error. The influence of pool construction error on PAF variability can be calculated by the difference in their sample variances. A similar strategy can be used to assess the influence of technical error.

Pooling design

Pooling extreme individuals has been routinely used for association tests (Bader et al. 2001) and involves assigning animals to pools based on their high (or low) phenotypic values. This strategy mirrors that of case-control studies for binary traits. Random pooling, on the other hand, assigns animals to pools completely at random.

K-means clustering has been used as a pooling strategy in medical studies (Mitchell et al. 2014) where performing biological assays on individuals may not be economical. In such cases, the K-means algorithm can be applied to a set of predictors (e.g., body mass index, age, etc.) to form groups (pools) of individuals to which the assay is then performed. The K-means algorithm minimizes the within cluster (pool) sum of squares (Friedman et al. 2001, p. 462; Clarke et al. 2009, p. 411)

$$S_w(C)=\sum _{k=1}^K{n}_k\sum _{C(i)=k}\left|\right|x_i-{\overline{x}}_k|{|}^2,$$

where C(i) = k indicates the i^th observation of variable x allocated to cluster (pool) k, and $\overline{\mathbf{x}}=[{\overline{x}}_1,\dots ,{\overline{x}}_K]$ is the vector of means associated to the clusters. Because the algorithm minimizes $S_w(C)$, observations (individuals) within a given cluster will be more similar. Thus, clusters can be selected based on the estimated cluster means (or centers $\overline{\mathbf{x}}$). Note that pool (cluster) size is variable when using the K-means algorithm such that the elements of G (corresponding to pools) may vary (wildly) in scale. Still, K-means may prove useful for allocating individuals to pools when binary or categorical traits are of interest. In such cases, some animals will have exactly the same phenotypic value. Grouping (pooling) can then be achieved by using a similarity measure based on continuous covariates or correlated traits.

Genomic relationship matrix using pooled and individual genotypes

Let X and Z be N × P and K × P matrices of genotypes for individual animals (sires) and allele frequencies for pools, respectively, N and K be the number of individuals and pools, respectively, and P be the number of loci. We scaled X by 0.5 such that elements $x_{ij}\in \{0.0,0.5,1.0\}$ and $z_{ij}\in (0,1)$. In addition, let $n=N+K$.

$$\mathbf{M}=\left[\begin{array}{c}\mathbf{X}\\ \mathbf{Z}\end{array}\right],\hspace{0.17em}{\widehat{\mu }}_j=\frac{1}{n}\sum _{i=1}^n{M}_{ij},$$

where M_ij was the genotype (or PAF) value for the i^th individual at the j^th SNP, and $\widehat{\mathbf{\mu }}=[{\widehat{\mu }}_1,\dots ,{\widehat{\mu }}_P]$ was a $1\times P$ vector of estimated allele frequencies. Moreover, genomic relationships derived from pooled and individual genotypes are proportional to those obtained from individual genotypes (Appendix C). In Keele et al. (2015), a genomic relationship matrix for pools was constructed as

$$\mathbf{G}=\frac{(\mathbf{M}-1_n\widehat{\mathbf{\mu }}){(\mathbf{M}-1_n\widehat{\mathbf{\mu }})}^{\mathsf{T}}}{{\widehat{\mathbf{\mu }}}^{\mathsf{T}}(1_P-\widehat{\mathbf{\mu }})},$$

where $1_n$ and $1_P$ were $n\times 1$ and $1\times P$ vectors of ones, respectively. We redefine G to encompass both pools and individual animals, and further scale it by 2 such that its elements were on an equivalent scale as G when constructed from individual genotypes. Thus, as described in Reverter et al. (2016), Bell et al. (2017), and Baller et al. (2020), G is made of three distinct blocks

$$\mathbf{G}=\left[\begin{array}{cc}{\mathbf{G}}_{ss}& {\mathbf{G}}_{s\overline{p}}\\ {\mathbf{G}}_{\overline{p}s}& {\mathbf{G}}_{\overline{p}\overline{p}}\end{array}\right]$$ (5)

where ${\mathbf{G}}_{ss},\hspace{0.17em}{\mathbf{G}}_{\overline{p}\overline{p}}$, and ${\mathbf{G}}_{s\overline{p}}$ correspond to genomic relationships among sires, among pools, and between sires and pools, respectively. Moreover, from equation (15, in Appendix B) it can be seen that the diagonal elements of the ${\mathbf{G}}_{\overline{p}\overline{p}}$ are $\simeq \frac{1}{n_k}$ even in the presence of pool construction error. Due to multicollinearity among markers and because the number of markers is considerably larger than the number of genotyped individuals and pools, the G matrix constructed in this way can be ill conditioned such that its inverse cannot be calculated (Cai et al. 2011; Keele et al. 2015). To address the poor conditioning of G, its diagonal values were multiplied by 1.01. Multiplying diagonal elements by a small constant substantially increases the size of those eigenvalues that are closer to zero, while leaving larger eigenvalues relatively unchanged. Modifying diagonal elements with different larger values than 1.01 (increments of 0.01 up to 1.09 were tested) had little impact on solutions. The correlation between these solutions were close to 1 (>0.999). An alternative methodology consists of finding the nearest (in the Frobenius norm) positive semi-definite matrix to a matrix with negative or close to zero eigenvalues (Higham 2002). Applying this approach to G in our study, also resulted in high correlations between solutions (>0.999). Moreover, the condition number of G was always lower when modifying the diagonal elements as compared to computing its nearest positive definite form as in Higham (2002). Therefore, the strategy of modifying the diagonal elements of G with a small multiplier was chosen.

Scaling

Elements in ${\mathbf{G}}_{ss}$ are on a different scale than those in ${\mathbf{G}}_{s\overline{p}}$ or ${\mathbf{G}}_{\overline{s}\overline{s}}$. Therefore, to determine if scaling had an effect on the resulting accuracy of GEBV the GBLUP model was fitted (i) with G and (ii) as recommended by Bell et al. (2017), with the elements of G scaled as $G_{ij}^{\star }=G_{ij}/\sqrt{G_{ii}{G}_{jj}}$.

Simulation

The effect of pool construction error was tested by building pools and (i) setting each individual weight ${\pi }_i=1/n_k$, or (ii) drawing π_i from a Dirichlet distribution with ${\alpha }_1=10$ as described earlier. Thus, (i) serves as a “control” treatment. Array-based technical error was tested by using three error variances (${\sigma }_T^2=\{{0.125}^2,{0.625}^2,{1.125}^2\}$) when simulating PAF in (3).

The pooling strategies tested were (i) random pooling, (ii) pooling on extreme observations (individuals), and (iii) K-means pooling. Out of the total population in the last generation (10,700 animals) 3000 individuals were pooled into 600, 300, 120, 60, or 30 pools (i.e., $n_k=\{5,10,25,50,100\}$). For i) a random subset (sampled without replacement) of 5, 10, 25, 50, or 100 animals from the population was allocated to each pool. For (ii) observations were ranked based on their phenotypic values and pools of size 5, 10, 25, 50, or 100 were constructed, drawing from the extremes of their distribution (tails). In all cases, individual animals were not placed in more than one pool. Additionally, in all simulations, the number of sires on which GEBV were predicted was fixed to 30; sires evaluated were randomly drawn from the previous generation, ignoring phenotypes.

For extreme or random pooling, the total number of individuals for pool construction was fixed to 3000. For K-means, such was not possible as pool sizes were necessarily unpredictable. Instead, all progeny from the last simulated generation were first allocated, using the K-means algorithm based on their phenotypes, to either 800, 500, 300, 200, or 100 initial pools. These were then sorted by their means (i.e., aggregate phenotypes) and 600, 300, 120, 60, or 30 pools with the most extreme values were selected. As with extreme pooling, half of the pools were selected from the bottom (low phenotypic means) of the distribution and half from the top (high phenotypic values) such that the total number of individuals was approximately 3000 (Table 1).

Table 1.

 Summary statistics for pool (cluster) size when using K-means clustering on phenotype

Selection scenario	Number of pools	Mean	SD	Min	Max	Size 1a	nb
Random mating	30	104.9	47.0	13.2	189.0	0.033	3148
	60	53.5	20.9	8.4	106.0	0.075	3210
	120	26.5	11.1	5.2	61.8	0.078	3180
	300	10.9	5.3	1.7	31.5	0.169	3283
	600	5.3	3.0	1.0	18.9	2.280	3168
High phenotypes	30	108.1	48.0	12.6	192.0	0.067	3242
	60	53.0	20.7	8.4	105.0	0.053	3181
	120	26.6	11.1	5.1	61.7	0.049	3194
	300	10.9	5.2	1.8	31.6	0.137	3268
	600	5.3	3.0	1.0	18.9	2.270	3178

^aPercentage of pools (clusters) of size 1.

^bTotal number of individuals allocated to pools.

This procedure was carried out for both selection strategies (random mating and selection for high phenotypes).

Genomic and pedigree BLUP

The effect of construction and technical error, pool size, design, scaling, and selection scenario on the accuracy of genomic prediction was tested. For each of the simulation scenarios, sire EBV from GBLUP (and pedigree BLUP; PBLUP) using pooled information were calculated in R (R Core Team 2020). The model fitted was

$$\overline{\mathbf{y}}=1\mu +\mathbf{Q}\mathbf{a}+\mathbf{e},$$ (6)

$$\mathbf{a}\sim \mathsf{Normal}(0,\hspace{0.17em}\mathbf{G}{\sigma }_A^2),$$ (7)

$$\mathbf{e}\sim \mathsf{Normal}(0,\hspace{0.17em}\mathbf{R}{\sigma }_e^2),$$

$$\mathbf{R}=\text{diag}\left(\frac{1}{n_1},\dots ,\frac{1}{n_K}\right),$$

where $\overline{\mathbf{y}}$ was the ($K\times 1$) vector of aggregate phenotypes (mean phenotype for a given pool), μ was an intercept, $1$ was a ($K\times 1$) vector of ones, Q was a ($K\times (N+K)$) incidence matrix for the additive effects of the form $\mathbf{Q}=[0_N\hspace{0.17em}{\mathbf{I}}_K]$ such that $0_N$ was a K × N matrix of zeros, and ${\mathbf{I}}_K$ was a K × K identity matrix. The K was the number of pools ($K=\{600,300,120,60,30\}$) for all designs and $\mathbf{a}=[{\mathbf{a}}_s\hspace{0.17em}{\mathbf{a}}_{\overline{p}}]$ was the ($(N+K)\times 1$) vector of GEBV for sires and pools, respectively. In addition, $\mathbf{R}=\text{diag}\left(1/n_1,\dots ,1/n_K\right)$, because $\mathit{Var}[{\overline{y}}_1]={\sigma }_e^2/n_1,\hspace{0.17em}{\sigma }_A^2=0.3$ was the additive variance, ${\sigma }_e^2=0.7$ was the residual variance of the aggregate phenotypes, and G was as in (5). The mixed model equations related to model (6) were constructed directly in R using the Matrix package (Bates and Maechler 2019). A total of 100 replications for each combination of the levels of the six factors considered in this study were conducted.

As additional benchmarks for comparison, GBLUP assuming individual genotypes were available on the progeny subset was fitted. The hybrid G in (7) was replaced by G (3000 × 3000) as defined in VanRaden (2008), with solutions obtained in R as with model (6). In addition, PBLUP were obtained assuming that all progeny in the last generation had complete pedigree information available. The ancestors of the sires selected, therefore, were also included in the pedigree. As with GBLUP, phenotypic information on sires was ignored. The vector of estimated breeding values (a) had dimensions $(N=10700+30+m)\times 1$, where m is the number of ancestors of the sires selected with the assumption that $\mathbf{a}\sim \text{Normal}(0\hspace{0.17em}\mathbf{A}{\sigma }_A^2)$. Also R was replaced by ${\mathbf{I}}_n$, where n = 10,700 because PBLUP for grouped (pooled) data was not considered in this study. Pedigree BLUP was also fitted in R as with model (6).

Measures of accuracy and bias

Accuracy was defined as $\mathit{Cor}({\mathbf{a}}_s,{\widehat{\mathbf{a}}}_s)$, where ${\mathbf{a}}_s$ was the sire true (simulated) breeding value, and ${\widehat{\mathbf{a}}}_s$ was the sire estimated breeding value. Sire breeding values were estimated by (i) GBLUP with hybrid G using PAF from their progeny, (ii) GBLUP with G constructed from individually genotyped progeny, or (iii) PBLUP. In all cases, the (randomly) chosen sires all had genotypes but no phenotypic information.

In addition to accuracy as defined above, sire GEBV theoretical accuracy (r) based on prediction error variance was calculated as $r_i=\sqrt{1-C^{ii}({\sigma }_e^2/{\sigma }_A^2})$, where Cⁱⁱ was the i^th diagonal element of the (generalized) inverse of the ${\mathbf{C}}^{22}$ block of the inverse of the coefficient matrix of the mixed model equations. Variability in r (SD) also was obtained because it can be used to determine sire representation in a given pool, where higher variability indicates poorer representation.

Because 30 sires were randomly chosen from the 240 available in the last generation along with a subset of all progeny, the chosen sires may not be well represented in pools. This potentially results in bias of the GEBV. As such, mean proportional bias, defined as the mean (TBV–EBV)/TBV was also calculated. Lastly, given that TBV were known (simulated), as an additional measure of performance, the mean TBV of sires based on their top (or bottom) 10%, 20%, and 40% GEBV was calculated.

Differences in accuracy and bias due to pool construction error, level of technical error, pool size (or number of pools), pooling design, scaling, and selection scenario were tested by using ANOVA. Despite pool size being variable for K-means, on average, the number of individuals allocated to pools was similar to random and extreme pooling (Table 1). K-means pooling was therefore categorized using the same levels (as 5 levels of pools) as random and extreme designs. The model fitted was

$$y_{\mathit{ijklmnp}}=\mu +F_i+T_j+H_k+D_l+V_m+B_n+e_{\mathit{ijklmnp}},$$ (8)

where y_ijklmnp was the accuracy for a given combination of the factors in the model, μ was the overall intercept, F_i was the effect of construction error in a pool (i = 1, 2 for presence and absence, respectively), T_j was the effect of technical error (j = 1, 2, 3 for σ_T = 0.125, 0.625, 1.125, respectively), H_k was the effect of number of pools ($k=1,2,3,4,5$, for 30, 60, 120, 300, 600, respectively), D_l represented the effect of the pooling design (l = 1, 2, 3 for random, extreme and K-means, respectively), V_m was the effect of scaling G (m = 1, 2 for unscaled and scaled as correlation, respectively), B_n was the type of selection scenario (n = 1, 2 for random mating or phenotypic selection), and e_ijklmnp was the residual. In addition to the main effects described, all interactions (up to the six-way) were also included in the final model.

Model (8) was fitted using the lm function in R, and estimated marginal means were calculated using the emmeans package (Lenth 2019). Inspection of the residuals appeared to show a lack of Normality. A generalized linear model, assuming a Beta distribution also was fitted and estimated marginal means and significance were compared with the linear model under the Normality assumption. Despite small numerical differences, the conclusions drawn from both models were the same; for ease of understanding only results from the linear model are presented.

Results and discussion

Mean linkage disequilibrium (LD) measured as r² (not to be confused with accuracy squared) calculated for adjacent markers (on a chromosome by chromosome basis and then averaged across the genome) was 0.266 (SD 0.003) for the random mating scenario, and was 0.281 (SD 0.003) for the selection scenario. Minimum and maximum LD values for the random mating scenario were 0.201 (SD 0.012) and 0.339 (SD 0.008), respectively, while for the selection scenario were 0.215 (SD 0.012) and 0.361 (SD = 0.009), respectively. The values calculated for the simulated sheep population are similar to those values reported by Kijas et al. (2014) for the Poll Dorset (0.279) breed, which has similar N_e to that simulated in this study. As such, our results are relevant to other sheep populations with similar levels of LD.

Accuracy of GEBV for the different effects and scenarios is described in the following sections. Although up to six-way interactions were included in model (8), only those between pool size and pooling design, and pooling design and selection scenario, were statistically significant (P < 0.001). While significant, their overall effect on accuracy was minimal (i.e., main effects could still be compared unambiguously). As such, marginal means (main effects) for the remaining effects are still presented when relevant.

Pool construction error

Pool construction error, due to (random) unequal animal representation, did not have a substantial effect on accuracy of sire GEBV (P = 0.837). Mean accuracy was 0.528 (SE 0.001) and 0.522 (SE 0.001) for the presence or absence of construction error, respectively. As shown in Appendix A, even when animals were unequally represented in a pool, the expected value of PAF was still unbiased. As such, and with ${\alpha }_1=10$ as in our simulation, variability due to construction error did not lead to a large change in prediction accuracy. With our specified value of ${\alpha }_1=10$ the coefficient of variation ranged from 28.0% to 31.4%. That is, even though there was considerable variability in PAF, there was little impact on accuracy of sire GEBV.

An estimate of pool construction error reported by Macgregor (2007) on pools of 384 individuals ranged from 0.00015 (or SD = 0.0122) to 0.00018 (or SD = 0.0134), which represented about 12.5% of the total variability in PAF. In Earp et al. (2011), the mean estimate of variability due to pool construction error was 0.000029 (SD = 0.0054) based on pools of 74 to 446 individuals, accounting for about 7.5% of total PAF variability. In our study, estimates of pool construction error (based on 20 simulated replicates; Supplementary Figure S1) ranged from 0.00075 (SD = 0.027) to 0.0000045 (SD = 0.0067) for pools of 5–100 individuals, respectively. These estimates represented about 6–8% of the total PAF variation. In Abrams et al. (2021), using pools of 50 steers, construction error (measured as animal SD) ranged from 0.0020 to 0.0040 depending on the methodology used. With those estimates, solving for α₁ using the equation for the variance in the Dirichlet distribution (2) results in values ranging from 19 to 121. Because larger values for α₁ imply less variation, clearly our value of 10 was conservative. The similarity between our estimates of pool construction error (based on stochastic simulation) and those reported in the literature from experimental data suggest that our methodology for simulating pool construction error was appropriate. However, in Macgregor (2007) and Earp et al. (2011), the authors assumed that pool construction error was additive (i.e., linear) with respect to allele frequency. Such is not the case in our specification of pool construction error.

As with accuracy, pool construction error did not have a substantial effect on sire theoretical accuracy (r). Mean sire r was 0.622 (SD 0.062) and 0.624 (SD 0.063) for the presence or absence of construction error, respectively. Because random variability in animal representation in a pool is unbiased, the expected value of the diagonal elements of ${\mathbf{G}}_{pp}$ was still $1/n_k$. Thus, on average, the effect on sire r is expected to be negligible. To our knowledge, the effects of pool construction error on sire r have not been reported in the literature before.

Pool construction error did not have a significant effect on mean proportional bias (P = 0.745). Because PAF were on average centered at the underlying true allele frequency (p_j), it is reasonable that its effect on bias was minimal. Such would be the case even when animals are unequally represented in a pool.

If DNA pooling is intended to be used routinely for prediction of GEBV then it would be important to establish replicability and reproducibility protocols and measures, so that extraneous sources of error can be minimized. For instance, with the goal of standardizing laboratory practices, if the DNA of individually genotyped animals is available, pools can be constructed with different levels of representation. Construction error could then be assessed by regressing PAF on genotype (number of copies of A allele) as in Kuehn et al. (2018). An estimate of α₁ can then be obtained as in Minka (2012). This would provide some sense of the variability in the constructed PAF in a given lab. Mitigating procedures can then be implemented if those errors are larger than expected based on a long-term average.

Technical error

Technical error had little effect on accuracy of sire GEBV (P = 0.857). Mean accuracy was 0.533 (SE 0.001), 0.529 (SE 0.001), and 0.512 (SE 0.001) for ${\sigma }_T=$ 0.125, 0.625, and 1.125, respectively. While there was a slight decrease in accuracy as the level of technical error increased, its overall effect was small.

Values of σ_T > 0.625 considerably shifted PAF such that the diagonal values of G corresponding to pools differed from their expected value (Figure 3). However, the distribution of the off-diagonal values was not substantially influenced by the levels of technical error variance. Recall that elements of G were scaled using estimated allele frequencies (i.e., ${\widehat{\mathbf{\mu }}}^{\mathsf{T}}(1_P-\widehat{\mathbf{\mu }})$). The effect of technical error variance on the off-diagonal values of the G among sires was then only due to scaling. Moreover, changes to the off-diagonal values of G between sires and pools, while present, were less important.

Figure 3.

Diagonal values of G constructed from 240 individually genotyped sires and pools ($i=1,\dots ,240$ for each) in the presence of three levels of technical error (σ_T). Pool size set to 25 such that the expected value of the diagonals for pools is 0.04 (dashed black line).

Given that prediction of genetic merit of sires are made by using genomic relationships, which as given by (14) to (16) are (on average) a function of pedigree relationships, these do not strongly depend on allele frequencies. Moreover, based on central limit theorem, the effect of array-based error was mitigated by the relatively large number of markers used in this study. The negative impact of technical error should be more evident when few markers are used for predictions of genetic merit. Considerable changes in GEBV due to allele frequency scaling were found in Forni et al. (2011). However, they found that the correlation between GEBV from these different scaling methods was high ($\ge 0.986$). As such, lack of differences between levels of technical error in our study were perhaps unsurprising. Overall, it appears that despite changes in GEBV due to changes in allele frequency resulting from technical error, accuracy is minimally impacted and predictions remained reliable.

Given the relative shift in PAF due to ${\sigma }_T^2$, technical error might play a more substantial role when there is interest in genome wide association studies using PAF (Earp et al. 2011). If such was the case, allele frequency corrections could be made as in Peiris et al. (2011) if the particular frequencies of heterozygous animals are available, or as in Wen and Stephens (2010) in a more general manner. The latter has the appealing property of using LD information for correcting allele frequencies. The effect of PAF correction factors on accuracy of GEBV is beyond the scope of the current study.

While accuracy was only slightly impacted by technical error, increasing technical error variance, in general, resulted in a reduction in sire theoretical accuracy. Mean sire r was 0.645 (SD 0.073), 0.616 (SD 0.059), and 0.609 (SD 0.047), for ${\sigma }_T=$ 0.125, 0.625, and 1.125, respectively. While genomic relationships were maintained to some extent when increasing technical error, such that accuracy was not substantially changed, the resulting change in the structure of G (and the mixed model equations) would be more substantial. Therefore, sire r was impacted.

As with pool construction error, technical error did not impact mean proportional bias (P = 0.899). To our knowledge, the effect of array-variability on GEBV bias has not been reported in the literature. Because the deviations in PAF due to technical error variance are averaged across markers and pools (central limit theorem), they resulted in minimal bias. In addition, in averaging bias estimates across sires and replicates, the individual effects of technical error on GEBV estimates may be less evident. As pointed out earlier, off-diagonal elements of G between pools and sires were not substantially affected by technical error variance, such that predictions of GEBV remained accurate.

Estimates of technical (array-based) error reported by Macgregor (2007) were obtained by genotyping pools of 384 individuals three times. They concluded that most (about 87.5%) of variability in PAF was due to technical error in contrast to construction error. Earp et al. (2011) reported similar findings. In our study, the trend was similar. With pool sizes above 10 technical error was larger than construction error irrespective of the value of σ_T. However, with the smallest pools, pool construction error exceeded technical error (Supplementary Figure S1). As a distinction to Macgregor (2007) and Earp et al. (2011), in our study technical error had a nonlinear effect on allele frequency. This nonlinear effect was more evident when ${\sigma }_T=1.125$.

Although not equivalent, the extent of technical error could be assessed by measuring the reproducibility and repeatability of laboratory results. For instance, correlations between relative allele signal scores from different runs of the same pooled sample ranged from 0.974 to 0.986 (Kirov et al. 2006). While the correlations were high, the authors reported considerable spread and weak reproducibility for a large number of markers. Still, the resulting errors in allele frequency estimates from pools were less than 4.6% for 99% of the results. The simulated errors in our study are considerably larger than those reported by Kirov et al. (2006). However, the proportion of variability partitioned into construction and technical error in Kirov et al. (2006) is not clear as the authors only considered technical error.

Pool size (number of pools)

Estimated marginal means for pool size and pooling design are provided in Figure 4. Although a significant interaction between pool size (or number of pools) and design was found (P < 0.001), accuracy of pool size was consistent for the levels of design (i.e., accuracy was always higher for smaller pools). However, the change (decrease) in accuracy due to increasing pool size (or number of pools) was more substantial for random pooling. For extreme and K-means pooling accuracy for pools of 10 and 25 individuals was 0.711 to 0.715 and 0.695 to 0.701, respectively; this was greater than 0.95 the accuracy for pools of size 5. On the other hand, for random pooling, accuracy for pools of 10 and 25 individuals was 0.703 and 0.351 times the accuracy of pools of 5 individuals, respectively.

Figure 4.

Estimated marginal means for accuracy of GEBV of sires with different pooling strategies (design) and pool size (n_k).

For non-random pooling, accuracy was only slightly higher for smaller pools, perhaps reflecting more informative relationships to sires as compared to those of larger pools; larger pools would be regressed toward the average population relationship, resulting in lower accuracy (Figure 4). A similar trend was observed in Alexandre et al. (2019), where only a slight decrease in accuracy was observed as pool size increased from 5 to 100 individuals, with a plateau at approximately 0.75. The values obtained in our study are slightly smaller, perhaps due to lower heritability ($h^2=0.3$ as compared to $h^2=0.4$).

Another difference from Alexandre et al. (2019) was that in our study, a random subset of 30 sires was picked from the 240 sires in the last generation, and not all progeny were used to construct pools. Instead, a subset representing approximately 30% was chosen either at random or from the tails of the distribution. This was done to mimic commercial practices in small ruminants where pedigree information may be unknown, and financial resources may be too limited to genotype all potential sires. As such, in some cases progeny in pools were unrelated to the set of sires chosen, thereby reducing the number of progeny per sire included in the evaluation. On average, out of the 30 sires (randomly) chosen, irrespective of pooling design, about 0.7 and 11.8 were represented in pools of 5 and 100 individuals, respectively. In the same manner, the number of progeny per sire in a given pool ranged from 0.8 for pools of 5 individuals to 15.0 for pools of 100 individuals (Figure 5).

Figure 5.

Estimated marginal means for sire representation (number of sires per pool and number of progeny by sire).

As with technical error, r of sire GEBV substantially decreased when pool size increased (or number of pools decreased for K-means). Mean sire r was 0.715 (SD 0.040), 0.654 (SD 0.039), 0.600 (SD 0.029), 0.579 (SD 0.020), and 0.568 (SD 0.015) for pools of 5, 10, 25, 50, and 100 individuals (or 600, 300, 120, 60, and 30 pools), respectively.

In Alexandre et al. (2019), pools of 10 individuals were found to retain about 80% of sire accuracy, while still being economical. In this study, for pools of 50 or 100 individuals, accuracy of sire GEBV was 0.81 and 0.79 times, respectively, that of pools of 5 individuals. Thus, even for larger pools, accuracy (theoretical and based on correlation) of sire GEBV remained high enough to select sires with a moderate degree of certainty despite pedigree relationships with progeny forming the pools being unknown. As such, constructing larger pools may represent a compromise between cost and uncertainty.

Variability (SD) in sire r was 0.020, 0.017, 0.013, 0.011, and 0.010 for pools of size 5, 10, 25, 50, and 100, respectively. Although perhaps counter-intuitive at first, lower variability in sire r is expected because there is a higher probability that a sire would be represented in pools of larger size. The same was true in (Alexandre et al. 2019). As such, while higher accuracy may be obtained in pools of smaller size, the resulting sire r will likely be more variable.

Pool size, overall, did not affect proportional bias (P = 0.555). Still, bias was more evident when pools were constructed with 5 or 10 individuals as compared to larger pools when constructed non-randomly (Figure 6). However, bias increased with pool size for random pooling. Averaging across extreme and K-means designs, sire GEBV were overestimated by about 6.6% for pools of five individuals which may be due to sires not being well represented. On the other hand, sire GEBV were underestimated by about 2.4% for pools of 100 individuals. Bias in GEBV has been documented when sires have no progeny genotyped (Tsuruta et al. 2019) and when only a small proportion of individuals were genotyped (Gowane et al. 2019). In our study, some sires did not have progeny in pools (i.e., not genotyped or included in the evaluation) particularly when small pools were constructed. As such, the existence of bias is expected. Bias in genomic prediction using single step for type traits in dairy cattle ranged from 0.75 (underestimation) for feet and leg conformation to 1.20 (overestimation) for rear teat placement (Tsuruta et al. 2021). Although to our knowledge bias in GEBV predictions using PAF has not been reported elsewhere, it seems consistent with that found with individual genotyping.

Figure 6.

Mean proportional sire GEBV bias, defined as (TBV–GEBV)/TBV, for different pooling designs and pool sizes.

In Figure 7, the mean TBV of the sires ranked as in the top 10%, 20%, and 40% on GEBV is shown relative to pooling design and pool size. The corresponding bottom 40%, 20%, and 10% can be found in the Supplementary Figures S2 and S3 (Supplementary File S4). An interaction was detected (P = 0.012). Irrespective of pool size, when GEBV were predicted using non-random pooling, there was clear delineation among ranking categories. However, with random pooling, such delineation was only apparent for pools of less than 25 individuals. This distinction was likely due to the more accurate prediction of GEBV with non-random pooling (Baller et al. 2020). Therefore, when resources are limited, larger pools provide a useful option as long as they are constructed in a designed fashion.

Figure 7.

Mean true breeding value for sires with the top 40%, 20%, and 10% GEBV for different pooling strategies (design) and pool sizes.

Pooling design

Extreme and random pooling

Although a significant interaction between design and selection scenario was present (P = 0.011), accuracy for the selection scenario was consistent across levels of design. Averaging across levels of selection scenario, mean accuracy was 0.704 (SE 0.001) and 0.174 (SE 0.001) for extremes, and random pooling, respectively. Sham et al. (2002) reported that for quantitative (continuous) traits and in the absence of experimental error the optimal selection criterion for pooling a random sample of unrelated individuals is to select individuals from the tails of the distribution. In our analyses, with a population under phenotypic selection, pools at the right tail (higher phenotypic values) of the distribution resulted in animals being marginally more related to sires than in those pools at the left tail of the distribution (lower phenotypic values; Figure 8). No such pattern was observed with random pooling.

Figure 8.

Average (across 50 replicates) genetic (pedigree) relationships between sires and animals allocated in pools for different pool sizes (n_k) and random (red) and extreme (blue) pooling designs.

There was considerable variability in the accuracy of sire GEBV obtained from pools, especially when constructed at random. Accuracy SD was 0.07 and 0.22 for extreme and random pooling, respectively. This likely reflected that the sets of sires were chosen at random, and in some instances these sires did not have many progeny in the constructed pools. This leads to low accuracy GEBV.

Theoretical accuracy of sire GEBV was 0.624 (SD 0.063) and 0.623 (SD 0.063) for extreme and random pooling, respectively. Variability in sire r was 0.014 in both cases. While sire GEBV accuracy was defined as the overall correlation between simulated and estimated breeding values, sire r instead reflected genomic relationships between sires and their progeny in pools, as well as their phenotypes. Due to sires being randomly chosen, the extent of those genomic relationships would be less affected by pooling design. The sire r of extreme and random pooling being virtually the same is then perhaps reasonable.

Pooling design affected mean proportional bias (P = 0.012). Averaging across levels of pool size, using extreme pooling resulted in overestimation of GEBV by 3.4%, while using random pooling resulted in underestimation of GEBV by 8.2%. As noted earlier, overestimation of GEBV may be due to only a subset of individuals being genotyped and evaluated (Tsuruta et al. 2019), irrespective of pooling design. Underestimation of GEBV has been reported when dams are absent in the reference population (Ma et al. 2015).

Mean TBV for sires with the top 40%, 20%, and 10% GEBV for the selection scenario were provided in Figure 7. Across pool sizes, the mean TBV for the sires with the top 10% GEBV was 8.46 for extreme pooling and 7.74 for random pooling. For comparison, the simulated mean TBV for the top 10% (under phenotypic selection), was 8.36. As mentioned earlier, pool size influenced GEBV more substantially with random pooling.

K-means

Accuracy of sire GEBV for K-means was 0.694 (SE 0.001). Because K-means pools also were constructed based on extreme phenotypes it is unsurprising that their accuracies were similar. In addition, accuracy SD was 0.08, as with extreme pooling.

Theoretical accuracy of sire GEBV was similar for K-means as with random and extreme pooling at 0.623 (SD 0.062). Variability in GEBV sire r was, however, slightly higher (0.015), perhaps reflecting the variability in pool size.

For K-means pooling, GEBV were overestimated by about 1.5%. Mean proportional bias was similar to extreme pooling, although it varied with the levels of pool size. The mean TBV of sires based on the top 10% ranking on GEBV was 8.44, also quite similar to that for extreme pooling.

While the average number of individuals in a given pool resembled that of extreme or random pooling, pool size can considerably vary when using K-means (Table 1). For instance, when K = 600, pool sizes of 1 individual were generated. In practice, however, pools of smaller size and similar phenotypic values (means) can be combined into larger pools so that the economic advantages of pooling are maintained.

For continuous traits, it is easier (and practical) to construct pools using extreme pooling because pool size can easily be fixed. In some cases, however, there is interest in obtaining GEBV for discrete (binary and categorical) traits (McDaneld et al. 2014; Keele et al. 2015, 2016). In such instances, some individuals will have the same “extreme” value of a phenotype (e.g., all 1’s or all 0’s). Although outside of the scope of this study, the K-means algorithm could be used on a correlated trait, or a continuous covariate associated to the trait, to distinctly select individuals and allocate them into pools (Guan and Li 2014; Vargas Jurado et al. 2018). In addition, K-means clustering could be particularly useful when multiple traits and continuous covariates are available on an individual.

Scaling

Scaling the hybrid G as a correlation matrix had little impact on sire GEBV accuracy (P = 0.760). Averaging across levels of the other effects, mean accuracy was 0.526 (SE 0.001) and 0.523 (SE 0.001) for unscaled and scaled hybrid G matrix, respectively. While sire GEBV and pool GEBV may differ in scale, the relative rank of those GEBV was maintained and thus the accuracies obtained were similar. Still, scaling may be an issue when there is interest in combining different sources of information (sires, dams, progeny, pools), or if the GEBV from pooling are to be compared with those obtained by other means (e.g., PBLUP).

The approach used in our study, and in Bell et al. (2017), differs from the one described in Reverter et al. (2016) where PAF were converted (or discretized) into 0, 1 and 2 based on PAF values and then a hybrid G matrix was constructed as in VanRaden (2008). While that approach attempts to address scaling issues, it was population specific, as the SNP “categories” for PAF were obtained by examining clusters based on the PAF and individual DNA samples.

Selection scenario

Despite the interaction between selection scenario and pooling design, selection scenario by itself had little impact on accuracy of sire GEBV (P = 0.242). Averaging over the other effects, mean accuracy was slightly lower for the random mating scenario at 0.516 (SE 0.001) as compared to the selection for high phenotypes scenario at 0.533 (SE 0.001). The small difference in performance was likely due to including accuracy from random pooling in calculating these means, which lowered the overall effect of selection. Differences were greater when looking at the conditional means of selection scenario by pooling design (Table 2).

Table 2.

 Estimated marginal means for accuracy of GEBV of sires with different pooling design and selection strategies

Selection scenario	n	Pooling design	Estimate	SE
High phenotypes	100	Extremes	0.721	0.001
	100	K-means	0.712	0.001
	100	Random	0.181	0.001
Random mating	100	Extremes	0.687	0.001
	100	K-means	0.680	0.001
	100	Random	0.167	0.001

The increase in accuracy when selecting for high phenotypes likely reflected the sires being more closely related to individuals allocated to pools (Figure 4). However, accuracy was only about 1.049, 1.047, and 1.084 times higher for extreme, K-means and random pooling, respectively, when selecting for high phenotypes compared to random mating. Thus, even in industries were high selection pressure is not necessarily the norm (e.g., small ruminants), DNA pooling provides a sensible measure of accuracy for selecting sires.

Similar to accuracy based on correlation, sire r was only marginally higher in the selection scenario at 0.625 (SD 0.064) compared to the random mating scenario at 0.621 (0.061). This likely reflected that in either case the number of progeny per sire was similar (about 15 progeny per sire). Although sires may be better connected when selected for high (or low) phenotypes, this did not translate into a substantial increase in their representation in pools. Thus, a considerable increase in accuracy (theoretical or based on correlation) should not be expected.

Other considerations

To our knowledge, the effect of pool construction and technical error on measures of accuracy and bias of GEBV obtained from GBLUP from pooled DNA samples has not been reported in the literature. For a trait with moderate heritability ($h^2=0.3$) the accuracy of GEBV obtained by using GBLUP on individually genotyped sires and progeny was 0.879 (0.071), while it was 0.781 (0.046) for PBLUP. For comparison, with pools of 5 individuals, accuracy of sire GEBV was 0.730 (0.01) for extreme and K-means pooling. This represents a reduction of accuracy of 0.149 (or 17%). Assuming a genotyping cost of $25 and a DNA extraction cost of $3 per sample (Abrams et al. 2021), the cost of genotyping 3000 individuals would be $84,000 compared to $24,000 for genotyping 600 pools of 5 individuals. In this case, there is a reduction of more than 66.7% of the cost of genotyping and only a reduction of 17% in accuracy when pooling. Further reduction in genotyping costs are achieved with larger pools. For instance, the cost of genotyping 120 pools of 25 individuals represents 14% of the cost of individual genotyping with only a 0.03 (or 4%) additional loss in accuracy. Moreover, because TBV for sires with the top (or bottom) 10, 20, and 40% GEBV were clearly delineated (when pooling extreme phenotypes), and with biases similar to those obtained with GBLUP (based on individual genotypes), selection decisions can be confidently made based on GEBV obtained by pooling.

In our study, the accuracy of GEBV was approximately 0.10 greater than $\sqrt{h^2}$ and comparable to phenotypic selection on breeding candidates. In situations where it is not feasible to phenotype potential breeding individuals, pooling offers an advantage over other methods. Because the accuracy of PBLUP was only marginally higher than that obtained by using PAF, it highlights the benefit of using pooling for prediction of genetic merit in cases where limited resources are available.

As shown in the current study, the scale of GEBV from sires and pools differs. While reasonable sire GEBV accuracies were obtained by using DNA pooling and GBLUP, integrating pooling into conventional genomic evaluations has been only recently addressed (Baller et al. 2020). In a single-step setting, combining pools and individual genotypes required modification of standard strategies, but provided accurate estimates of GEBV for sires and individuals related to pools.

Recent developments (Baller et al. 2020) have made it easier to incorporate DNA pooling for large scale genomic prediction. While such infrastructure is already available for the dairy and beef cattle industries, among others, such is not the case for small ruminants. With few exceptions (Auvray et al. 2014; Swan et al. 2014), routine genotyping of individual animals or using pooled genotypes for the estimation of breeding values is not the norm in the sheep and goat industries. Based on the results from our current and previous studies, DNA pooling is an economical alternative to individual genotyping that would allow producers and breeders to obtain reasonably accurate and reliable breeding values for sires.

Conclusions

Overall, accuracy of sire GEBV based on pooled DNA from their progeny was moderate when extreme or K-means pooling was used. On the other hand, random pooling performed poorly. With extreme or K-means pooling, variability in accuracy (theoretical or based on correlation) was still observed. Perhaps due to that variability in accuracy, there were no substantial effects of scaling the G matrix, pool size (or number of pools for K-means), construction error or technical error variance. In this study, K-means resulted in similar GEBV accuracy to that of extreme pooling. In those scenarios where the costs and practicalities of individual genotyping make it prohibitive, such as in small ruminants, DNA pooling appears to offer a viable option for incorporating genomic information into genetic evaluation.

Data availability

The R code used to simulate the population used during the current study, and to produce Figures 1 and 2 is available as Supplementary Files S1 to S3. In addition, other R code used in the present study is available at https://github.com/napovargas/G3Sim (last accessed 21 July 2021).

Supplementary material is available at G3 online.

Appendix

Appendix A

Under the assumption that technical error is not present (i.e., U_jk equals ${\mu }_{jk})$, the expected value of the PAF for a given locus is given by

$$\begin{array}{l}{\theta }_{jk}=\frac{1}{2}\sum _{i=1}^{n_k}{\pi }_i{X}_{ij},\\ E\left[{\theta }_{jk}\right]=E\left[\frac{1}{2}\sum _{i=1}^{n_k}{\pi }_i{X}_{ij}\right],\\ =\frac{1}{2}\sum _{i=1}^{n_k}E\left[{\pi }_i{X}_{ij}\right],\\ =\frac{n_k}{2}E\left[{\pi }_i\right]E\left[X_{ij}\right],\\ =\frac{n_k}{2}\frac{1}{n_k}2p_j\\ =p_j.\end{array}$$

(9)

Step (9) is due to the assumption that variations in animal contributions to a pool happen randomly and independently of genotype values or allele frequencies. If technical error is present then the expected value of the PAF can be approximated by either numerical integration (for specific values of p and σ_T), or by a Taylor series. Recall that the definition of technical error on the logit scale is

$$\begin{array}{l}{U}_j\sim \mathsf{Normal}\left(\text{log}\hspace{0.17em}\left(\frac{p_j}{1-p_j}\right),{\sigma }_T^2\right),\\ W_j=\frac{e^{U_j}}{1+e^{U_j}}.\end{array}$$

The expected value of W_j can be approximated by using a second-order Taylor series of the inverse logit function $g\left(U_j\right)$ about the mean $E\left[U_j\right]=\mu$, given by

$$\begin{array}{l}E\left[W_j\right]\approx E\left[g(\mu )+g^{\prime }(\mu )(U_j-\mu )+\frac{1}{2}{g}^{″}(\mu ){(U_j-\mu )}^2\right],\\ \approx \frac{e^{\mu }}{1+e^{\mu }}+\frac{e^{\mu }}{{(1+e^{\mu })}^2}(U_j-\mu )-\frac{1}{2}\frac{e^{\mu }(1-e^{\mu })}{{(1+e^{\mu })}^3}{(U_j-\mu )}^2\end{array}$$

Given that $E[U_j-\mu ]=0$ and $E\left[{(U_j-\mu )}^2\right]={\sigma }_T^2$ by definition, the second term above drops and we are left with

$$\begin{array}{l}E\left[W\right]\approx \frac{e^{\mu }}{1+e^{\mu }}-\frac{1}{2}\frac{e^{\mu }(1-e^{\mu })}{{(1+e^{\mu })}^3}{\sigma }_T^2,\\ \approx \frac{\frac{p_j}{1-p_j}}{\frac{1-p_j+p_j}{1-p_j}}-\frac{1}{2}\frac{\frac{p_j}{1-p_j}\left(\frac{1-p_j-p_j}{1-p_j}\right)}{{\left(\frac{1-p_j+p_j}{1-p_j}\right)}^3}{\sigma }_T^2,\\ \approx p_j-\frac{1}{2}{p}_j(1-2p_j)(1-p_j){\sigma }_T^2\\ \approx p_j\left[1-\left(\frac{1-3p_j+2p_j^2}{2}\right){\sigma }_T^2\right]\end{array}$$

which is given by substituting $\mu =\text{log}\hspace{0.17em}\left(\frac{p_j}{1-p_j}\right)$ and simplifying. Note that μ is undefined for values p_j = 0 or p_j = 1. The same strategy is applied to approximate the variance. However, only a first-order approximation follows, as 3rd and 4th moments are needed for higher-order approximations

$$\begin{array}{l}Var\left[W_j\right]\approx \mathit{Var}\left[g\right(U_j\left)\right],\\ \approx \mathit{Var}\left[g(\mu )+g^{\prime }(\mu )(U_j-\mu )\right],\\ \approx {\left[g\prime (\mu )\right]}^{2}Var\left[U_j\right],\\ \approx {\left[\frac{e^{\mu }}{{(1+e^{\mu })}^2}\right]}^2{\sigma }_T^2,\\ \approx \left[\frac{2p_j}{{(1-p_j)}^3}\right]{\sigma }_T^2.\end{array}$$

Appendix B

The calculations below assume no technical error (for simplicity). Let X_ik and θ_jk be the genotype (or PAF value) of the k^th marker and i^th (or j^th) individual (or pool). Then, the covariance among (individual) sires is

$$\begin{array}{l}Cov\left[\frac{1}{2}{X}_{ik},\frac{1}{2}{X}_{jk}\right]=\frac{1}{4}Cov[X_{ik},X_{jk}],\\ =\frac{1}{4}{a}_{ij}2p_k(1-p_k),\\ =\frac{1}{2}{a}_{ij}{p}_k(1-p_k).\end{array}$$

The covariance among sires and pools is given by

$$\begin{array}{l}Cov\left[\frac{1}{2}{X}_{ik},{\theta }_{jk}\right]=\mathit{Cov}\left[\frac{1}{2}{X}_{ik},\frac{1}{2}\sum _{l=1}^{n_j}{\pi }_l{X}_{lk}\right],\hspace{0.17em}\text{definition}\hspace{1em}\text{of}\hspace{1em}{\theta }_j\\ =\frac{1}{4}\sum _{l=1}^{n_j}Cov\left[X_{ik},\pi X_{lk}\right],\hspace{0.17em}\text{because}\hspace{1em}{\pi }_l\stackrel{\mathit{iid}}{\sim }\mathsf{Dir}\left(\mathbf{\alpha }\right)\\ \simeq \frac{1}{4}\frac{1}{n_j}\sum _{l=1}^{n_j}Cov\left[X_{ik},X_{lk}\right],\end{array}$$

(10)

$$\begin{array}{l}\simeq \frac{1}{2n_j}\sum _{l=1}^{n_j}{a}_{il}{p}_k(1-p_k),\\ \simeq \frac{1}{2}{\overline{a}}_{i·}{p}_k(1-p_k)\end{array}$$

(11)

where n_j is the number of individuals in pool j. Finally, the covariance among pools is

$$\begin{array}{l}Cov\left[{\theta }_{ik},{\theta }_{jk}\right]=\mathit{Cov}\left[\frac{1}{2}\sum _{l=1}^{n_i}{\pi }_l{X}_{lk},\frac{1}{2}\sum _{l^{\prime }=1}^{n_j}{\pi }_{l^{\prime }}{X}_{l^{\prime }k}\right],\hspace{0.17em}\text{definition}\hspace{1em}\text{of}\hspace{1em}{\theta }_j\\ =\frac{1}{4}\sum _{l=1}^{n_i}\sum _{l^{\prime }=1}^{n_j}Cov\left[\pi X_{lk},\pi X_{l^{\prime }k}\right],\hspace{0.17em}\text{because}\hspace{1em}{\pi }_l,{\pi }_{l^{\prime }}\stackrel{\mathit{iid}}{\sim }\mathsf{Dir}\left(\mathbf{\alpha }\right)\\ \simeq \frac{1}{4}\frac{1}{n_i}\frac{1}{n_j}\sum _{l=1}^{n_i}\sum _{l^{\prime }=1}^{n_j}Cov\left[X_{lk},X_{l^{\prime }k}\right],\end{array}$$

(12)

$$\begin{array}{l}\simeq \frac{1}{2n_i{n}_j}\sum _{l=1}^{n_i}\sum _{l^{`}=1}^{n_j}{a}_{l{l}^{`}}{p}_k(1-p_k),\\ \simeq \frac{1}{2}{\overline{a}}_{..}{p}_k(1-p_k)\end{array}$$

(13)

where as before, n_i and n_j are the number of individuals in pool i and j, respectively. In addition, equations (10) and (12) use the following results (Bohrnstedt and Goldberger 1969)

$$\begin{array}{l}Cov[X,UY]=E\left[U\right]\mathit{Cov}[X,Y],\\ \\ Cov[VX,UY]=E\left[V\right]E\left[U\right]\mathit{Cov}[X,Y],\end{array}$$

and the assumption that π_l (or ${\pi }_{l^{\prime }})$ and X_ik (or $X_{l^{\prime }k}$) are independent. The justification for this assumption is that pool construction error occurs randomly for every individual in a given pool and independently of their genotype values. This assumption was also made in Macgregor (2007) when approximating pool construction error.

In the case of $\mathit{Cov}[{\theta }_{ik},{\theta }_{ik}]$ (i.e., $l=l^{\prime }$), we get

$$\begin{array}{l}Cov[{\theta }_{ik},{\theta }_{ik}]\simeq \frac{1}{2n_i^2}\underset{n_i}{\underset{\underset{n_i}{⏟}}{\sum _{l=1}^{n_i}\sum _{l=1}^{n_i}{a}_{ll}}}{p}_k(1-p_k),\\ \simeq \frac{1}{2n_i^2}{n}_i{p}_k(1-p_k),\\ \simeq \frac{1}{2n_i}{p}_k(1-p_k).\end{array}$$

Appendix C

With individual genotypes, Habier et al. (2007) showed that

$$\begin{array}{l}E\left[{\mathbf{x}}_i^{\mathsf{T}}{\mathbf{x}}_j\right]=\sum _{k=1}^{P}E\left[x_{ik}{x}_{jk}\right],\\ =\sum _{k=1}^{P}Cov[x_{ik},x_{jk}]+E\left[x_{ik}\right]E\left[x_{jk}\right],\\ =2a_{ij}\sum _{k=1}^P{p}_k(1-p_k)+4\sum _{k=1}^P{p}_k^2\end{array}$$

where ${\mathbf{x}}_i$ was the i^th row of the matrix of genotypes X so that $E\left[\mathbf{X}{\mathbf{X}}^{\mathsf{T}}\right]=\mathbf{A}\left[2\sum _{k=1}^P{p}_k\right(1-p_k\left)\right]+\mathbf{J}{\mathbf{J}}^{\mathsf{T}}4\sum _{k=1}^P{p}_k^2$ with ${\mathbf{J}}^{\mathsf{T}}=[1,\dots ,1]$, and A was the numerator relationship matrix with individual elements given by a_ij. In the case of individually genotyped sires and pools (even when construction error is present) the following relationships are possible, (i) among sires, (ii) among pools, and (iii) between sires and pools. It can be shown that

$$\sum _{k=1}^{P}E\left[\frac{1}{2}{x}_{ik}\frac{1}{2}{x}_{jk}\right]=\frac{1}{2}{a}_{ij}\sum _{k=1}^P{p}_k(1-p_k)+\sum _{k=1}^P{p}_k^2,$$ (14)

$$\sum _{k=1}^{P}E\left[\frac{1}{2}{x}_{ik}{\theta }_{jk}\right]\simeq \frac{1}{2}{\overline{a}}_{i·}\sum _{k=1}^P{p}_k(1-p_k)+\sum _{k=1}^P{p}_k^2,$$ (15)

$$\sum _{k=1}^{P}E\left[{\theta }_{ik}{\theta }_{jk}\right]\simeq \frac{1}{2}{\overline{a}}_{..}\sum _{k=1}^P{p}_k(1-p_k)+\sum _{k=1}^P{p}_k^2,$$ (16)

where ${\overline{a}}_{i·}$ denotes the average relationship between individual i and the individuals in pool j, and ${\overline{a}}_{··}$ is the average additive relationship between individuals in pool i and individuals in pool j. The second and first terms of the right hand side in (15) and (16) are derived in Appendices A and B, respectively.