Animal Science Journal
Volume 95, Issue 1 e13928
SHORT COMMUNICATION
Open Access

Prediction of response to truncated selection based on BLUP of breeding values and its prediction accuracy

Masahiro Satoh, 

Corresponding Author

Masahiro Satoh

Graduate School of Agricultural Science, Tohoku University, Sendai, Japan

Correspondence

Masahiro Satoh, Graduate School of Agricultural Science, Tohoku University, Sendai, Miyagi 980-8572, Japan.

Email: [email protected]

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First published: 24 February 2024
https://doi.org/10.1111/asj.13928
Citations: 1

Abstract

Three methods of predicting the response to truncated selection based on BLUP of breeding values (BVs) were compared under conditions in which the phenotypic values for the progenies of selected animals were not available. The following methods were used to predict the response to selection: (1) based on the mean of estimated breeding values (EBV) in the candidate population for selection ( ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ ), (2) based on the variance of EBV in the candidate population for selection ( ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ ), and (3) based on diagonal elements of the inverse matrix on the left-hand side of the mixed model equation ( ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ ). The deviation of the average BV of the selected animals from the average BV of the candidate population for selection was taken as the true response to selection. The pedigree information and phenotypic values used for comparison were generated by Monte Carlo computer simulation. The results showed that ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ had the smallest absolute mean error and ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ had the smallest root-mean-square error. We concluded that it is desirable to use ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ or ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ to predict the response to truncated selection based on BLUP of BVs. However, in the population where selection is ongoing, the prediction accuracy of selection response is likely to be affected by the distortion of the distribution and the Bulmer effect for ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ .

1 INTRODUCTION

The change produced by selection that chiefly interests us is the change of the population mean. This is called the response to selection and is the difference in mean phenotypic value between the offspring of the selected parents and the whole of the parental generation before selection (Falconer, 1981). However, if the environment differs between generations, it is necessary to correct for this effect. Therefore, in such cases, the breeding value (BV) estimated by the BLUP method is generally used to predict the response to selection.

Prediction of response to selection is important in choosing breeding programs and for estimating the expected economic benefits (Dekkers, 1992; Verrier et al., 1991). The classical formula for predicting response to truncated phenotypic selection (Falconer, 1981) is
Δg = i ¯ · σ g · r gp $$ \Delta \mathrm{g}=\overline{i}\cdotp {\upsigma}_{\mathrm{g}}\cdotp {\mathrm{r}}_{\mathrm{g}\mathrm{p}} $$
where i ¯ $$ \overline{i} $$ is the intensity of selection, σ g $$ {\upsigma}_{\mathrm{g}} $$ is the additive genetic standard deviation in the candidate population for selection, and r gp $$ {\mathrm{r}}_{\mathrm{gp}} $$ is the correlation coefficient between BVs and phenotypic values in the candidate population for selection (accuracy of selection). If the selection is based on BLUP ( g ̂ $$ \hat{\mathrm{g}} $$ ) of the breeding values instead of the phenotypic value, then r gp $$ {\mathrm{r}}_{\mathrm{gp}} $$ is r g g ̂ $$ {\mathrm{r}}_{\mathrm{g}\hat{\mathrm{g}}} $$ .

Theoretical methods have been proposed to approximate the responses to selection based on selection index and BLUP methods (e.g., Dekkers, 1992; Wray & Hill, 1989). These methods predict the response to selection by giving population size, selection rate, genetic parameters, and so on. On the other hand, when actual pedigree information and phenotypic records are available, the predicted response to selection based on the BLUP method is the deviation of the mean of estimated breeding values (EBVs) in the selected animals from the mean of EBVs in the candidate population for selection. There are also several possible methods of predicting response to selection based on the above-quoted Falconer's equation.

In this study, the prediction accuracies of three different methods for predicting responses to truncated selection based on the BLUP method were compared using data generated by Monte Carlo computer simulation. The estimation accuracies of fixed effects were also examined.

2 METHODS

Consider the following equations for a mixed model:
y = Xb + Zg + e , E g e = 0 , var g e = G 0 0 R = σ g 2 A 0 0 σ e 2 I , $$ \mathbf{y}=\mathbf{Xb}+\mathbf{Zg}+\mathbf{e},\kern0.5em E\left[\begin{array}{c}\mathbf{g}\\ {}\mathbf{e}\end{array}\right]=\mathbf{0},\mathit{\operatorname{var}}\left[\begin{array}{c}\mathbf{g}\\ {}\mathbf{e}\end{array}\right]=\left[\begin{array}{cc}\mathbf{G}& \mathbf{0}\\ {}\mathbf{0}& \mathbf{R}\end{array}\right]=\left[\begin{array}{cc}{\upsigma}_{\mathrm{g}}^2\mathbf{A}& \mathbf{0}\\ {}\mathbf{0}& {\upsigma}_{\mathrm{e}}^2\mathbf{I}\end{array}\right], $$
where y, b, g, and e are vectors of phenotypic records, fixed effects, breeding values, and random errors; X and Z are the incidence matrices that relate y to b and y to g; σ e 2 $$ {\upsigma}_{\mathrm{e}}^2 $$ is the random environmental variance; and A and I are the additive relationship and incidence matrices, respectively. Let b ̂ $$ \hat{\mathbf{b}} $$ and g ̂ $$ \hat{\mathbf{g}} $$ be vectors of estimated values of b and predicted values of g. These are obtained by solving the following mixed model equation (MME) (Henderson, 1973).
X ′ R − 1 X X ′ R − 1 Z Z ′ R − 1 X Z ′ R − 1 Z + G − 1 b ̂ g ̂ = X ′ R − 1 y Z ′ R − 1 y $$ \left[\begin{array}{cc}{\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{X}& {\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{Z}\\ {}{\mathbf{Z}}^{\prime }{\mathbf{R}}^{-1}\mathbf{X}& {\mathbf{Z}}^{\prime }{\mathbf{R}}^{-1}\mathbf{Z}+{\mathbf{G}}^{-1}\end{array}\right]\left[\begin{array}{c}\hat{\mathbf{b}}\\ {}\hat{\mathbf{g}}\end{array}\right]=\left[\begin{array}{c}{\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{y}\\ {}{\mathbf{Z}}^{\prime }{\mathbf{R}}^{-1}\mathbf{y}\end{array}\right] $$ (1)
The inverse matrix on the left-hand side of Equation (1) is
X ′ R − 1 X X ′ R − 1 Z Z ′ R − 1 X Z ′ R − 1 Z + G − 1 − 1 = C 11 C 12 C 21 C 22 $$ {\left[\begin{array}{cc}{\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{X}& {\mathbf{X}}^{\prime }{\mathbf{R}}^{-1}\mathbf{Z}\\ {}{\mathbf{Z}}^{\prime }{\mathbf{R}}^{-1}\mathbf{X}& {\mathbf{Z}}^{\prime }{\mathbf{R}}^{-1}\mathbf{Z}+{\mathbf{G}}^{-1}\end{array}\right]}^{-1}=\left[\begin{array}{cc}{\mathbf{C}}^{11}& {\mathbf{C}}^{12}\\ {}{\mathbf{C}}^{21}& {\mathbf{C}}^{22}\end{array}\right] $$ (2)
From Henderson (1975),
var b ̂ − b = C 11 $$ \mathit{\operatorname{var}}\left(\hat{\mathbf{b}}-\mathbf{b}\right)={\mathbf{C}}^{11} $$ (3)
and
var g ̂ = cov g ̂ g ′ = G − C 22 . $$ \mathit{\operatorname{var}}\left(\hat{\mathbf{g}}\right)=\mathit{\operatorname{cov}}\left(\hat{\mathbf{g}},{\mathbf{g}}^{\prime}\right)=\mathbf{G}-{\mathbf{C}}^{22}. $$ (4)

2.1 Root-mean-square error (RMSE) of fixed effect

2.1.1 RMSE calculated from estimates of fixed effect

Let t, b j * $$ {\mathrm{b}}_{\mathrm{j}}^{\ast } $$ , and b ̂ j * $$ {\hat{\mathrm{b}}}_{\mathrm{j}}^{\ast } $$ be the number of levels of the fixed effect, the true value at the jth level, and its estimated value in the solution of Equation (1). The root-mean-square error ( RMSE Fxd 1 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1} $$ ) at each level of the fixed effect is
RMSE Fxd 1 = ∑ k = 1 t b ̂ k * − b k * 2 t . $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1}=\sqrt{\frac{\sum_{\mathrm{k}=1}^{\mathrm{t}}{\left({\hat{\mathrm{b}}}_{\mathrm{k}}^{\ast }-{\mathrm{b}}_{\mathrm{k}}^{\ast}\right)}^2}{\mathrm{t}}}. $$

2.1.2 RMSE calculated from the inverse matrix of the left-hand side of MME

Let c j 11 $$ {\mathrm{c}}_{\mathrm{j}}^{11} $$ be the jth diagonal element of C 11 $$ {\mathbf{C}}^{11} $$ in Equations (2) and (3). The root-mean-square-error ( RMSE Fxd 2 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2} $$ ) at each level of the fixed effect is
RMSE Fxd 2 = ∑ k = 1 t c k 11 t . $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2}=\sqrt{\frac{\sum_{\mathrm{k}=1}^{\mathrm{t}}{\mathrm{c}}_{\mathrm{k}}^{11}}{\mathrm{t}}}. $$

2.2 Prediction of response to selection

Let us assume that the candidate population for selection consists of an equal number of males and females (N each) and the total average BV in the population is BV ¯ 0 $$ {\overline{\mathrm{BV}}}_0 $$ . Let BV ¯ S m $$ {\overline{\mathrm{BV}}}_{{\mathrm{S}}_{\mathrm{m}}} $$ and BV ¯ S f $$ {\overline{\mathrm{BV}}}_{{\mathrm{S}}_{\mathrm{f}}} $$ be the average BV of n m $$ {\mathrm{n}}_{\mathrm{m}} $$ males and n f $$ {\mathrm{n}}_{\mathrm{f}} $$ females selected from the population. The response to selection ( ∆ g 0 $$ \Delta {\mathrm{g}}_0 $$ ) is
∆ g 0 = BV ¯ S m + BV ¯ S f 2 − BV ¯ 0 . $$ \Delta {\mathrm{g}}_0=\frac{{\overline{\mathrm{BV}}}_{{\mathrm{S}}_{\mathrm{m}}}+{\overline{\mathrm{BV}}}_{{\mathrm{S}}_{\mathrm{f}}}}{2}-{\overline{\mathrm{BV}}}_0. $$

This was assumed to be the true response to selection.

2.2.1 Prediction of response to selection from the mean of EBV

Let EBV ¯ 0 $$ {\overline{\mathrm{EBV}}}_0 $$ be the average EBV in a candidate population for selection and EBV ¯ S m $$ {\overline{\mathrm{EBV}}}_{{\mathrm{S}}_{\mathrm{m}}} $$ and EBV ¯ S f $$ {\overline{\mathrm{EBV}}}_{{\mathrm{S}}_{\mathrm{f}}} $$ , respectively, be the average EBV of selected males and females. The response to selection ( ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ ) is
∆ g 1 = EBV ¯ S m + EBV ¯ S f 2 − EBV ¯ 0 . $$ \Delta {\mathrm{g}}_1=\frac{{\overline{\mathrm{EBV}}}_{{\mathrm{S}}_{\mathrm{m}}}+{\overline{\mathrm{EBV}}}_{{\mathrm{S}}_{\mathrm{f}}}}{2}-{\overline{\mathrm{EBV}}}_0. $$

2.2.2 Prediction of response to selection from the variance of EBV

The variance of EBV in a candidate population for selection ( σ g ̂ 2 $$ {\upsigma}_{\hat{\mathrm{g}}}^2 $$ ) is estimated from
σ g ̂ 2 = ∑ i = 1 2 N EBV i − EBV ¯ 0 2 2 N − 1 , $$ {\upsigma}_{\hat{\mathrm{g}}}^2=\frac{\sum_{\mathrm{i}=1}^{2\mathrm{N}}{\left({\mathrm{EBV}}_{\mathrm{i}}-{\overline{\mathrm{EBV}}}_0\right)}^2}{2\mathrm{N}-1}, $$
where EBV i $$ {\mathrm{EBV}}_{\mathrm{i}} $$ is the EBV in the ith animal. From Equation (4), σ g ̂ 2 $$ {\upsigma}_{\hat{\mathrm{g}}}^2 $$ is equal to covariance ( σ g g ̂ $$ {\upsigma}_{\mathrm{g}\hat{\mathrm{g}}} $$ ) between BV and EBV. Therefore, the response to selection ( ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ ) is
∆ g 2 = i ¯ · 1 − F ¯ 1 2 σ g · r g g ̂ = i ¯ · 1 − F ¯ 1 2 σ g ̂ , $$ \Delta {\mathrm{g}}_2=\overline{i}\cdotp {\left(1-\overline{\mathrm{F}}\right)}^{\frac{1}{2}}{\upsigma}_{\mathrm{g}}\cdotp {\mathrm{r}}_{\mathrm{g}\hat{\mathrm{g}}}=\overline{i}\cdotp {\left(1-\overline{\mathrm{F}}\right)}^{\frac{1}{2}}{\upsigma}_{\hat{\mathrm{g}}}, $$
where F ¯ $$ \overline{\mathrm{F}} $$ is the average inbreeding coefficient in the population.

2.2.3 Prediction of response to selection from the inverse of the left-hand side of MME

Let c i 22 $$ {\mathrm{c}}_{\mathrm{i}}^{22} $$ be the ith diagonal element of C 22 $$ {\mathbf{C}}^{22} $$ ,
var g ̂ i = cov g i g ̂ i = 1 + F i σ g 2 − c i 22 , $$ \mathit{\operatorname{var}}\left({\hat{\mathrm{g}}}_{\mathrm{i}}\right)=\mathit{\operatorname{cov}}\left({\mathrm{g}}_{\mathrm{i}},{\hat{\mathrm{g}}}_{\mathrm{i}}\right)=\left(1+{\mathrm{F}}_{\mathrm{i}}\right){\upsigma}_{\mathrm{g}}^2-{\mathrm{c}}_{\mathrm{i}}^{22}, $$
where g i $$ {\mathrm{g}}_{\mathrm{i}} $$ , g ̂ i $$ {\hat{\mathrm{g}}}_{\mathrm{i}} $$ , and F i $$ {\mathrm{F}}_{\mathrm{i}} $$ are BV, EBV, and the inbreeding coefficient of the ith animal. σ g ̂ 2 $$ {\upsigma}_{\hat{\mathrm{g}}}^2 $$ and σ g g ̂ $$ {\upsigma}_{\mathrm{g}\hat{\mathrm{g}}} $$ in a candidate population for selection from Equations (2) and (4)
σ g ̂ 2 = σ g g ̂ ≅ ∑ i = 1 2 N var g ̂ i 2 N = 1 + F ¯ σ g 2 − c ¯ 22 $$ {\upsigma}_{\hat{\mathrm{g}}}^2={\upsigma}_{\mathrm{g}\hat{\mathrm{g}}}\cong \frac{\sum_{\mathrm{i}=1}^{2\mathrm{N}}\mathit{\operatorname{var}}\left({\hat{\mathrm{g}}}_{\mathrm{i}}\right)}{2\mathrm{N}}=\left(1+\overline{\mathrm{F}}\right){\upsigma}_{\mathrm{g}}^2-{\overline{\mathrm{c}}}^{22} $$
where c ¯ 22 $$ {\overline{\mathrm{c}}}^{22} $$ is the mean of the diagonal elements of C 22 $$ {\mathbf{C}}^{22} $$ corresponding to each animal in the population. Therefore, the response to selection ( ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ ) is
∆ g 3 = i ¯ · 1 − F ¯ 1 2 σ g · r g g ̂ ≅ i ¯ · 1 − F ¯ 1 + F ¯ 1 2 · 1 + F ¯ σ g 2 − c ¯ 22 1 2 . $$ \Delta {\mathrm{g}}_3=\overline{i}\cdotp {\left(1-\overline{\mathrm{F}}\right)}^{\frac{1}{2}}{\upsigma}_{\mathrm{g}}\cdotp {\mathrm{r}}_{\mathrm{g}\hat{\mathrm{g}}}\cong \overline{i}\cdotp {\left(\frac{1-\overline{\mathrm{F}}}{1+\overline{\mathrm{F}}}\right)}^{\frac{1}{2}}\cdotp {\left[\left(1+\overline{\mathrm{F}}\right){\upsigma}_{\mathrm{g}}^2-{\overline{\mathrm{c}}}^{22}\right]}^{\frac{1}{2}}. $$

2.3 Comparison of response to selection using computer simulation

To compare the response to selection using different prediction methods ( ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ , ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ , ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ ), a Monte Carlo computer simulation was used to generate pedigree and phenotypic records. The base population (G0) consisted of n m $$ {\mathrm{n}}_{\mathrm{m}} $$ males and n f $$ {\mathrm{n}}_{\mathrm{f}} $$ females that were all assumed to be unrelated, unselected, and non-inbred. Single trait of six separate generations (G0 to G5) were simulated. One male was mated at random to five females, a mated female produced two males and two females, and n m $$ {\mathrm{n}}_{\mathrm{m}} $$ males and n f $$ {\mathrm{n}}_{\mathrm{f}} $$ females were randomly selected in each generation. An infinitesimal additive genetic model (Bulmer, 1980) was assumed. The additive genetic value ( g i $$ {\mathrm{g}}_{\mathrm{i}} $$ ) of the ith animal was generated using the following equation:
g i = g S i + g D i 2 + 1 2 − F S i + F D i 4 · σ g · ε i $$ {\mathrm{g}}_{\mathrm{i}}=\frac{{\mathrm{g}}_{{\mathrm{S}}_{\mathrm{i}}}+{\mathrm{g}}_{{\mathrm{D}}_{\mathrm{i}}}}{2}+\sqrt{\frac{1}{2}-\frac{{\mathrm{F}}_{{\mathrm{S}}_{\mathrm{i}}}+{\mathrm{F}}_{{\mathrm{D}}_{\mathrm{i}}}}{4}}\cdotp {\upsigma}_{\mathrm{g}}\cdotp {\upvarepsilon}_{\mathrm{i}} $$
where g S i $$ {\mathrm{g}}_{{\mathrm{S}}_{\mathrm{i}}} $$ and g D i $$ {\mathrm{g}}_{{\mathrm{D}}_{\mathrm{i}}} $$ are the additive genetic values for the sire and dam of the ith animal, respectively; F S i $$ {\mathrm{F}}_{{\mathrm{S}}_{\mathrm{i}}} $$ and F D i $$ {\mathrm{F}}_{{\mathrm{D}}_{\mathrm{i}}} $$ are the inbreeding coefficients of the sire and dam, respectively; and ε i $$ {\upvarepsilon}_{\mathrm{i}} $$ is the standard normal deviate of the ith animal.
The phenotypic record ( y i $$ {\mathrm{y}}_{\mathrm{i}} $$ ) of the ith animal was simulated using the following equation:
y i = g i + e i $$ {\mathrm{y}}_{\mathrm{i}}={\mathrm{g}}_{\mathrm{i}}+{\mathrm{e}}_{\mathrm{i}} $$
where e i $$ {\mathrm{e}}_{\mathrm{i}} $$ is the random environmental effect for the ith animal with a mean of zero and variance of σ e 2 $$ {\upsigma}_{\mathrm{e}}^2 $$ .

The phenotypic variance of a trait with heritability of 0.2 or 0.5 was set at 100. Fixed effect was assumed to be mean (one level), sex (two levels), or generation (five levels), and the magnitude of every level was set at zero. The population sizes ( n m : n f $$ {\mathrm{n}}_{\mathrm{m}}:{\mathrm{n}}_{\mathrm{f}} $$ ) were assumed to be 10 : 50 $$ \left(10:50\right) $$ , 20 : 100 $$ \left(20:100\right) $$ , 30 : 150 $$ \left(30:150\right) $$ , 40 : 200 $$ \left(40:200\right) $$ , or 50 : 250 $$ \left(50:250\right) $$ . Thus, the selection intensities in each population are 1.2640, 1.2700, 1.2765, 1.2765, and 1.2765 (Falconer, 1981). Animals in G0 had no records and G1 through G5 had one record for all animals. One thousand replicates were simulated for each combination ( 2 × 3 × 5 = 30 $$ 2\times 3\times 5=30 $$ ) of heritability, fixed effect, and population size.

RMSE Fxd 1 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1} $$ and RMSE Fxd 2 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2} $$ were obtained based on the data generated using the simulation. When ∆ g 0 $$ \Delta {\mathrm{g}}_0 $$ was taken as the true value, the mean errors and RMSEs of ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ to ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ were calculated as follows:
The i th mean error = ∑ k = 1 rpt ∆ g ik − ∆ g 0 k rpt and RMSE = ∑ k = 1 rpt ∆ g ik − ∆ g 0 k 2 rpt $$ \mathrm{The}\ {i}^{\mathrm{th}}\ \mathrm{mean}\ \mathrm{error}=\frac{\sum_{k=1}^{rpt}\left(\Delta {\mathrm{g}}_{ik}-\Delta {\mathrm{g}}_{0k}\right)}{rpt}\ \mathrm{and}\ \mathrm{RMSE}=\sqrt{\frac{\sum_{k=1}^{rpt}{\left(\Delta {\mathrm{g}}_{ik}-\Delta {\mathrm{g}}_{0k}\right)}^2}{rpt}} $$
where ∆ g ik $$ \Delta {\mathrm{g}}_{ik} $$ is ∆ g i $$ \Delta {\mathrm{g}}_i $$ at kth iteration and rpt (= 1000) is the number of iterations. The mean errors and RMSEs were used as criteria of the prediction accuracy in the response to selection.

3 RESULTS

Table 1 shows the mean RMSE and its standard error (SE) of fixed effect over 1000 replicates. No obvious differences or trends were found between RMSE Fxd 1 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1} $$ and RMSE Fxd 2 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2} $$ . The SE for RMSE Fxd 2 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2} $$ was smaller than that for RMSE Fxd 1 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1} $$ . The RMSE tended to rise with increasing number of levels of fixed effect. The larger the population size, the smaller the RMSE. The RMSE was smaller for a trait with low heritability than with high heritability.

TABLE 1. Mean RMSE and its standard error (SE) of the fixed effect over 1000 replicates when one of the three kinds of fixed effects is included in the mixed model.
Population size Fixed effect h2 = 0.2 h2 = 0.5
RMSEFxd1 RMSEFxd2 RMSEFxd1 RMSEFxd2
10 sires: 50 dams Mean 0.917 (1.26E-03) 0.892 (2.96E-06) 2.172 (3.28E-03) 1.817 (7.67E-07)
Sex 0.945 (1.19E-03) 0.982 (2.88E-06) 1.986 (2.58E-03) 1.891 (7.33E-07)
Genertion 1.787 (1.96E-03) 1.763 (1.99E-05) 3.369 (3.94E-03) 3.093 (1.37E-05)
20 sires: 100 dams Mean 0.463 (6.81E-04) 0.446 (1.07E-06) 0.935 (1.43E-03) 0.908 (2.86E-07)
Sex 0.488 (6.17E-04) 0.491 (1.03E-06) 0.896 (1.14E-03) 0.946 (2.72E-07)
Generation 0.881 (9.38E-04) 0.884 (7.09E-06) 1.596 (1.83E-03) 1.549 (5.10E-06)
30 sires: 150 dams Mean 0.288 (4.41E-04) 0.297 (5.71E-07) 0.608 (8.04E-04) 0.606 (1.54E-07)
Sex 0.345 (4.56E-04) 0.327 (5.71E-07) 0.654 (9.28E-04) 0.631 (1.50E-07)
Generation 0.613 (6.16E-04) 0.590 (3.91E-06) 1.039 (1.16E-03) 1.033 (2.76E-06)
40 sires: 200 dams Mean 0.222 (3.14E-04) 0.223 (3.55E-07) 0.440 (6.52E-04) 0.454 (9.76E-08)
Sex 0.239 (3.12E-04) 0.246 (3.82E-07) 0.443 (6.18E-04) 0.473 (9.86E-08)
Generation 0.404 (4.13E-04) 0.443 (2.63E-06) 0.757 (9.12E-04) 0.775 (1.82E-06)
50 sires: 250 dams Mean 0.180 (2.41E-04) 0.179 (2.67E-07) 0.390 (5.27E-04) 0.363 (7.13E-08)
Sex 0.196 (2.55E-04) 0.196 (2.81E-07) 0.390 (5.25E-04) 0.378 (7.14E-08)
Generation 0.351 (3.62E-04) 0.354 (1.88E-06) 0.655 (7.61E-04) 0.620 (1.33E-06)
  • Note: Parentheses denote SE of RMSE.
  • Abbreviations: RMSEFxd1, RMSE calculated from solutions of MME; RMSEFxd2, RMSE calculated from the inverse of the left-hand side of MME.

Table 2 shows the mean of response to selection and its SE over 1000 replicates for each prediction method. Since no difference among three fixed effects was observed, the results are presented here with the fixed effect as mean only. The SE was, in increasing order ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ , followed by ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ , and was largest for ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ . The response to selection increased as the population size increased.

TABLE 2. Mean true response to selection (Δg0) and predicted responses to selection (Δg1-Δg3) and their standard errors (SE) over 1000 replicates under conditions of different heritabilities and population sizes.
Heritability Population size Δg0 Δg1 Δg2 Δg3
Mean (se) Mean (se) Mean (se) Mean (se)
0.2 10 sires: 50 dams 3.265 (8.9E-04) 3.249 (5.4E-04) 3.088 (4.9E-04) 3.474 (1.9E-05)
20 sires: 100 dams 3.434 (6.7E-04) 3.444 (4.3E-04) 3.355 (3.8E-04) 3.529 (1.0E-05)
30 sires: 150 dams 3.473 (5.3E-04) 3.452 (3.4E-04) 3.410 (3.1E-04) 3.558 (6.8E-06)
40 sires: 200 dams 3.497 (4.5E-04) 3.485 (3.0E-04) 3.451 (2.8E-04) 3.564 (5.0E-06)
50 sires: 250 dams 3.510 (3.8E-04) 3.510 (2.7E-04) 3.483 (2.5E-04) 3.567 (4.1E-06)
0.5 10 sires: 50 dams 6.538 (1.1E-03) 6.562 (8.7E-04) 6.223 (7.5E-04) 6.805 (1.6E-05)
20 sires: 100 dams 6.861 (8.4E-04) 6.857 (6.7E-04) 6.674 (6.0E-04) 6.953 (8.4E-06)
30 sires: 150 dams 6.938 (6.9E-04) 6.951 (5.4E-04) 6.848 (4.9E-04) 7.028 (5.6E-06)
40 sires: 200 dams 6.982 (6.2E-04) 6.972 (4.9E-04) 6.899 (4.3E-04) 7.047 (4.4E-06)
50 sires: 250 dams 6.996 (5.4E-04) 6.995 (4.3E-04) 6.934 (3.9E-04) 7.059 (3.6E-06)
  • Note: Parentheses denote SE of RMSE.
  • Abbreviations: Δg0, true response to selection; Δg1, predicted response to selection calculated from the mean of EBV; Δg2, predicted response calculated from the variance of EBV; Δg3, predicted response calculated from the inverse of the left-hand side of MME.

Table 3 shows the average mean error and RMSE over 1000 replicates. The absolute value of the mean error for ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ was the smallest and the absolute values for ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ and ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ were, respectively, all negative and all positive. The absolute values of the mean errors tended to become larger as the population size decreased. The RMSE was smallest for ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ , followed by ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ and ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ . For all prediction methods, the RMSE tended to become larger as the population size decreased. In increasing order, RMSE was smallest for ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ , followed by ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ and ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ .

TABLE 3. Average mean error and RMSE over 1000 replicates of deviations of predicted responses to selection (Δg1 to Δg3) from true response to selection (Δg0).
Heritability Population size Mean error RMSE
Δg1 Δg2 Δg3 Δg1 Δg2 Δg3
0.2 10 sires: 50 dams −0.015 −0.176 0.209 0.338 0.268 0.494
20 sires: 100 dams 0.011 −0.078 0.095 0.306 0.261 0.401
30 sires: 150 dams −0.020 −0.063 0.085 0.247 0.226 0.357
40 sires: 200 dams −0.011 −0.046 0.067 0.233 0.215 0.323
50 sires: 250 dams 0.013 −0.090 0.090 0.281 0.224 0.387
0.5 10 sires: 50 dams 0.024 −0.316 0.267 0.361 0.232 0.566
20 sires: 100 dams −0.004 −0.187 0.092 0.291 0.219 0.431
30 sires: 150 dams −0.001 −0.028 0.057 0.220 0.205 0.295
40 sires: 200 dams −0.010 −0.083 0.065 0.246 0.219 0.364
50 sires: 250 dams −0.001 −0.062 0.063 0.238 0.209 0.345
  • Abbreviations: Δg0, true response to selection; Δg1, predicted response to selection from the mean of EBV; Δg2, predicted response to selection from the variance of EBV; Δg3, predicted response to selection from the inverse of the left-hand side of MME.

4 DISCUSSION

One method of estimating genetic trends is to use the BLUP of BVs (Henderson, 1973), which is a development of the method for separating genetic effects from environmental effects (Henderson et al., 1959). Estimation of genetic trends is important to provide breeders with information that will enable them to develop more efficient selection programs in the future (Béjar et al., 1993). This method is widely used from laboratory animals to large livestock (e.g., Abdallah & McDaniel, 2000; Saad et al., 2020; Satoh et al., 1997; Tomiyama et al., 2011). When estimating responses to selection using this method, animals for which phenotypic values have been obtained are targeted. On the other hand, it is possible to predict the response to selection even if the phenotypic values of animals are not available. In this study, when the phenotypic values for progenies of selected animals are not available, three methods of predicting the response to selection were compared in terms of the accuracy of estimating fixed effects and the accuracy of predicting responses to truncated selection based on EBV.

In the estimation of fixed effect, there was no obvious difference between RMSE Fxd 1 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1} $$ and RMSE Fxd 2 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2} $$ , and the SE was smaller for RMSE Fxd 2 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}2} $$ than for RMSE Fxd 1 $$ {\mathrm{RMSE}}_{\mathrm{Fxd}1} $$ . However, since the SE was very small compared with the RMSE, it was considered to have no obvious impact on the accuracy of the estimation of the fixed effect.

The absolute value of the mean error was the smallest for ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ and the RMSE was the smallest for ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ . Since ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ predicts the response to selection using the diagonal elements of the inverse of the left-hand side of MME, it requires more computing time than ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ and ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ . In addition, the mean error was not the smallest and the RMSE was the largest for ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ . Since ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ and ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ predict the response to selection based on the accuracy of selection, it is likely that ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ underestimates and ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ overestimates the variance of the EBV. In particular, the low prediction accuracy of ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ may be due to the use of the diagonal elements of the inverse matrix of the mixed model equations. Essentially, the diagonal elements are the prediction error variances of BVs in each individual. The use of these averages in this study may be one of the reasons for the low prediction accuracy. We therefore consider it desirable to use ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ or ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ in a randomly selected population if the phenotypic records for progenies of selected animals are not available. It should be noted, however, that ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ was slightly biased downward from its true value. Although the RMSE increased with decreasing population size, the response to selection based on EBV can be predicted, even in small populations.

The formula for ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ replaces the BV in the formula for the true value ( ∆ g 0 $$ \Delta {\mathrm{g}}_0 $$ ) with the estimated value EBV and can be used even when generations overlap or when the distribution is not normal. For ∆ g 2 $$ \Delta {\mathrm{g}}_2 $$ and ∆ g 3 $$ \Delta {\mathrm{g}}_3 $$ , if the BVs or EBVs are not normally distributed, the prediction accuracy of selection response may decrease, and it may be difficult to calculate the selection intensity in a population with overlapping generations. Thus, in the population where selection is ongoing, the prediction accuracy of selection response is likely to be affected by the distortion of the distribution and the Bulmer effect (Bijma, 2012; Gorjanc et al., 2015). The ∆ g 1 $$ \Delta {\mathrm{g}}_1 $$ is, therefore, considered to be the most practical for such a population.

ACKNOWLEDGMENTS

This work was supported by JSPS KAKENHI Grant Numbers JP22H02491.

    CONFLICT OF INTEREST STATEMENT

    The author declares that he has no conflicts of interest.

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