Linear Molecular and Genomic Eigen Selection Index Methods

The three main linear phenotypic eigen selection index methods are the eigen selection index method (ESIM), the restricted ESIM (RESIM) and the predetermined proportional gain ESIM (PPG-ESIM). The ESIM is an unrestricted index, but the RESIM and PPG-ESIM allow null and predetermined restrictions respectively to be imposed on the expected genetic gains of some traits, whereas the rest remain without any restrictions. These indices are based on the canonical correlation, on the singular value decomposition, and on the linear phenotypic selection indices theory. We extended the ESIM theory to the molecular-assisted and genomic selection context to develop a molecular ESIM (MESIM), a genomic ESIM (GESIM), and a genome-wide ESIM (GW-ESIM). Also, we extend the RESIM and PPG-ESIM theory to the restricted genomic ESIM (RGESIM), and to the predetermined proportional gain genomic ESIM (PPG-GESIM) respectively. The latter ﬁ ve indices use marker and phenotypic information jointly to predict the net genetic merit of the candidates for selection, but although MESIM uses only statistically signi ﬁ cant markers linked to quantitative trait loci, the GW-ESIM uses all genome markers and phenotypic information and the GESIM, RGESIM, and PPG-GESIM use the genomic estimated breeding values and the phenotypic values to predict the net genetic merit. Using real and simulated data, we validated the theoretical results of all ﬁ ve indices.


The Molecular Eigen Selection Index Method
The molecular eigen selection index method (MESIM) is very similar to the linear molecular selection index (LMSI) described in Chap. 4; thus, it uses the same set of information to predict the net genetic merit of individual candidates for selection, and therefore needs the same set of conditions as those of the LMSI. The only difference between the two indices is how the vector of coefficients is obtained and the assumption associated with the vector of economic weights. Thus, although the LMSI obtains the vector of coefficients according to the linear phenotypic selection index (LPSI) described in Chap. 2 and assumes that the economic weights are known and fixed, the MESIM assumes that the economic weights are unknown and fixed and obtains the vector of coefficients according to the ESIM theory.

The MESIM Parameters
In the MESIM context, the net genetic merit can be written as where g 0 ¼ g 1 . . . g t ½ is the vector of true breeding values, t is the number of traits, w 0 1 ¼ w 1 Á Á Á w t ½ is a vector of unknown economic weights associated with g, w 0 2 ¼ 0 1 Á Á Á 0 t ½ is a null vector associated with the vector of marker score values s 0 ¼ s 1 s 2 . . . s t ½ , w 0 ¼ w 0 1 w 0 2 ½ and a 0 ¼ g 0 s 0 ½ (Chap. 4 for details). The MESIM index can be written as where y 0 ¼ y 1 Á Á Á y t ½ is the vector of phenotypic values; s 0 ¼ s 1 s 2 . . . s t ½ is the vector of marker scores; β 0 y and β s are vectors of phenotypic and marker score weight values respectively, β 0 ¼ β 0 y β 0 G Â Ã and t 0 ¼ y 0 s 0 ½ . The objectives of the MESIM are the same as those of the ESIM (see Chap. 7 for details).
Let Var H ð Þ ¼ w 0 Ψ M w ¼ σ 2 H be the variance of H, Var I ð Þ ¼ β 0 T M β ¼ σ 2 I the variance of I, and Cov(H, I ) ¼ w 0 Ψ M β the covariance between H and I, where are block matrices of size 2t Â 2t (t is the number of traits) of covariance matrices where P, S M , and C are covariance matrices t Â t of phenotypic (y), marker score (s), and genetic breeding (g) values respectively. Let ρ HI ¼ correlation between H and I, and the heritability of I respectively; then, the MESIM selection response can be written as where k I is the standardized selection differential (or selection intensity) associated with MESIM; σ H ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi w 0 Ψ M w p and σ I ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi β 0 T M β p are the standard deviations of the variance of H and I respectively. It is assumed that k I is fixed, and that matrices T M and Ψ M are known; therefore, we can maximize R by maximizing ρ HI (Eq. 8.3) with respect to vectors w and β, or by maximizing h 2 I (Eq. 8.4) only with respect to vector β.
Maximizing h 2 I only with respect to β is simpler than maximizing ρ HI with respect to w and β; however, in the latter case the maximization process of ρ HI gives more information associated with MESIM parameters than when h 2 I is maximized only with respect to β (see Chap. 7,Eq. 7.13,for details). In this subsection, we maximize ρ HI with respect to vectors w and β similar to the ESIM in Chap. 7, Sect. 7.1.1. Thus, we omit the steps and details of the maximization process of ρ HI .
We maximize ρ HI ¼ with respect to vectors w and β under the restrictions σ 2 H ¼ w 0 Ψw, σ 2 I ¼ β 0 Tβ, and 0 < σ 2 H , σ 2 I < 1, where σ 2 H is the variance of H ¼ w 0 a and σ 2 I is the variance of I ¼ β 0 t. Thus, it is necessary to maximize the function f β; w; μ; ϕ ð Þ¼w 0 Ψβ À 0:5μ β 0 Tβ À σ 2 I À Á À 0:5ϕ w 0 Ψw À σ 2 H À Á ð8:5Þ with respect to β, w, μ, and ϕ, where μ and ϕ are Lagrange multipliers. The derivatives of Eq. (8.5) with respect to β, w, μ, and ϕ are: ¼ À10:09 À10:31 À2:53 À4:39 ½ respectively, whereas the estimated LMSI accuracy was b ρ :009 0:006 À0:402 0:628 ½ and then the estimated MESIM index was b I M ¼ À0:009 PHT þ 0:006 EHT À 0:402 S PHT þ 0:628 S EHT , where S PHT and S EHT denote the marker scores associated with PHT and EHT respectively. The estimated MESIM expected genetic gain, selection response, and accuracy were b The inner product of the estimated LMSI and MESIM vector of coefficients were 1.221 and 0.556 respectively, whence the estimated LMSI selection response (20.41) divided by 1.221 was 16.716, and the estimated MESIM selection response (6.573) divided by 0.556 was 11.821. That is, the estimated LMSI selection response was higher than the estimated MESIM selection response for this data set. Similar results were found when we compared the estimated LMSI expected genetic gain per trait with the estimated MESIM expected genetic gain per trait. Finally, Fig. 8.1 presents the frequency distribution of the 247 estimated MESIM values for the real data set described earlier, which approaches normal distribution, as we would expect. Now with a selection intensity of 10% (k I ¼ 1.755), we compare the LMSI and MESIM efficiency using the simulated data set described in Sect. 2.8.1 of Chap. 2 for four phenotypic selection cycles, each with four traits (T 1 , T 2 , T 3 and T 4 ), 500 genotypes, and four replicates of each genotype. The economic weights for T 1 , T 2 , T 3 , and T 4 were 1, À1, 1, and 1 respectively. For this data set, we did not use the linear The estimated selection responses of the linear marker, combined genomic and genome-wide selection indices (LMSI, CLGSI, and GW-LMSI respectively; see Fig. 8.1 Frequency distribution of 247 estimated molecular eigen selection index method (MESIM) values for one selection cycle in an environment for a real maize (Zea mays) F 2 population with 195 molecular markers and two traits, plant height (PHT, cm) and ear height (EHT, cm), and their associated marker scores S PHT and S EHT respectively Chaps. 4 and 5 for details) for four simulated selection cycles when their vectors of coefficients were normalized, are presented in Table 8.1. Also, in this table the selection responses of the estimated linear molecular, genomic, and genome-wide eigen selection index methods (MESIM, GESIM, and GW-ESIM respectively; details in Sect. 8.2) are shown for four simulated selection cycles. The average of the estimated LMSI selection response was 2.22, whereas the average of the estimated MESIM selection response was 1.69. The estimated LMSI selection response was higher than that of the MESIM. Table 8.2 presents the estimated LMSI and MESIM expected genetic gains for four traits (T1, T2, T3, and T4) and their associated marker scores (S1, S2, S3, and S4) for four simulated selection cycles. The averages of the estimated LMSI Chapter 11 presents RIndSel, a user-friendly graphical unit interface in JAVA that is useful for estimating the LMSI and ESIM parameters and selecting parents for the next selection cycle.

The Linear Genomic Eigen Selection Index Method
The linear genomic eigen selection index method (GESIM) is based on the standard CLGSI described in Chap. 5, and uses genomic estimated breeding values (GEBVs) and phenotypic values jointly to predict the net genetic merit. Thus, conditions for constructing a valid GESIM are the same as those for constructing the CLGSI. Also, the MESIM theory described in Sect. 8.1 is directly applied to the GESIM and only minor changes are necessary in GESIM theory. For example, instead of marker scores, the GESIM uses GEBVs to predict the net genetic merit; thus, the details of the estimation process are the same as for the MESIM.

The GESIM Parameters
In the GESIM context, the net genetic merit can be written as where g 0 ¼ g 1 . . . g t ½ is the vector of true breeding values, t is the number of traits, w 0 is a vector of unknown economic weights associated with g, w 0 The estimator of γ is the GEBV (see Chap. 5 for additional details). The GESIM index can be written as where y 0 ¼ y 1 Á Á Á y t ½ is the vector of phenotypic values; β 0 y and β γ are vectors of weights of phenotypic and genomic breeding values weights respectively; 2t Â 2t (t is the number of traits) of covariance matrices and P, Γ, and C are covariance matrices of phenotypic (y), genomic (γ), and genetic (g) values respectively. Then, ρ HI ¼ is the correlation between H ¼ w 0 α and I ¼ β 0 f and the GESIM selection response can be written as where k I is the standardized selection differential (or selection intensity) associated with the GESIM and σ H ¼ ffiffiffiffiffiffiffiffiffiffiffiffi w 0 Aw p is the standard deviation of the variance of H. It is assumed that k I is fixed, and that matrices Φ and A are known; then, we can maximize R by maximizing ρ HI with respect to vectors w and β under the restrictions σ 2 H ¼ w 0 Aw, σ 2 I ¼ β 0 Φβ, and 0 < σ 2 H , σ 2 I < 1; similar to the MESIM. It can be shown that the vector w in the GESIM context is and that the net genetic merit can be written as H G ¼ w 0 G α. The correlation between and the GESIM index vector of coefficients that maximizes ρ H G I can be obtained from the equation where I 2t is an identity matrix of size 2t Â 2t (t is the number of traits); the optimized GESIM index can be written as By Eqs. (8.19) and (8.20), GESIM accuracy can be written as is the square of the canonical correlation between H G and I G , and β G is the canonical vector associated with λ 2 The maximized GESIM selection response and expected genetic gain per trait are
The estimated GESIM vector of coefficients, selection response, accuracy, and expected genetic gain per trait were b β 0 G 1 ¼ À0:207 0:029 0:041 0:820 0:337 0:411 Now, we compare the estimated CLGSI and GESIM selection response and expected genetic gain per trait using the simulated data set described in Sect. 2.8.1 of Chap. 2 for four phenotypic selection cycles, each with four traits (T 1 , T 2 , T 3 and T 4 ), 500 genotypes, and four replicates per genotype. The economic weights of T 1 , T 2 , T 3 , and T 4 were 1, À1, 1, and 1 respectively and the selection intensity for both For this data set, the averages of the estimated CLGSI and GESIM selection responses were 0.68 and 2.74 (Table 8.1) respectively. The estimated CLGSI selection response was lower than the estimated GESIM selection response. Table 8.3 presents the estimated CLGSI and GESIM expected genetic gain for four traits (T1, T2, T3, and T4) and their associated genomic estimated breeding values (GEBV1, GEBV2, GEBV3, and GEBV4) for four simulated selection cycles. The averages of the estimated CLGSI expected genetic gains for the four traits and their associated GEBVs were 7.45, À3.35, 2.68, 1.09, 7.13, À3.68, 3.13, and 2.69 respectively, whereas the averages of the estimated GESIM expected genetic gains for the four traits and their associated GEBVs were 8.18, À3.08, 2.27, 0.71, 7.46, À3.53, 2.86, and 2.39 respectively. The estimated CLGSI and GESIM expected genetic gains per trait were very similar.

The Genome-Wide Linear Eigen Selection Index Method
The MESIM requires regressing phenotypic values on marker coded values to predict the marker score values for each individual candidate for selection, and then combining the marker scores with phenotypic information using the MESIM

The GW-ESIM Parameters
In the GW-ESIM context, the net genetic merit can be written as is the vector of unknown economic weights associated with the breeding values; w 0 The GW-ESIM (I ) index combines the phenotypic value and all the marker information of individuals to predict Eq. (8.24) values in each selection cycle and can be written as where β 0 y and β m are vectors of phenotypic and marker weights respectively; Zw be the variance of I ¼ β 0 q and H ¼ w 0 z respectively, and σ HI ¼ w 0 Zβ the covariance between I and H, where Q ¼ Var are block matrices of size (t + N) Â (t + N) (t is the number of traits and N is the number of markers) where P ¼ Var(y), M ¼ Var(m), C ¼ Var(g), and G M ¼ cov (y, m) ¼ cov (g, m) are covariance matrices of phenotypic (y), coded marker (m), and genetic (g) values respectively, whereas G M is the covariance matrix between y and m, and between g and m (for details see Chap. 4); w and β were defined earlier. Note that although the size of matrices P and C are t Â t, the sizes of matrices M and G M are N Â N and N Â t respectively. Thus, if the number of markers is very high, the size of matrices M and G M could also be very high.
In Chap. 4 we described matrix M as where (1 À 2θ ij ) and θ ij (i, j¼ 1, 2, . . ., N¼ number of markers) are the covariance (or correlation) and the recombination frequency between the ith and jth marker respectively, whereas matrix G M can be written as where (1 À 2r ik )α qk (i¼ 1, 2, . . ., N, k¼ 1, 2, . . ., N Q ¼ number of quantitative trait loci (QTL), q ¼ 1, 2, . . ., t) is the covariance between the qth trait and the ith marker; r ik is the recombination frequency between the ith and kth QTL, and α qk is the effect of the kth QTL over the qth trait.
be the correlation between I ¼ β 0 q and H ¼ w 0 x; then, the GW-ESIM selection response can be written as where k I is the standardized selection differential (or selection intensity) associated with GW-ESIM and σ H ¼ ffiffiffiffiffiffiffiffiffiffiffiffi w 0 Xw p is the standard deviation of the variance of H. Assuming that k I is fixed, and that matrices Q and X are known, we can maximize R (Eq. 8.28) by maximizing ρ HI with respect to vectors w 0 and β under the restrictions σ 2 H ¼ w 0 Xw, σ 2 I ¼ β 0 Qβ, and 0 < σ 2 H ,σ 2 I < 1, similar to the MESIM and GESIM. It can be shown that vector w can be written as where I (t + N ) is an identity matrix of size (t + N ) Â (t + N ) and I W ¼ β 0 W q is the optimized GW-ESIM. The accuracy of the GW-ESIM can be written as is the square of the canonical correlation between H W and I W .
The maximized GW-ESIM selection response and expected genetic gain per trait are and respectively, where β W is the first eigenvector of Eq. (8.30).

Estimating GW-ESIM Parameters
In Chap. 2, Eqs. (2.22) to (2.24), we described the restricted maximum likelihood methods to estimate matrices C and P, which can be denoted by b C and b P. In Chap. 4, we described how to estimate matrices M and G M , which can be denoted by b M and b G M . With these estimates, we constructed the block estimated matrices as , whence we obtained the equation j ¼ 1, 2, . . ., (t + N ), where (t + N ) is the number of traits and markers in the GW-ESIM index. Similar to the MESIM, we obtained estimators b β W 1 and b λ 2 W 1 of the first eigenvector β W 1 and the first eigenvalue b λ 2 W 1 respectively, from equation These results allow the GW-ESIM index selection response and its expected genetic gain per trait to be estimated as respectively, whereas the estimator of GW-ESIM accuracy is b λ W 1 .

Numerical Examples
We compare the estimated GW-LMSI and GW-ESIM selection responses using the simulated data set described in Sect. 2.8.1 of Chap. 2, with a selection intensity of 10% (k I ¼ 1.755). Table 8.1 presents the estimated GW-LMSI selection response for four simulated selection cycles when their vectors of coefficients are normalized, whence it can be seen that the average estimated GW-LMSI selection response was 0.87. Table 8.1 also presents the estimated GW-ESIM selection response for four simulated selection cycles; the average of the estimated GW-ESIM selection responses was 0.93. Thus, for this data set, the estimated GW-LMSI and selection responses were very similar.

The Restricted Linear Genomic Eigen Selection Index Method
The restricted linear genomic eigen selection index method (RGESIM) is based on the restricted linear phenotypic ESIM (RESIM) theory described in Chap. 7. In the RESIM, the breeder's objective is to improve only (t À r) of t (r < t) traits, leaving r of them fixed. The same is true for RGESIM, but in this case, we should impose 2r restrictions, i.e., we need to fix r traits and their associated r GEBV to obtain results similar to those obtained with the RESIM (see Chap. 7 for details). This is the main difference between the RGESIM and the RESIM. It can be shown that Cov(I, α) ¼ Aβ is the covariance between the breeding value vector (α 0 ¼ [g 0 γ 0 ]) and the GESIM index (I ¼ β 0 f). In the RGESIM, we want some covariances between the linear combinations of α (U 0 G α) and I ¼ β 0 f to be zero, i.e., Cov G is a matrix 2(t À 1) Â 2t of 1s and 0s (1 indicates that the trait and its associated GEBV are restricted, and 0 indicates that the trait and its GEBV have no restrictions). We can solve this problem by maximizing β 0 Aβ ffiffiffiffiffiffiffi ffi β 0 Φβ p with respect to vector β under the restriction U 0 G Aβ ¼ 0 and β 0 β ¼ 1 similar to the RESIM, or by maximizing the correlation between H ¼ w 0 α and , with respect to vectors w 0 and β under the restrictions U 0 G Aβ ¼ 0, σ 2 H ¼ w 0 Aw, σ 2 I ¼ β 0 Φβ and 0 < σ 2 H , σ 2 I < 1, as we did for the GESIM.

The RGESIM Parameters
To obtain the RGESIM vector of coefficients, we maximize the function ] is a vector of Lagrange multipliers. The derivatives of function f(β, v 0 ) with respect to β and v 0 can be written as , I 2t is an identity matrix of size 2t Â 2t, and ; therefore, the maximized correlation between I RG and H RG or RGESIM accuracy can be written as where w 0 RG Aw RG is the variance of H RG . Hereafter, to simplify the notation, we write Eq. (8.42) as λ RG .
The maximized selection response and the expected genetic gain per trait of the RGESIM are and respectively, where β RG is the first eigenvector of matrix K RG Φ À1 A.

Estimating RGESIM Parameters
In Sect. 8.2, we indicated how to estimate matrices P, Γ, and C using phenotypic and genomic information, whence we can estimate matrices . Those methods are also valid for the RGESIM. This means that the SVD methods described for estimating MESIM parameters are also valid for estimating RGESIM parameters.

Numerical Examples
With a selection intensity of 10% (k I ¼ 1.755), we compare the CRLGSI (for details see Chap. 6) versus the RGESIM theoretical results using a real maize (Zea mays) We have indicated that the main difference between the RLPSI and the CRLGSI is the matrix U 0 C , on which we now need to impose two restrictions: one for the trait and another for its associated GEBV. Consider the data set described earlier and suppose that we restrict the trait GY (ton ha À1 ) and its associated GEBV GY ; then, matrix U 0 C should be constructed as U 0 A was described in Chap. 6, and is also valid for estimating RGESIM parameters. The estimated CRLGSI vector of coefficients is b A wis the estimated CLGSI vector of coefficients (Chap. 6). Let w 0 ¼ [5 À 0.1 À 0.1 0 0 0] be the vector of economic weights and suppose that we restrict trait GY and its associated GEBV GY ; in this case, U 0 C1 ¼ 1 0 0 0 0 0 0 0 0 1 0 0 ! , and according to matrices b P, b C, and b Γ described earlier, b β 0 CR ¼ 0:076 À0:004 À0:018 2:353 À0:096 À0:082 ½ was the estimated CRLGSI vector of coefficients and the estimated CRLGSI was b I CR ¼ 0:076GY À 0:004EHT À 0:018PHT þ 2:353GEBV GY À 0:096GEBV EHT À 0:082GEBV PHT where GEBV GY , GEBV EHT , and GEBV PHT are the GEBVs associated with the traits GY, EHT, and PHT respectively. The same procedure is valid for two or more restrictions. The estimated CRLGSI selection response and expected genetic gain per β CR q 0 À3:53 À6:03 0 À2:93 À4:87 ½ respectively, whereas the estimated CR , the trait GY and its associated GEBV GY have null values, as we would expect.
The estimated RGESIM selection response and expected genetic gain per β RG q 0 À3:28 À6:03 0 À2:93 À5:40 ½ respectively, whereas the estimated Fig. 8.3 presents the frequency distribution of the 244 estimated RGESIM index values for two null restrictions on traits GY and EHT and their associated GEBV GY and GEBV EHT , for one selection cycle in an environment for a real maize (Zea mays) F 2 population with 233 molecular markers. Note that the frequency distribution of the estimated RGESIM index values approaches the normal distribution. Now we compare the estimated CRLGSI and RGESIM selection responses and expected genetic gains per trait using the simulated data set described in Sect. 2.8.1 of Chap. 2. We used that data set for four phenotypic selection cycles (C2, C3, C4, and C5), each with four traits (T 1 , T 2 , T 3 , and T 4 ), 500 genotypes, and four replicates per genotype. The economic weights for T 1 , T 2 , T 3 , and T 4 were 1, À1, 1, and Fig. 8.3 Frequency distribution of the 244 estimated restricted genomic eigen selection index method (RGESIM) values for two null restrictions on traits grain yield (GY) and EHT and their associated genomic estimated breeding values (GEBVs), GEBV GY and GEBV EHT respectively, for one selection cycle in an environment for a real maize (Zea mays) F 2 population with 233 molecular markers. Note that the frequency distribution of the estimated RGESIM index values approaches normal distribution 1 respectively. For this data set, matrix F was an identity matrix of size 8 Â 8 for all four selection cycles.
Columns 2, 3, and 4 (from left to right) of Table 8.4 present the estimated CRLGSI selection responses when their vectors of coefficients are normalized and the estimated RGESIM and selection responses for one, two, and three restrictions for four simulated selection cycles. The averages of the estimated CRLGSI selection responses of the traits and their associated GEBVs for each of the three null restrictions were 3.24 for one restriction, 4.08 for two restrictions, and 5.06 for three restrictions, whereas the averages of the estimated RGESIM selection responses were 3.08 for one restriction, 2.79 for two restrictions, and 3.23 for three restrictions. Note that although for one restriction the selection response was similar for both indices, for two and three restrictions the CRLGSI selection responses were greater than the RGESIM selection responses. Table 8.5 presents the estimated CRLGSI and RGESIM expected genetic gains per trait for four traits (T1, T2, T3, and T4) and their associated GEBVs (in this case denoted by G1, G2, G3, and G4 to simplify the notation) in four simulated selection cycles and for one, two, and three null restrictions in four simulated selection cycles. Note that the null values of the traits and their restricted GEBVs are not shown in Table 8.5 with the aim of simplifying the table. The averages of the estimated CRLGSI expected genetic gains for the three traits and their associated GEBVs were À2.60, 2.16, 2.84, À1.21, 0.67, and 1.02 for one restriction; 2.74, 3.23, 0.78, Table 8.4 Estimated combined null restricted linear genomic selection index (CRLGSI) and estimated combined predetermined proportional gain linear genomic selection index (CPPG-LGSI) selection responses for one, two, and three restrictions when their vectors of coefficients are normalized for four simulated selection cycles Cycle CRLGSI response for one, two and three null restrictions CPPG-LGSI response for one, two and three predetermined restrictions

The Predetermined Proportional Gain Linear Genomic Eigen Selection Index Method
The predetermined proportional gain linear genomic eigen selection index method (PPG-GESIM) theory is based on the predetermined proportional gain linear phenotypic ESIM (PPG-ESIM) described in Chap. 7. In the PPG-ESIM, the vector of PPG (predetermined proportional gain) imposed by the breeder was However, because the PPG-GESIM uses phenotypic and GEBV information jointly to predict the net genetic merit, the vector of PPG All T1, T2, G1, and G2 expected genetic gains were null c All T1, T2, T3, G1, G2, and G3 expected genetic gains were null imposed by the breeder (d PG ) should be twice the standard vector d 0 , that is, The maximized correlation between I PG and H PG , or PPG-GESIM accuracy, is where w 0 PG Aw PG is the variance of H PG . Hereafter, to simplify the notation, we write Eq. (8.47) as λ PG .
The maximized selection response and the expected genetic gain per trait of the PPG-GESIM are and respectively, where β PG is the first eigenvector of Eq. (8.45).

Numerical Examples
The process for estimating PPG-ESIM parameters is similar to the method described for estimating RGESIM parameters. With a selection intensity of 10% (k I ¼ 1.755), we compare the combined predetermined proportional gain linear genomic selection index (CPPG-LGSI) and PPG-GESIM results using the real maize (Zea mays) F 2 population with 244 genotypes, 233 molecular markers, and three traits-GY (ton ha À1 ), EHT (cm), and PHT The estimated CPPG-LGSI vector of coefficients was b be the estimated block matrices and d 0 PG ¼ 7 À3 3:5 À1:5 ½ the vector of PPG imposed by the breeder on the traits GY and EHT, and their associated genomic estimated breeding values (GEBV GY and GEBV EHT ), and let b I PG ¼ 0:001GY À 0:05EHT þ 0:029PHT þ 0:975GEBV GY þ 0:154GEBV EHT À 0:157GEBV PHT where GEBV GY , GEBV EHT , and GEBV PHT are the GEBVs associated with the traits GY, EHT, and PHT respectively. The estimated PPG-GESIM selection response, accuracy, and expected genetic gain per trait were b 0:01 À1:00 À3:56 0 À0:46 À3:98 ½ respectively. Fig. 8.4 presents the frequency distribution of the 244 estimated PPG-GESIM index values for two predetermined restrictions on the traits GY and EHT and their associated GEBVs (GEBV GY and GEBV EHT ), for one selection cycle in an environment for a real maize (Zea mays) F 2 population with 233 molecular markers. Note that the frequency distribution of the estimated PPG-GESIM index values approaches normal distribution. Now, with a selection intensity of 10% (k I ¼ 1.755) and a vector of predetermined restrictions d 0 PG ¼ 7 À3 5 3:5 À1:5 2:5 ½ , we compare the estimated CPPG-LGSI and PPG-GESIM selection responses and expected genetic gains per Fig. 8.4 Frequency distribution of the 244 estimated predetermined proportional gain genomic eigen selection index method (PPG-GESIM) values for two predetermined restrictions on the traits GY and EHT and their associated GEBVs, GEBV GY and GEBV EHT , for one selection cycle in an environment for a real maize (Zea mays) F 2 population with 233 molecular markers trait using the simulated data set described in Sect. 2.8.1 of Chap. 2. Traits T1, T2, and T3 and their associated GEBVs (GEBV1, GEBV2, and GEBV3 respectively) were restricted, but trait T4 and its associated GEBV4 were not restricted. For this data set, matrix F was an identity matrix of size 8 Â 8 for all four selection cycles. Table 8.6 presents the estimated CPPG-LGSI selection responses when their vectors of coefficients are normalized, and the estimated PPG-GESIM selection responses for one, two, and three predetermined restrictions for four simulated selection cycles. The averages of the estimated CPPG-LGSI selection responses were 5.08 for one restriction, 3.42 for two restrictions, and 1.60 for three restrictions, whereas the averages of the estimated PPG-GESIM selection responses were 1.96 for one restriction, 4.14 for two restrictions, and 5.46 for three restrictions. For this data set, when the number of restrictions increases, the estimated CPPG-LGSI The selection intensity was 10% (k I ¼ 1.755) and the vector of predetermined restrictions was d 0 PG ¼ 7 À3 5 3:5 À1:5 2:5 ½ . Trait T4 and its associated GEBV4 were not restricted selection response tends to decrease, whereas the estimated PPG-GESIM selection response increases. Tables 8.7 presents the estimated CPPG-LGSI and PPG-GESIM expected genetic gains for one, two, and three predetermined restrictions respectively, for four simulated selection cycles. The averages of the estimated CPPG-LGSI expected genetic gains for the four traits and their four associated GEBVs were 8.28, À4.12, 3.23, 2.23, 4.14, À2.26, 1.71, and 1.01 for one restriction; 8.43, À3.61, 3.28, 2.13, 4.22, À1.81, 1.72, and 0.93 for two restrictions; and 5.81, À2.49, 4.15, 2.26, 2.90, À1.24, 2.07, and 0.89 for three restrictions. On the other hand, the averages of the estimated PPG-GESIM expected genetic gains for the four traits and their four associated GEBVs were 6.97, À1.31, 1.78, 0.52, 5.64, À1.74, 1.75, and 0.58 for one restriction; 6.93, À2.73, 1.29, 0.85, 5.75, À2.55, 1.49, and 0.79 for two restrictions, and 8.12, À3.27, 2.99, 1.13, 2.19, À1.15, 1.30, and 0.45 for three Table 8.7 Estimated PPG-GESIM expected genetic gains for one, two, and three restricted traits (T1, T2, and T3) and for one, two, and three restricted GEBVs (GEBV1, GEBV2, and GEBV3) for four simulated selection cycles The selection intensity was 10% (k I ¼ 1.755) and the vector of predetermined restrictions was d 0 PG ¼ 7 À3 5 3:5 À1:5 2:5 ½ . Trait T4 and its associated GEBV4 were not restricted restrictions. These results indicate that the estimated CPPG-LGSI expected genetic gains for the four traits and their four associated GEBVs were generally higher than the estimated PPG-GESIM expected genetic gains for the four traits and their four associated GEBVs.
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