## Abstract

The traditional molecular selection index (MSI) employed in marker-assisted selection maximizes the selection response by combining information on molecular markers linked to quantitative trait loci (QTL) and phenotypic values of the traits of the individuals of interest. This study proposes an MSI based on an eigenanalysis method (molecular eigen selection index method, MESIM), where the first eigenvector is used as a selection index criterion, and its elements determine the proportion of the trait's contribution to the selection index. This article develops the theoretical framework of MESIM. Simulation results show that the genotypic means and the expected selection response from MESIM for each trait are equal to or greater than those from the traditional MSI. When several traits are simultaneously selected, MESIM performs well for traits with relatively low heritability. The main advantages of MESIM over the traditional molecular selection index are that its statistical sampling properties are known and that it does not require economic weights and thus can be used in practical applications when all or some of the traits need to be improved simultaneously.

MARKER-ASSISTED selection (MAS) is an important breeding tool in which molecular marker alleles linked to quantitative trait loci (QTL) that control phenotypic variables of important traits are selected. Marker-assisted selection can be more efficient than selecting individuals on the basis of phenotypic trait values. Progeny of specific progenitors can be selected on the basis of molecular markers as long as these are associated with breeding values of the traits under consideration. This is one form of MAS (Dekkers and Dentine 1991; Arus and Moreno-Gonzalez 1993). Another form of MAS is based on the molecular selection index (MSI) proposed by Lande and Thompson (1990). In MSI the selection response is maximized by combining information on molecular markers linked to QTL and the phenotypic values of the traits of interest.

To construct an MSI, it is necessary to identify the linkage between the molecular marker and the QTL, the estimated effect of the QTL linked to the molecular marker (MQTL effect), and the combination of MQTL effects and phenotypic information that allows genotypes to be classified and selected using a selection index. The MQTL effects can be identified and estimated through the linkage disequilibrium that arises when crossing inbred lines or divergent populations (Zhang and Smith 1992, 1993; Xie and Xu 1998). The MSI depends on various factors, such as number and density of molecular markers associated with QTL, population size, trait heritability, additive genetic variances that can be explained by molecular markers, and precision of the estimated effect of gene substitution (Dekkers and Dentine 1991; Moreau *et al*. 2000).

The MSI is an application of the selection index methodology proposed by Smith (1936), in which MQTL effects are incorporated. As proposed by Lande and Thompson (1990), the MSI performs a linear regression of phenotypic values on the coded values of the molecular markers such that selected molecular markers are those statistically linked to QTL that explain most of the variability in regression models. The coefficient of regression of the molecular marker is the MQTL effect. Statistical models and methods for mapping QTL and estimating their MQTL effects have been developed (Jansen 2003). Several authors have pointed out the effectiveness of the MSI in inbred populations with large population sizes and traits with low heritability values (Zhang and Smith 1992, 1993; Gimelfarb and Lande 1994, 1995; Whittaker 2003) when only one trait (and its associated molecular score) is considered.

The selection index theory was originally developed by Smith (1936) and generalized by Kempthorne and Nordskog (1959) for a restrictive selection index. The standard selection index is defined as a linear combination of the observed phenotypic values of the traits of interest with the traits' previously defined economic weights. Selection indexes are based on improving one trait by incorporating information on related traits (Wei *et al*. 1996; Falconer and Mackay 1997) or incorporating information on MQTL effects by means of the MSI; other selection indexes are based on improving several traits simultaneously, which requires assigning economic weights to each trait, as proposed by Smith (1936).

Moreau *et al*. (2000) and Whittaker (2003) found that the MSI is more effective than Smith's selection index only in early generation testing and has the additional disadvantage of increased costs due to molecular marker evaluation. Selection intensity must also be considered because it affects genetic marker means and the ability to detect QTL (Wu *et al*. 2000). Furthermore, since selection increases the frequency of the QTL's favorable allele, as well as the allele of the molecular marker linked to it, total variability in the selected sample is reduced (Mackinnon and Georges 1992).

The MSI has the same advantages and disadvantages as Smith's selection index; it is simple to use but its sampling statistical properties and selection response are unknown, except in the case of two traits (Hayes and Hill 1980). Even for two traits, the statistical properties of Smith's selection index and its selection response, obtained using the delta method, are difficult to use and evaluate (Harris 1964); furthermore, it is not easy to consistently assign economic weights to the traits.

Recently, Cerón-Rojas *et al*. (2006) developed a selection index based on eigenanalyses of the phenotypic variance–covariance (or correlation) matrix of the traits of interest (called the eigen selection index method, ESIM). The authors showed that ESIM does not require economic weights or estimates of the genotypic variances–covariances. In ESIM the elements of the first eigenvector determine the proportion each trait contributes to the selection index, and the first eigenvalue is used in the selection response. From a theoretical perspective, Cerón-Rojas *et al*. (2006) demonstrated that selection responses from Smith's selection index and from ESIM are the same, except for differences in selection index coefficients due to the different estimation methods. In addition, the ESIM of Cerón-Rojas *et al*. (2006) allows constructing a function to estimate gains (or losses) between selection cycles and predicting the selection response for future selection cycles. Following the restrictive selection index of Kempthorne and Nordskog (1959), Cerón-Rojas *et al*. (2008) developed a restrictive ESIM (RESIM) that facilitates maximizing the genetic progress of some characters while leaving the others unchanged.

In this article we develop a molecular selection index (molecular eigen selection index method, MESIM) based on the RESIM of Cerón-Rojas *et al*. (2008) and the molecular selection index developed by Lande and Thompson (1990), using the selection index methodology proposed by Smith (1936), in which MQTL effects are incorporated. Simulated data were generated for comparing the selection response based on various selection indexes: (1) MESIM *vs*. Lande and Thompson (1990), (2) RESIM *vs*. the restrictive selection index of Kempthorne and Nordskog (1959), and (3) ESIM *vs*. the Smith selection index (Smith 1936). Practical and theoretical properties of estimators from , RESIM, ESIM, the Lande and Thompson molecular selection index, the Smith selection index, and the restrictive selection index of Kempthorne and Nordskog are discussed. The efficiency of MESIM, the Lande and Thompson molecular selection index, ESIM, the Smith selection index, and the restrictive selection index of Kempthorne and Nordskog is evaluated using the genotypic means of the selected individuals. The theory of RESIM is described in Cerón-Rojas *et al*. (2008).

## THEORY OF SELECTION INDEXES

#### Smith's selection index:

Details of Smith's selection index (SI) are given in Cerón-Rojas *et al*. (2006, 2008). A brief description follows. Smith's selection index is based on the linear combinations(1)where is the vector of the phenotypic values and is the vector of coefficients of , is the breeding value, is the vector of genotypic values, and is the vector of economic weights. The phenotypic values (*j* = 1, 2, … , *q*) are modeled as , where is the genotypic value of the *j*th trait and is the environmental component. Assuming that and are independent and that represents only additive effects, denotes the breeding value (Hazel 1943; Kempthorne and Nordskog 1959). Hence, selection based on leads to a selection response(2)where and are the variance–covariance matrices of genotypic and phenotypic values, respectively, is the standardized selection differential, is the covariance between and , is the variance of , is the variance of , and is the correlation between and .

In Smith's selection index, the vector (where the subscript S denotes Smith's method and is the inverse of the phenotypic variance–covariance matrix, ) allows us to construct the SI, , that maximizes the correlation with the breeding value .

#### Molecular selection index:

Lande and Thompson (1990) extended Equation 1 to include the case where information on QTL associated with molecular markers is available and denoted the molecular selection index as(3)where is a vector of phenotypic weights, is the vector of weights of the molecular score, is the vector of phenotypic values, and , where each (*j* = 1, 2, … , *N*; *N* = number of molecular scores) is the *j*th molecular score given by the the sum of the products of the estimated additive effect of the QTL linked to the molecular marker (MQTL effects) multiplied by the coded values of their corresponding molecular markers. The response to this molecular selection index may be written as(4)where has been defined as in Equation 2, is the variance of the breeding value (), is a vector of economic weights (in the standard molecular selection index, is a vector of zeros), is a vector containing phenotypic () and molecular () weight scores, and are the variance–covariance matrices defined in Equation 2, and is the variance–covariance matrix of the molecular scores when two or more traits are considered (Lande and Thompson 1990). Only statistically significant additive MQTL effects are included in .

The vector allows constructing the molecular selection index that has maximum correlation () with (the subscript MSI in denotes Lande and Thompson's molecular selection index method). In , (Equation 3). The variance of is and the maximized selection response can be written as . Estimators of and ( and ) for various traits are obtained directly from the estimators of , , and (, , and ) and from the vector .

## MESIM

Using a concept similar to that of Kempthorne and Nordskod (1959), which maximizes the selection response (Equation 2) by maximizing the square of the correlation between *Y* and *Z* (Equation 1) and utilizing basic concepts from Cerón-Rojas *et al*. (2008), it can be shown that Equation 4 is maximized by maximizing . The key point when maximizing is that the variances (or standard deviations) of and are constants in each selection cycle. Thus, the selection of genotypes can be done using either or . Because of this fact, when maximizing it is possible to impose restrictions and such that, in MESIM, it is required to maximizewith respect to , , , and , where is the vector of coefficients, is the vector of economic weights, and and are Lagrange multipliers. In MESIM it is assumed that is not a vector of constants.

When is derived with respect to and (appendix) and the result is set to the null vector, it follows that(5)(6)Because the two restrictions and , when Equation 5 is multiplied by and Equation 6 is multiplied by , the result is . Hence, maximizes under the restrictions and .

The following task is to determine the vector that allows constructing that maximizes its correlation with . The appendix shows that the required is the solution to the equality(7)where . Thus, for , the value that maximizes under restrictions and is the first eigenvalue () of matrix , and the vector that allows constructing (with maximum correlation with ) is the first eigenvector of matrix .

Let and be the first (largest) eigenvalue and its corresponding **Q** eigenvector, respectively; then, the selection index in the context of is and, because , the maximized selection response can be written as . From it is possible to determine the -coefficients of (Equation 3), . Although the partial derivatives of are obtained with respect to and , in estimating and , the vector of economic weights () is not required because and are obtained directly from matrix .

Note that when information on the QTL linked to the molecular markers is not incorporated into the selection index, *i.e*., when , , and , then Equation 7 can be written as(8)from which it is evident that . Equation 8 can be considered a variant of the procedure developed by Cerón-Rojas *et al*. (2006) for cases where the assumption of ESIM () is relaxed.

As indicated by Ceron-Rojas *et. al*. (2008), the maximized selection response, or , gives a general theoretical assessment of the gain for all traits considered simultaneously but does not provide genetic gains per trait at each selection cycle. Alternatively, the expected selection response (Baker 1986; Van Vleck 1993) determines the expected genetic gain per trait per selection cycle . However, estimates the expected value of the genetic gains with low precision; thus in our simulated data we used the genotypic means of the selected individuals and the regression of the genotypic means of the selected individuals on the selection cycles for evaluating the efficiency of MESIM, RESIM, ESIM, the Lande and Thompson molecular selection index, the restrictive selection index of Kempthorne and Nordskog, and the Smith selection index on the response to selection.

Matrix is square and nonsymmetric of order (where is the total number of variables: phenotypic and molecular scores):(9)Therefore, it is not possible to construct a subset of orthogonal vectors from Equation 7. However, orthogonal vectors from **Q** can be calculated by means of singular value decomposition (SVD) (Mardia *et al*. 1982). Using SVD, can be written as(10)where the columns of matrix () are the left singular vector of , and the columns of matrix () are the right singular vector of is a diagonal matrix with the square root of the eigenvalues (singular values) of or (the eigenvalues of and are the same).

The problem now is to determine the following: From where should the first singular vector for constructing be taken, from or from ? Note that Equation 10 can be written as , from which it is evident that if is the first singular value of , and and are its associated left and right first singular vectors, respectively, then , from which . Let ; then is a linear transformation of . The estimators of and are obtained from , such that . According to Anderson (2003), and are the maximum-likelihood estimators of the eigenvector and the eigenvalue of , respectively, and are asymptotically consistent and unbiased. The estimators of , , , and are , , , and , respectively, so . These results allow estimating as . Asymptotically, .

When only one trait and its molecular scores are considered,and

When , thenwhere and are the phenotypic and genotypic variances of the trait, respectively, is the variance of the molecular score associated with the trait under selection, and .

## SIMULATED DATA

We have simulated genotypes from a population with the aim of comparing theoretical and practical results from MESIM, RESIM, ESIM, the restrictive selection index of Kempthorne and Nordskog (1959), the Smith (1936) selection index, and the Lande–Thompson (Lande and Thompson 1990) molecular selection index. The simulator system used in this study, developed by Wang *et al*. (2004), has two main engines, QU-GENE and QuCim, which require different input data. To simulate a population, the input file for QU-GENE should contain the genetic structure of the genotypes for each specific trait: *i.e*., number of genes (or QTL); gene effect for each trait including additive, dominance, and epistasis; linkage among the genes in one chromosome; and trait heritability, etc. Component QU-GENE can generate genotypes making up populations of cross-pollinated or self-pollinated species or create different environmental conditions where the simulated genotypes will be evaluated. On the other hand, the input file for QuCim must have the type of crosses and the selection method to be used in each breeding strategy. Selection methods that can be simulated in QuCim include mass selection, pedigree system, bulk population system, backcross breeding, top-cross breeding, doubled-haploid breeding, marker-assisted selection for one trait, and many combinations and modifications of these (Wang *et al*. 2004). The simulator provides, for each genotype in the population, the true genotypic value as well as the phenotypic value of the traits under study.

#### Generating a doubled-haploid population for selection:

The original data were taken from an actual doubled-haploid maize mapping population of 236 genotypes with five traits; QTL for all five traits were mapped. The five traits measured were male flowering time (MFL) (days), female flowering time (FFL) (days), plant height (PHT) (centimeters), ear height (EHT) (centimeters), and 100-kernel weight (HKF) (grams). This data file was used to generate 200 doubled-haploid genotypes that form the reference population (cycle 0). Using a selection pressure of 10% (*k* = 1.755), 20 genotypes were selected under MESIM, the Lande–Thompson selection index, ESIM, RESIM, the Smith selection index, and the restrictive selection index of Kempthorne and Nordskog. These 20 selected doubled haploids were then crossed in diallel fashion, and a new population of 200 doubled haploids was generated. This was repeated during five selection cycles for all five traits. The efficiency of the indexes was compared, using the mean genotypic value and the regression of the mean genotypic value of the selected genotypes on the selection cycles. We used phenotypic, genotypic, and molecular score variance–covariance matrices for estimating the singular vectors and singular values.

We also generated populations on the basis of selection of individual traits with the objective of comparing MESIM and the Lande–Thompson (Lande and Thompson 1990) molecular selection index method for the simultaneous selection of five traits.

#### Sign of the coefficients, economic weights, and expected genetic gains:

When using MESIM, ESIM, and RESIM, it is often necessary to change the sign of the coefficients of the first singular eigenvector to select the genotypes according to the desired genetic advance; that is, for traits such as MFL, FFL, PHT, and EHT, the signs are always negative (decreasing the mean genotypic value), whereas for HKF the signs are always positive (increasing the mean genotypic value).

Concerning the economic weights for the Lande–Thompson molecular selection index, the restrictive selection index of Kempthorne and Nordskog (1959), and the Smith (1936) selection index, economics weights were assigned following Smith *et al*. (1981). Then, one set had coefficients of 1 or −1, and the other had the heritability of each trait multiplied by 1 or −1, depending on the trait. Therefore, for MFL, FFL, PHT, EHT, and HKF, the first set of coefficients was −1, −1, −1, −1, and 1, respectively, whereas the second set of coefficients was −, , , , and (0.27); all economic weights of the molecular markers associated with the traits were equal to zero. All five traits were simultaneously selected under MESIM, the Lande–Thompson selection index, ESIM, and the Smith selection index, whereas for the restrictive selection index of Kempthorne and Nordskog and RESIM, the traits that were unchanged were MFL and PHT.

Furthermore, MESIM and the Lande–Thompson selection index were compared when traits were selected individually. When selection was performed on individual traits, the Lande–Thompson molecular selection index based on heritabilities as economic weights was not applied, and only the index based on coefficients 1 and −1 (depending on the trait of interest), and 0 for the economic weights, was employed.

## RESULTS AND DISCUSSION

The genotypic means under MESIM and the Lande–Thompson selection index when selection is practiced on traits individually (not simultaneously on various traits) are shown in Table 1. Because genetic variability became exhausted, only two selection cycles were run. The MESIM-selected genotypes had better genotypic means than those selected under the Lande–Thompson index for all five traits. To clarify the interpretation of the MESIM, consider, for example, the first selection cycle on the individual-trait MFL. The estimated phenotypic, genotypic, and molecular score variances in the original population were , , and , respectively, from whichThe first singular value and its associated singular vector are and , respectively. However, because MFL decreases, it is necessary to multiply the elements of by −1 such that the selection index in the context of MESIM is , where MFL denotes the trait of interest, and is the molecular score associated with MFL. In this case, the total expected genetic response can be partitioned into two components, the coefficient related to the phenotypic values *per se* and those related to the molecular scores. Value −0.233 is the phenotypic coefficient, and −0.972 is the molecular score coefficient.

When selection is practiced on all five traits simultaneously, then economic weights −1, −1, −1, −1, and 1 for each trait are used; the heritability of the traits is also used as weights. The Lande–Thompson molecular selection index is denoted as Lande–Thompson 1 when −1, −1, −1, −1, and 1 are used as economic weights, and when heritabilities are used as economic weights, it is denoted Lande–Thompson 2. Similarly, the standard Smith selection index is denoted as Smith 1 in the first case and Smith 2 in the second case; and the Kempthorne–Nordskog restricted selection indexes are denoted as KN1 and KN2, respectively.

For the trait HKF, the selection gain per cycle for MESIM (0.50 g) was greater than that obtained by Lande–Thompson 1 (0.21 g) and Lande–Thompson 2 (0.31 g) (Table 2). However, for MFL, the opposite was true; that is, Lande–Thompson 1 (−0.91 days) and Lande–Thompson 2 (−0.83 days) under both sets of economic weights were more effective than MESIM (−0.71 days) for maturity (Table 2). Comparing the genotypic means when individual traits are selected (Table 1) with those obtained when five traits are simultaneously selected (Table 2), it is evident that the genotypic means are higher when only one trait is under selection. Correlations between traits play an important role in the correlated response of other traits.

Regarding the Smith SI and ESIM, the genotypic means of the selected genotypes are shown in Table 3. In this case, for four of the five traits, MFL, FFL, EHT, and HKF, the selection gain per cycle of ESIM was greater than that obtained with the Smith SI. Concerning KN restricted (R)SI and RESIM (keeping MFL and PHT unchanged), the genotypic means of the selected genotypes are shown in Table 4. For HKF, the selection gain per cycle for RESIM (0.48 g) was greater than that obtained using KN1 RSI (0.27 g) and KN2 RSI (0.19 g). However, for FFL, the opposite was true; that is, KN1 RSI (−1.05 days) and KN2 RSI (−1.10 days) under both sets of economic weights were more effective than RESIM (−0.92 days) for maturity. The effective selection gain per cycle estimated as the linear regression of the mean genotypic trait value on the selection cycle is also shown in the last row of Tables 3 and 4.

Figures 1–3⇓⇓ show the genotypic means for HKF, FFL, and MFL for five selection cycles when the genotypes are selected under different selection indexes. Increasing trends in the genotypic means of the selected genotypes for the five selection cycles under MESIM, Lande–Thompson 1 and 2, ESIM, Smith 1 and 2, RESIM, and Kempthorne–Nordskog for HKF are shown in Figure 1. Clearly, MESIM selected genotypes with higher HFK in all cycles. For FFL (Figure 2) ESIM was the best in all cycles, whereas MESIM was better than Lande–Thompson 1 and 2 in the last three cycles. For MFL, Figure 3 shows that MESIM results are similar to those of Lande–Thompson 1 and 2. However, ESIM is still the selection index that gave the highest response to selection. Furthermore, note that since MFL was unchanged when applying the restrictive selection indexes (RESIM, KN1, and KN2), their genotypic means did not change over the selection cycles and stayed around the mean of cycle 0 (Figure 3).

As previously indicated, the molecular selection indexes (MESIM and Lande–Thompson) depend on the heritability of each trait. According to Lande and Thompson (1990), Zhang and Smith (1992, 1993), Gimelfarb and Lande (1994, 1995), and Whittaker (2003), the molecular selection index is expected to be more efficient than the standard selection indexes (*i.e*., ESIM and Smith's selection index) when the heritability of the trait is low. Figure 1 shows the genotypic means of HKF with a heritability of 0.27, whereas Figures 2 and 3 depict the genotypic means of the selected genotypes for FFL and MFL), with heritabilities of 0.46 and 0.51, respectively. This would explain why MESIM was more efficient than the other indexes for selecting the genotypes with the highest genotypic means. Detailed descriptions of ESIM, RESIM, and the Smith selection index can be found in Cerón- Rojas *et al*. (2008). For the other traits, the gains of MESIM over Lande–Thompson 1 and 2 are not as clear as for HKF and FFL (Tables 2–4⇑⇑). However, when traits are selected individually, the genotypic mean obtained for MESIM is higher than that achieved by Lande–Thompson for most traits (Table 1).

It is worth noting that when the eigenvectors are obtained from the variance–covariance phenotypic and genotypic matrices, then MESIM, ESIM, and RESIM assign weights proportional to the heritability of the trait; that is, the higher the heritability, the more weight, and vice versa. As mentioned by Cerón-Rojas *et al*. (2006), a solution would be to use the phenotypic and genotypic correlation matrices. Another solution would be to use the inverse of and thus give more weight to traits with low heritability. The latter solution for constructing MESIM comes naturally from Equation 7, since and can be written as , from which . Then the equation to obtain the eigenvectors is , in which casewhen only one trait and its molecular scores are considered, thenand when ,from which it is evident that traits with low heritability will have higher weights.

Finally, it is worth noting that although MESIM, ESIM, and RESIM may occasionally not to turn out to be the indexes with the highest selection gains, they have the statistical properties of the principal components. According to Okamoto (1969), these are optimal properties established in terms of maximization and minimization. Thus the first component has the largest variance and the smallest loss of information (Rao 1964). On the other hand, statistical properties of other selection indexes are unknown.

This research found that MESIM has three advantages over Lande–Thompson 1 and 2: first, it can be used to solve practical problems faced by breeders attempting to select plants or animals for the next generation when no estimates of economic weights are available. Even if economic weights are available, in practice it is very unlikely that they would maximize the derivative of with respect to and to (under the imposed restrictions). Furthermore, if two breeders are interested in improving, say, *n* traits, it is very unlikely that they would assign the same weights to them. Second, estimates of MESIM have known statistical sampling properties, but estimates for the Lande–Thompson molecular selection index are unknown. Third, results from MESIM using simulated data show that realized genetic gains for various traits simultaneously are similar to, or higher than, those obtained by Lande and Thompson (1990).

## CONCLUSIONS

This research presents a molecular selection index based on principles developed by Cerón-Rojas *et al*. (2008). Simulated results show that when genotypes are selected on the basis of individual traits, MESIM increased the response to selection over the Lande–Thompson index. When several traits are selected simultaneously, MESIM outperformed Lande–Thompson for traits with low heritability. For traits with high heritability, ESIM performed very well. One of the most important results of MESIM is that is the maximum-likelihood estimate of , whereas is an estimate of , whose sampling properties are unknown. MESIM can be considered a generalization of ESIM (Cerón-Rojas *et al*. 2006) when information on QTL is incorporated through molecular markers. The sampling properties of ESIM (and therefore of MESIM) and its selection response are known, and its estimators showed desirable statistical properties such as consistency and asymptotic unbiasedness.

It should be pointed out that MESIM is more general than ESIM (Cerón-Rojas *et al*. 2006) because the basic underlying assumption made in ESIM, , is relaxed in MESIM. MESIM maximizes the selection response by maximizing the square of the correlation between and , , which is the same as maximizing . This basic idea, used for developing a restrictive selection index (Cerón-Rojas *et al*. 2008), is valid for MESIM when no restrictions are imposed on any of the traits.

Some advantages of MESIM over MSI should be pointed out: (1) the sampling properties of MESIM, , are known and easy to evaluate; (2) the MESIM eigenvalue and eigenvector are estimated by the maximum-likelihood method; and (3) a restrictive SI can be developed from MESIM when only some markers and/or traits are used. In summary, the results of this study indicate that MESIM is a generalization of ESIM when information on QTL linked to molecular markers is incorporated.

The availability of abundant molecular markers can help to achieve faster breeding progress than with traditional breeding methods or marker-assisted selection by means of genomewide selection (Bernardo and Yu 2007). The MESIM could be a valid option for a genomewide selection method because the serious problem of parameter identification created by the collinearity of the markers is overcome by the singular value decomposition method of MESIM. Furthermore, MESIM naturally performs cross-product between all trait–environment combinations and markers; thus it implicitly introduces estimates of particular epistatic interactions into the seletion index. Further research on the use of MESIM in genomewide selection is required.

## APPENDIX: THEORETICAL DERIVATION OF MESIM

The procedure shown below is a slight modification of that used by Cerón-Rojas *et al*. (2008) within the context of a restricted selection index method based on eigenanalysis (RESIM). In this case, must be maximized under the restrictions and ; *i.e*., we should maximizewith respect to , , , and , where is the vector of coefficients, is the vector of economic weights, and and are Lagrange multipliers. In MESIM it is assumed that is not a vector of constants.

When is derived with respect to , , , and , and the result is set to the null vector, it follows that(A1)(A2)(A3)(A4)where Equations A3 and A4 denote the restrictions imposed for the maximization of with respect to and . Because the restrictions and , when Equation A1 is multiplied by and Equation A2 is multiplied by , both equations can be written asClearly, . Therefore, maximizes under the restrictions and .

The following problem is to determine the vector , which allows constructing the selection index that has maximum correlation with . Because , Equations A1 and A2 can be written as(A5)(A6)Multiplying Equation A5 by , we obtain that , from which is computed.

Substitute, in Equation A6, for and get , from which Equation 7 (see the text) is obtained,where , and and are the eigenvalue and the eigenvector of , respectively. Thus, for , the values that maximize under the restrictions and are the eigenvalues () of the matrix and its eigenvector vector, , that allows constructing the index that maximizes its correlation with .

## Acknowledgments

The authors are grateful to Jiankang Wang for his valuable assistance and help when running the simulation software QU-GENE. The authors thank the associate editor and two anonymous reviewers for their comments and suggestions, which significantly improved the quality of this article. The authors are thankful to BIMBO-Mexico for partially funding this research.

## Footnotes

Communicating editor: R. W. Doerge

- Received April 28, 2008.
- Accepted June 18, 2008.

- Copyright © 2008 by the Genetics Society of America