mesim: Molecular Eigen Selection Index Method (MESIM)

In selection.index: Analysis of Selection Index in Plant Breeding

Molecular Eigen Selection Index Method (MESIM)

Description

Implements the MESIM by maximising the squared accuracy through the generalised eigenproblem combining phenotypic data with molecular marker scores. Unlike Smith-Hazel LPSI, **no economic weights are required**.

Usage

mesim(pmat, gmat, S_M, S_Mg = NULL, S_var = NULL, selection_intensity = 2.063)

Arguments

pmat	Phenotypic variance-covariance matrix (n_traits x n_traits).
gmat	Genotypic variance-covariance matrix (n_traits x n_traits).
S_M	Covariance between phenotypes and marker scores (n_traits x n_traits). This represents Cov(y, s) where s are marker scores. Used in the phenotypic variance matrix T_M.
S_Mg	Covariance between genetic values and marker scores (n_traits x n_traits). This represents Cov(g, s). Used in the genetic variance matrix Psi_M. If NULL (default), uses S_M, which is appropriate when assuming Cov(e, s) ~= 0 (errors uncorrelated with markers), so Cov(y,s) ~= Cov(g,s).
S_var	Variance-covariance matrix of marker scores (n_traits x n_traits). Represents Var(s). If NULL (default), uses S_M for backward compatibility, but providing the actual Var(s) is more statistically rigorous.
selection_intensity	Selection intensity constant k_I (default: 2.063 for 10% selection).

Details

Eigenproblem (Section 8.1):

(\mathbf{T}_M^{-1}\mathbf{\Psi}_M - \lambda_M^2 \mathbf{I}_{2t})\boldsymbol{\beta}_M = 0

where:

\mathbf{T}_M = \begin{bmatrix} \mathbf{P} & \mathrm{Cov}(\mathbf{y},\mathbf{s}) \\ \mathrm{Cov}(\mathbf{s},\mathbf{y}) & \mathrm{Var}(\mathbf{s}) \end{bmatrix}

\mathbf{\Psi}_M = \begin{bmatrix} \mathbf{C} & \mathrm{Cov}(\mathbf{g},\mathbf{s}) \\ \mathrm{Cov}(\mathbf{s},\mathbf{g}) & \mathrm{Var}(\mathbf{s}) \end{bmatrix}

Theoretical distinction:

• T_M uses phenotypic covariances: Cov(y, s) provided via S_M

• Psi_M uses genetic covariances: Cov(g, s) provided via S_Mg

• Since y = g + e, if Cov(e, s) ~= 0, then Cov(y, s) ~= Cov(g, s)

• Chapter 8.1 assumes marker scores are pure genetic predictors, so S_M can be used for both (default behavior when S_Mg = NULL)

The solution \lambda_M^2 (largest eigenvalue) equals the maximum achievable index heritability.

Key metrics:

R_M = k_I \sqrt{\boldsymbol{\beta}_{M}^{\prime}\mathbf{T}_M\boldsymbol{\beta}_{M}}

\mathbf{E}_M = k_I \frac{\mathbf{\Psi}_M\boldsymbol{\beta}_{M}}{\sqrt{\boldsymbol{\beta}_{M}^{\prime}\mathbf{T}_M\boldsymbol{\beta}_{M}}}

Value

Object of class "mesim", a list with:

summary

Data frame with coefficients and metrics.

b_y

Coefficients for phenotypic data.

b_s

Coefficients for marker scores.

b_combined

Combined coefficient vector.

E_M

Expected genetic gains per trait.

sigma_I

Standard deviation of the index.

hI2

Index heritability (= leading eigenvalue).

rHI

Accuracy r_{HI}.

R_M

Selection response.

lambda2

Leading eigenvalue (maximised index heritability).

selection_intensity

Selection intensity used.

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 8.1.

Examples

## Not run: 
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Simulate marker score matrices (in practice, compute from data)
S_M <- gmat * 0.7 # Cov(y, s) - phenotype-marker covariance
S_Mg <- gmat * 0.65 # Cov(g, s) - genetic-marker covariance
S_var <- gmat * 0.8 # Var(s) - marker score variance

# Most rigorous: Provide all three covariance matrices
result <- mesim(pmat, gmat, S_M, S_Mg = S_Mg, S_var = S_var)
print(result)

# Standard usage: Cov(g,s) defaults to Cov(y,s) when errors uncorrelated
result_standard <- mesim(pmat, gmat, S_M, S_var = S_var)

# Backward compatible: Chapter 8.1 simplified notation
result_simple <- mesim(pmat, gmat, S_M)

## End(Not run)

selection.index documentation built on March 9, 2026, 1:06 a.m.