lmsi: Linear Marker Selection Index (LMSI)

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

Linear Marker Selection Index (LMSI)

Description

Implements the LMSI which combines phenotypic information with molecular marker scores from statistically significant markers (Lande & Thompson, 1990). The index is I = b_y' * y + b_s' * s, where y are phenotypes and s are marker scores.

Usage

lmsi(
  phen_mat = NULL,
  marker_scores = NULL,
  pmat,
  gmat,
  G_s = NULL,
  wmat,
  wcol = 1,
  selection_intensity = 2.063,
  GAY = NULL
)

Arguments

phen_mat	Matrix of phenotypes (n_genotypes x n_traits). Can be NULL if G_s is provided directly (theoretical case where covariance structure is known without needing empirical data).
marker_scores	Matrix of marker scores (n_genotypes x n_traits). These are computed as s_j = sum(x_jk * beta_jk) where x_jk is the coded marker value and beta_jk is the estimated marker effect for trait j. Can be NULL if G_s is provided directly.
pmat	Phenotypic variance-covariance matrix (n_traits x n_traits).
gmat	Genotypic variance-covariance matrix (n_traits x n_traits).
G_s	Genomic covariance matrix explained by markers (n_traits x n_traits). This represents Var(s) which approximates Cov(y, s) when markers fully explain genetic variance. If provided, phen_mat and marker_scores become optional as the covariance structure is specified directly. If NULL, computed empirically from marker_scores and phen_mat.
wmat	Economic weights matrix (n_traits x k), or vector.
wcol	Weight column to use if wmat has multiple columns (default: 1).
selection_intensity	Selection intensity k (default: 2.063 for 10% selection).
GAY	Optional. Genetic advance of comparative trait for PRE calculation.

Details

Mathematical Formulation:

The LMSI maximizes the correlation between the index I_{LMSI} = \mathbf{b}_y^{\prime}\mathbf{y} + \mathbf{b}_s^{\prime}\mathbf{s} and the breeding objective H = \mathbf{w}^{\prime}\mathbf{g}.

Combined covariance matrices:

\mathbf{P}_L = \begin{bmatrix} \mathbf{P} & \text{Cov}(\mathbf{y}, \mathbf{s}) \\ \text{Cov}(\mathbf{y}, \mathbf{s})^{\prime} & \text{Var}(\mathbf{s}) \end{bmatrix}

\mathbf{G}_L = \begin{bmatrix} \mathbf{G} \\ \mathbf{G}_s \end{bmatrix}

where \mathbf{P} is the phenotypic variance, \text{Cov}(\mathbf{y}, \mathbf{s}) is the covariance between phenotypes and marker scores (computed from data), \text{Var}(\mathbf{s}) is the variance of marker scores, \mathbf{G} is the genotypic variance, and \mathbf{G}_s represents the genetic covariance explained by markers.

Index coefficients:

\mathbf{b}_{LMSI} = \mathbf{P}_L^{-1} \mathbf{G}_L \mathbf{w}

Accuracy:

\rho_{HI} = \sqrt{\frac{\mathbf{b}_{LMSI}^{\prime} \mathbf{G}_L \mathbf{w}}{\mathbf{w}^{\prime} \mathbf{G} \mathbf{w}}}

Selection response:

R_{LMSI} = k \sigma_{I_{LMSI}} = k \sqrt{\mathbf{b}_{LMSI}^{\prime} \mathbf{P}_L \mathbf{b}_{LMSI}}

Expected genetic gain per trait:

\mathbf{E}_{LMSI} = k \frac{\mathbf{G}_L^{\prime} \mathbf{b}_{LMSI}}{\sigma_{I_{LMSI}}}

Value

List of class "lmsi" with components:

b_y

Coefficients for phenotypes (n_traits vector).

b_s

Coefficients for marker scores (n_traits vector).

b_combined

Combined coefficient vector [b_y; b_s] (2*n_traits vector).

P_L

Combined phenotypic-marker covariance matrix (2*n_traits x 2*n_traits).

G_L

Combined genetic-marker covariance matrix (2*n_traits x n_traits).

G_s

Genomic covariance matrix explained by markers (n_traits x n_traits).

rHI

Index accuracy (correlation between index and breeding objective).

sigma_I

Standard deviation of the index.

R

Selection response (k * sigma_I).

Delta_H

Expected genetic gain per trait (vector of length n_traits).

GA

Overall genetic advance in breeding objective.

PRE

Percent relative efficiency (if GAY provided).

hI2

Index heritability.

summary

Data frame with coefficients and metrics (combined view).

phenotype_coeffs

Data frame with phenotype coefficients only.

marker_coeffs

Data frame with marker score coefficients only.

coeff_analysis

Data frame with coefficient distribution analysis.

References

Lande, R., & Thompson, R. (1990). Efficiency of marker-assisted selection in the improvement of quantitative traits. Genetics, 124(3), 743-756.

Cerón-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Chapter 4.

Examples

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

# Simulate marker scores (in practice, computed from QTL mapping)
set.seed(123)
n_genotypes <- 100
n_traits <- ncol(gmat)
marker_scores <- matrix(rnorm(n_genotypes * n_traits, mean = 5, sd = 1.5),
  nrow = n_genotypes, ncol = n_traits
)
colnames(marker_scores) <- colnames(gmat)

# Simulate phenotypes
phen_mat <- matrix(rnorm(n_genotypes * n_traits, mean = 15, sd = 3),
  nrow = n_genotypes, ncol = n_traits
)
colnames(phen_mat) <- colnames(gmat)

# Economic weights
weights <- c(10, 5, 3, 3, 5, 8, 4)

# Calculate LMSI
result <- lmsi(phen_mat, marker_scores, pmat, gmat,
  G_s = NULL, wmat = weights
)
print(result$summary)

## End(Not run)

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