Linear Marker Selection Index (LMSI) — lmsi • selection.index

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.

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.

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

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

if (FALSE) { # \dontrun{
# 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)
} # }