Genome-Wide Linear Marker Selection Index (GW-LMSI)

Implements the GW-LMSI which uses all available genome-wide markers directly as predictors in the selection index. Unlike LMSI which uses aggregated marker scores per trait, GW-LMSI treats each individual marker as a separate predictor.

Usage

gw_lmsi(
  marker_mat,
  trait_mat = NULL,
  gmat,
  P_GW = NULL,
  G_GW = NULL,
  wmat,
  wcol = 1,
  lambda = 0,
  selection_intensity = 2.063,
  GAY = NULL
)

Arguments

marker_mat: Matrix of marker genotypes (n_genotypes x n_markers). Typically coded as -1, 0, 1 or 0, 1, 2.
trait_mat: Matrix of trait values (n_genotypes x n_traits). Used to compute G_GW if not provided.
gmat: Genotypic variance-covariance matrix (n_traits x n_traits).
P_GW: Marker covariance matrix (n_markers x n_markers). If NULL, computed as Var(marker_mat).
G_GW: Covariance between markers and traits (n_markers x n_traits). If NULL, computed as Cov(marker_mat, trait_mat).
wmat: Economic weights matrix (n_traits x k), or vector.
wcol: Weight column to use if wmat has multiple columns (default: 1).
lambda: Ridge regularization parameter (default: 0). If lambda > 0, uses P_GW + lambda*I for regularization. Automatic warnings issued when n_markers > n_genotypes (high-dimensional case) or when P_GW is ill-conditioned. Recommended values: 0.01-0.1 times mean(diag(P_GW)).
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 "gw_lmsi" with components:

b: Index coefficients for markers (n_markers).
P_GW: Marker covariance matrix (n_markers x n_markers).
G_GW: Covariance between markers and traits (n_markers 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.
lambda: Ridge regularization parameter used.
n_markers: Number of markers.
high_dimensional: Logical indicating if n_markers > n_genotypes.
condition_number: Condition number of P_GW (if computed).
summary: Data frame with metrics.

Details

Mathematical Formulation:

The GW-LMSI maximizes the correlation between the index $I_{GW} = \mathbf{b}_{GW}^{\prime}\mathbf{m}$ and the breeding objective $H = \mathbf{w}^{\prime}\mathbf{g}$.

Marker covariance matrix: $$\mathbf{P}_{GW} = Var(\mathbf{m})$$

Covariance between markers and traits: $$\mathbf{G}_{GW} = Cov(\mathbf{m}, \mathbf{g})$$

Index coefficients (with optional Ridge regularization): $$\mathbf{b}_{GW} = (\mathbf{P}_{GW} + \lambda \mathbf{I})^{-1} \mathbf{G}_{GW} \mathbf{w}$$

Accuracy: $$\rho_{HI} = \sqrt{\frac{\mathbf{b}_{GW}^{\prime} \mathbf{G}_{GW} \mathbf{w}}{\mathbf{w}^{\prime} \mathbf{G} \mathbf{w}}}$$

Selection response: $$R_{GW} = k \sqrt{\mathbf{b}_{GW}^{\prime} \mathbf{P}_{GW} \mathbf{b}_{GW}}$$

Expected genetic gain per trait: $$\mathbf{E}_{GW} = k \frac{\mathbf{G}_{GW}^{\prime} \mathbf{b}_{GW}}{\sigma_{I_{GW}}}$$

Note on Singularity Detection and Regularization: The function automatically detects problematic cases:

1. **High-dimensional case**: When n_markers > n_genotypes, P_GW is mathematically singular (rank-deficient). The function issues a warning and suggests an appropriate lambda value.

2. **Ill-conditioned case**: When P_GW has a high condition number (> 1e10), indicating numerical instability.

3. **Numerical singularity**: When P_GW has eigenvalues near zero.

Ridge regularization adds $\lambda$I to P_GW, ensuring positive definiteness. Recommended lambda values are 0.01-0.1 times the average diagonal element of P_GW. Users can also set lambda = 0 to force generalized inverse (less stable but sometimes needed).

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)
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Simulate marker data
set.seed(123)
n_genotypes <- 100
n_markers <- 200
n_traits <- ncol(gmat)

# Marker matrix (coded as 0, 1, 2)
marker_mat <- matrix(sample(0:2, n_genotypes * n_markers, replace = TRUE),
  nrow = n_genotypes, ncol = n_markers
)

# Trait matrix
trait_mat <- matrix(rnorm(n_genotypes * n_traits, mean = 15, sd = 3),
  nrow = n_genotypes, ncol = n_traits
)

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

# Calculate GW-LMSI with Ridge regularization
result <- gw_lmsi(marker_mat, trait_mat, gmat,
  wmat = weights, lambda = 0.01
)
print(result$summary)
} # }