Multistage Linear Genomic Selection Index (MLGSI)

Implements the two-stage Linear Genomic Selection Index where selection is based on GEBVs at both stages with covariance adjustments due to selection effects.

Usage

mlgsi(
  Gamma1,
  Gamma,
  A1,
  A,
  C,
  G1,
  P1,
  wmat,
  wcol = 1,
  selection_proportion = 0.1,
  use_young_method = FALSE,
  k1_manual = 2.063,
  k2_manual = 2.063,
  tau = NULL,
  reliability = NULL
)

Arguments

Gamma1: GEBV variance-covariance matrix for stage 1 traits (n1 x n1)
Gamma: GEBV variance-covariance matrix for all traits at stage 2 (n x n)
A1: Covariance matrix between GEBVs and true breeding values for stage 1 (n1 x n1)
A: Covariance matrix between GEBVs and true breeding values for stage 2 (n x n1)
C: Genotypic variance-covariance matrix for all traits (n x n)
G1: Genotypic variance-covariance matrix for stage 1 traits (n1 x n1)
P1: Phenotypic variance-covariance matrix for stage 1 traits (n1 x n1)
wmat: Economic weights vector or matrix (n x k)
wcol: Weight column to use if wmat has multiple columns (default: 1)
selection_proportion: Proportion selected at each stage (default: 0.1)
use_young_method: Logical. Use Young's method for selection intensities (default: FALSE). Young's method tends to overestimate intensities; manual intensities are recommended.
k1_manual: Manual selection intensity for stage 1
k2_manual: Manual selection intensity for stage 2
tau: Standardized truncation point
reliability: Optional reliability vector for computing A matrices

Value

List with components:

beta1 - Stage 1 genomic index coefficients
w - Economic weights (stage 2 coefficients)
stage1_metrics - List with stage 1 metrics (R1, E1, rho_HI1)
stage2_metrics - List with stage 2 metrics (R2, E2, rho_HI2)
Gamma_star - Adjusted genomic covariance matrix at stage 2
C_star - Adjusted genotypic covariance matrix at stage 2
rho_I1I2 - Correlation between stage 1 and stage 2 indices
k1 - Selection intensity at stage 1
k2 - Selection intensity at stage 2
summary_stage1 - Data frame with stage 1 summary
summary_stage2 - Data frame with stage 2 summary

Details

Mathematical Formulation:

Stage 1: The genomic index is $I_1 = \mathbf{\beta}_1' \mathbf{\gamma}_1$

Coefficients: $\mathbf{\beta}_1 = \mathbf{\Gamma}_1^{-1}\mathbf{A}_1\mathbf{w}_1$

Stage 2: The index uses economic weights directly: $I_2 = \mathbf{w}' \mathbf{\gamma}$

Adjusted genomic covariance matrix: $$\mathbf{\Gamma}^* = \mathbf{\Gamma} - u \frac{\mathbf{A}_1'\mathbf{\beta}_1\mathbf{\beta}_1'\mathbf{A}_1}{\mathbf{\beta}_1'\mathbf{\Gamma}_1\mathbf{\beta}_1}$$

Adjusted genotypic covariance matrix: $$\mathbf{C}^* = \mathbf{C} - u \frac{\mathbf{G}_1'\mathbf{b}_1\mathbf{b}_1'\mathbf{G}_1}{\mathbf{b}_1'\mathbf{P}_1\mathbf{b}_1}$$

where $u = k_1(k_1 - \tau)$

Accuracy at stage 1: $\rho_{HI_1} = \sqrt{\frac{\mathbf{\beta}_1'\mathbf{\Gamma}_1\mathbf{\beta}_1}{\mathbf{w}'\mathbf{C}\mathbf{w}}}$

Accuracy at stage 2: $\rho_{HI_2} = \sqrt{\frac{\mathbf{w}'\mathbf{\Gamma}^*\mathbf{w}}{\mathbf{w}'\mathbf{C}^*\mathbf{w}}}$

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Chapter 9, Section 9.4.

Examples

if (FALSE) { # \dontrun{
# Two-stage genomic selection example
# Stage 1: Select based on GEBVs for 3 traits
# Stage 2: Select based on GEBVs for all 7 traits

# Compute covariance matrices
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Simulate GEBV covariances (in practice, compute from genomic prediction)
set.seed(123)
reliability <- 0.7
Gamma1 <- reliability * gmat[1:3, 1:3]
Gamma <- reliability * gmat
A1 <- reliability * gmat[1:3, 1:3]
A <- gmat[, 1:3]

# Economic weights
weights <- c(10, 8, 6, 4, 3, 2, 1)

# Run MLGSI
result <- mlgsi(
  Gamma1 = Gamma1, Gamma = Gamma, A1 = A1, A = A,
  C = gmat, G1 = gmat[1:3, 1:3], P1 = pmat[1:3, 1:3],
  wmat = weights, selection_proportion = 0.1
)

print(result$summary_stage1)
print(result$summary_stage2)
} # }