mlgsi: Multistage Linear Genomic Selection Index (MLGSI)

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

Multistage Linear Genomic Selection Index (MLGSI)

Description

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

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}}}

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

References

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

Examples

## Not run: 
# 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)

## End(Not run)

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