selection.index (version 2.0.0)

mppg_lgsi: Multistage Predetermined Proportional Gain Linear Genomic Selection Index (MPPG-LGSI)

Description

Implements the two-stage Predetermined Proportional Gain LGSI where breeders specify desired proportional gains between traits at each stage using GEBVs.

Usage

mppg_lgsi(
  Gamma1,
  Gamma,
  A1,
  A,
  C,
  G1,
  P1,
  wmat,
  wcol = 1,
  d1,
  d2,
  U1 = NULL,
  U2 = NULL,
  selection_proportion = 0.1,
  use_young_method = FALSE,
  k1_manual = 2.063,
  k2_manual = 2.063,
  tau = NULL
)

Value

List with components similar to mlgsi, plus:

• beta_P1 - PPG genomic stage 1 coefficients

• beta_P2 - PPG genomic stage 2 coefficients

• b_P1 - PPG phenotypic stage 1 coefficients (used for C* adjustment)

• theta1 - Proportionality constant for stage 1

• theta2 - Proportionality constant for stage 2

• gain_ratios_1 - Achieved gain ratios at stage 1

• gain_ratios_2 - Achieved gain ratios at stage 2

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)

d1

Vector of desired proportional gains for stage 1 (length n1)

d2

Vector of desired proportional gains for stage 2 (length n)

U1

Constraint matrix for stage 1 (n1 x r1), optional

U2

Constraint matrix for stage 2 (n x r2), optional

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

Details

Mathematical Formulation:

The PPG genomic coefficients are: $$\mathbf{\beta}_{P_1} = \mathbf{\beta}_{R_1} + \theta_1 \mathbf{U}_1(\mathbf{U}_1'\mathbf{\Gamma}_1\mathbf{U}_1)^{-1}\mathbf{d}_1$$ $$\mathbf{\beta}_{P_2} = \mathbf{\beta}_{R_2} + \theta_2 \mathbf{U}_2(\mathbf{U}_2'\mathbf{\Gamma}\mathbf{U}_2)^{-1}\mathbf{d}_2$$

where proportionality constants are: $$\theta_1 = \frac{\mathbf{d}_1'(\mathbf{U}_1'\mathbf{\Gamma}_1\mathbf{U}_1)^{-1}\mathbf{U}_1'\mathbf{A}_1\mathbf{w}}{\mathbf{d}_1'(\mathbf{U}_1'\mathbf{\Gamma}_1\mathbf{U}_1)^{-1}\mathbf{d}_1}$$

Covariance Adjustment:

The genetic covariance matrix \(\mathbf{C}^*\) is adjusted using phenotypic PPG coefficients \(\mathbf{b}_{P1} = \mathbf{P}_1^{-1}\mathbf{G}_1\mathbf{P}_1^{-1}\mathbf{d}_1\), which reflect the same proportional gain constraints as the genomic coefficients \(\mathbf{\beta}_{P1}\). This ensures the adjustment reflects the actual PPG selection occurring at stage 1.

Important: When using custom U1 matrices (subset constraints), the phenotypic proxy \(\mathbf{b}_{P1}\) uses the standard Tallis formula (all traits constrained), while the genomic index \(\mathbf{\beta}_{P1}\) respects the U1 subset. This may cause \(\mathbf{C}^*\) to be slightly over-adjusted. For exact adjustment, use U1 = NULL (default, all traits constrained). Calculating the exact restricted phenotypic proxy would require implementing the full MPPG-LPSI projection matrix method for the phenotypic coefficients.

Note: Input covariance matrices (C, P1, Gamma1, Gamma) should be positive definite. Non-positive definite matrices may lead to invalid results or warnings.

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Chapter 9, Section 9.6.

Examples

if (FALSE) {
# Two-stage proportional gain genomic selection
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

reliability <- 0.7
Gamma1 <- reliability * gmat[1:3, 1:3]
Gamma <- reliability * gmat
A1 <- reliability * gmat[1:3, 1:3]
A <- gmat[, 1:3]

# Desired proportional gains
d1 <- c(2, 1, 1)
d2 <- c(3, 2, 1, 1, 1, 0.5, 0.5)

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

result <- mppg_lgsi(
  Gamma1 = Gamma1, Gamma = Gamma, A1 = A1, A = A,
  C = gmat, G1 = gmat[1:3, 1:3], P1 = pmat[1:3, 1:3],
  wmat = weights, d1 = d1, d2 = d2
)
}