selection.index (version 2.0.0)

mppg_lpsi: Multistage Predetermined Proportional Gain Linear Phenotypic Selection Index (MPPG-LPSI)

Description

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

Usage

mppg_lpsi(
  P1,
  P,
  G1,
  C,
  wmat,
  wcol = 1,
  d1,
  d2,
  stage1_indices = 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 mlpsi, plus:

• b_M1 - PPG stage 1 coefficients

• b_M2 - PPG stage 2 coefficients

• b_R1 - Restricted stage 1 coefficients

• b_R2 - Restricted stage 2 coefficients

• K_M1 - PPG projection matrix for stage 1

• K_M2 - PPG projection matrix for stage 2

• 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

P1

Phenotypic variance-covariance matrix for stage 1 traits (n1 x n1)

P

Phenotypic variance-covariance matrix for all traits at stage 2 (n x n)

G1

Genotypic variance-covariance matrix for stage 1 traits (n1 x n1)

C

Genotypic variance-covariance matrix for all traits (n x n)

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)

stage1_indices

Integer vector specifying which traits correspond to stage 1 (default: 1:nrow(P1))

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 (Chapter 9.3.1, Eq 9.17):

The PPG coefficients are computed using the projection matrix method: $$\mathbf{b}_{M_1} = \mathbf{b}_{R_1} + \theta_1 \mathbf{U}_1(\mathbf{U}_1'\mathbf{G}_1\mathbf{P}_1^{-1}\mathbf{G}_1\mathbf{U}_1)^{-1}\mathbf{d}_1$$ $$\mathbf{b}_{M_2} = \mathbf{b}_{R_2} + \theta_2 \mathbf{U}_2(\mathbf{U}_2'\mathbf{C}\mathbf{P}^{-1}\mathbf{C}\mathbf{U}_2)^{-1}\mathbf{d}_2$$

where:

• \(\mathbf{b}_{R_i} = \mathbf{K}_{M_i}\mathbf{b}_i\) are restricted coefficients

• \(\mathbf{K}_{M_i} = \mathbf{I} - \mathbf{Q}_{M_i}\) is the projection matrix

• \(\theta_i\) is the proportionality constant computed from \(\mathbf{d}_i\)

• \(\mathbf{U}_i = \mathbf{I}\) (all traits constrained)

$$\mathbf{b}_{M_1} = \mathbf{K}_{M_1} \mathbf{b}_1$$ $$\mathbf{b}_{M_2} = \mathbf{K}_{M_2} \mathbf{b}_2$$

where \(\mathbf{K}_{M_i}\) is computed to achieve proportional gains specified by \(\mathbf{d}_i\)

References

Tallis, G. M. (1962). A selection index for optimum genotype. Biometrics, 18(1), 120-122.

Examples

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

P1 <- pmat[1:3, 1:3]
G1 <- gmat[1:3, 1:3]
P <- pmat
C <- gmat

# Desired proportional gains
d1 <- c(2, 1, 1) # Trait 1 gains twice as much at stage 1
d2 <- c(3, 2, 1, 1, 1, 0.5, 0.5) # Different proportions at stage 2

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

result <- mppg_lpsi(
  P1 = P1, P = P, G1 = G1, C = C, wmat = weights,
  d1 = d1, d2 = d2
)
}