selection.index (version 2.0.0)

ppg_lgsi: Predetermined Proportional Gains Linear Genomic Selection Index (PPG-LGSI)

Description

Implements the PPG-LGSI where breeders specify desired proportional gains between traits rather than restricting specific traits to zero. This is genomic version of PPG-LPSI using GEBVs only.

Usage

ppg_lgsi(
  Gamma,
  d,
  wmat = NULL,
  wcol = 1,
  U = NULL,
  k_I = 2.063,
  L_G = 1,
  gmat = NULL,
  GAY = NULL
)

Value

List with:

• summary - Data frame with coefficients and metrics

• b - Vector of PPG-LGSI coefficients (\(\beta_{PG}\))

• E - Named vector of expected genetic gains per trait

• theta_G - Proportionality constant

• gain_ratios - Ratios of achieved to desired gains

Arguments

Gamma

GEBV variance-covariance matrix (n_traits x n_traits)

d

Vector of desired proportional gains (length n_traits or n_constraints). If length n_traits, constraints are applied to all traits. If length n_constraints, must provide U matrix.

wmat

Optional. Economic weights for GA/PRE calculation

wcol

Weight column to use if wmat has multiple columns (default: 1)

U

Optional. Constraint matrix (n_traits x n_constraints). If NULL, assumes d applies to all traits (U = I).

k_I

Selection intensity (default: 2.063)

L_G

Standardization constant (default: 1)

gmat

Optional. True genetic variance-covariance matrix for exact accuracy calculation. If NULL, uses Gamma as approximation.

GAY

Optional. Genetic advance of comparative trait for PRE calculation

Details

Mathematical Formulation (Chapter 6, Section 6.2):

Alternative form: \(beta_PG = beta_RG + theta_G * U * (U' * Gamma * U)^{-1} * d\)

Where: - beta_RG = Restricted index coefficients (from RLGSI) - theta_G = Proportionality constant - d = Vector of desired proportional gains

Proportionality constant: $$theta_G = (d' * (U' * Gamma * U)^{-1} * U' * Gamma * w) / (d' * (U' * Gamma * U)^{-1} * d)$$

Examples

if (FALSE) {
# Simulate GEBV variance-covariance matrix
set.seed(123)
n_traits <- 5
Gamma <- matrix(rnorm(n_traits^2), n_traits, n_traits)
Gamma <- (Gamma + t(Gamma)) / 2
diag(Gamma) <- abs(diag(Gamma)) + 2

# Desired proportional gains (e.g., 2:1:1:0:0 ratio)
d <- c(2, 1, 1, 0, 0)

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

result <- ppg_lgsi(Gamma, d, wmat = w)
print(result$summary)
print(result$gain_ratios) # Should be approximately proportional to d
}