Smith-Hazel Linear Phenotypic Selection Index — smith

Implements the optimal Smith-Hazel selection index which maximizes the correlation between the index I = b'y and the breeding objective H = w'g.

This is the foundational selection index method from Chapter 2.

Usage

smith_hazel(
  pmat,
  gmat,
  wmat,
  wcol = 1,
  selection_intensity = 2.063,
  GAY = NULL
)

Arguments

pmat: Phenotypic variance-covariance matrix (n_traits x n_traits)
gmat: Genotypic variance-covariance matrix (n_traits x n_traits)
wmat: Economic weights matrix (n_traits x k), or vector
wcol: Weight column to use if wmat has multiple columns (default: 1)
selection_intensity: Selection intensity constant (default: 2.063 for 10% selection)
GAY: Optional. Genetic advance of comparative trait for PRE calculation

Value

List with:

summary - Data frame with coefficients and metrics
b - Vector of Smith-Hazel index coefficients
w - Named vector of economic weights
Delta_G - Named vector of expected genetic gains per trait
sigma_I - Standard deviation of the index
GA - Total genetic advance
PRE - Percent relative efficiency
hI2 - Heritability of the index
rHI - Accuracy (correlation with breeding objective)

Details

Mathematical Formulation (Chapter 2):

Index coefficients: \(b = P^{-1}Gw\)

Where: - P = Phenotypic variance-covariance matrix - G = Genotypic variance-covariance matrix - w = Economic weights

Key metrics: - Variance of index: \(\sigma^2_I = b'Pb\) - Total genetic advance: \(R_H = i\sqrt{b'Pb}\) - Expected gains per trait: \(\Delta G = (i/\sigma_I)Gb\) - Heritability of index: \(h^2_I = b'Gb / b'Pb\) - Accuracy: \(r_{HI} = \sqrt{b'Gb / b'Pb}\)

Examples

if (FALSE) { # \dontrun{
# Calculate variance-covariance matrices
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Define economic weights
weights <- c(10, 8, 6, 4, 2, 1, 1)

# Build Smith-Hazel index
result <- smith_hazel(pmat, gmat, weights)
print(result)
summary(result)
} # }