Restricted Linear Phenotypic Eigen Selection Index (RESIM)

Extends ESIM by imposing null restrictions: genetic gains for $r$ selected traits are forced to zero while the index heritability for the remaining traits is maximised.

Usage

resim(
  pmat,
  gmat,
  restricted_traits = NULL,
  U_mat = NULL,
  selection_intensity = 2.063
)

Arguments

pmat: Phenotypic variance-covariance matrix (n_traits x n_traits).
gmat: Genotypic variance-covariance matrix (n_traits x n_traits).
restricted_traits: Integer vector of trait indices to restrict to zero genetic gain. Example: c(1, 3) restricts traits 1 and 3. Alternatively supply a custom restriction matrix via U_mat.
U_mat: Optional. Restriction matrix (n_traits x r) where each column defines one null restriction ($\mathbf{U}^{\prime}\mathbf{C}\mathbf{b} = 0$). Ignored if restricted_traits is provided.
selection_intensity: Selection intensity constant (default: 2.063).

Value

Object of class "resim", a list with:

summary: Data frame with b coefficients and key metrics.
b: Named numeric vector of RESIM coefficients.
Delta_G: Named vector of expected genetic gains per trait.
sigma_I: Index standard deviation.
hI2: Index heritability (leading eigenvalue of KP^(-1)C).
rHI: Index accuracy.
lambda2: Leading eigenvalue of the restricted eigenproblem.
K: Projection matrix used.
U_mat: Restriction matrix used.
restricted_traits: Integer vector of restricted trait indices.
implied_w: Implied economic weights.
selection_intensity: Selection intensity used.

Details

Projection matrix (Section 7.2): $$\mathbf{K} = \mathbf{I}_t - \mathbf{P}^{-1}\mathbf{C}\mathbf{U} (\mathbf{U}^{\prime}\mathbf{C}\mathbf{P}^{-1}\mathbf{C}\mathbf{U})^{-1} \mathbf{U}^{\prime}\mathbf{C}$$

Restricted eigenproblem: $$(\mathbf{K}\mathbf{P}^{-1}\mathbf{C} - \lambda_R^2 \mathbf{I}_t)\mathbf{b}_R = 0$$

Selection response and genetic gain: $$R_R = k_I \sqrt{\mathbf{b}_R^{\prime}\mathbf{P}\mathbf{b}_R}$$ $$\mathbf{E}_R = k_I \frac{\mathbf{C}\mathbf{b}_R}{\sqrt{\mathbf{b}_R^{\prime}\mathbf{P}\mathbf{b}_R}}$$

Implied economic weights: $$\mathbf{w}_R = \mathbf{C}^{-1}[\mathbf{P} + \mathbf{Q}_R^{\prime}\mathbf{C}]\mathbf{b}_R$$ where $\mathbf{Q}_R = \mathbf{I} - \mathbf{K}$.

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 7.2.

Examples

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

# Restrict traits 1 and 3 to zero genetic gain
result <- resim(pmat, gmat, restricted_traits = c(1, 3))
print(result)
summary(result)
} # }