mlpsi: Multistage Linear Phenotypic Selection Index (MLPSI)

In selection.index: Analysis of Selection Index in Plant Breeding

Multistage Linear Phenotypic Selection Index (MLPSI)

Description

Implements the two-stage Linear Phenotypic Selection Index where selection occurs at two different stages with covariance adjustments due to selection effects using Cochran/Cunningham's method.

Usage

mlpsi(
  P1,
  P,
  G1,
  C,
  wmat,
  wcol = 1,
  stage1_indices = NULL,
  selection_proportion = 0.1,
  use_young_method = FALSE,
  k1_manual = 2.063,
  k2_manual = 2.063,
  tau = NULL
)

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)
stage1_indices	Integer vector specifying which traits (columns of P and C) correspond to stage 1. Default is 1:nrow(P1), assuming stage 1 traits are the first n1 traits. Use this to specify non-contiguous traits, e.g., c(1, 3, 5) for traits 1, 3, and 5.
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 (used if use_young_method = FALSE)
k2_manual	Manual selection intensity for stage 2 (used if use_young_method = FALSE)
tau	Standardized truncation point (default: computed from selection_proportion)

Details

Mathematical Formulation:

Stage 1 index coefficients:

\mathbf{b}_1 = \mathbf{P}_1^{-1}\mathbf{G}_1\mathbf{w}

Stage 2 index coefficients:

\mathbf{b}_2 = \mathbf{P}^{-1}\mathbf{G}\mathbf{w}

Adjusted phenotypic covariance matrix (Cochran/Cunningham):

\mathbf{P}^* = \mathbf{P} - u \frac{Cov(\mathbf{y},\mathbf{x}_1)\mathbf{b}_1\mathbf{b}_1'Cov(\mathbf{x}_1,\mathbf{y})}{\mathbf{b}_1'\mathbf{P}_1\mathbf{b}_1}

Adjusted genotypic covariance matrix:

\mathbf{C}^* = \mathbf{C} - u \frac{\mathbf{G}_1'\mathbf{b}_1\mathbf{b}_1'\mathbf{G}_1}{\mathbf{b}_1'\mathbf{P}_1\mathbf{b}_1}

where u = k_1(k_1 - \tau)

Selection response: R_1 = k_1 \sqrt{\mathbf{b}_1'\mathbf{P}_1\mathbf{b}_1}, R_2 = k_2 \sqrt{\mathbf{b}_2'\mathbf{P}^*\mathbf{b}_2}

Value

List with components:

• b1 - Stage 1 index coefficients

• b2 - Stage 2 index coefficients

• stage1_metrics - List with stage 1 metrics (R1, E1, rho_H1)

• stage2_metrics - List with stage 2 metrics (R2, E2, rho_H2)

• P_star - Adjusted phenotypic covariance matrix at stage 2

• C_star - Adjusted genotypic covariance matrix at stage 2

• rho_12 - Correlation between stage 1 and stage 2 indices

• k1 - Selection intensity at stage 1

• k2 - Selection intensity at stage 2

• summary_stage1 - Data frame with stage 1 summary

• summary_stage2 - Data frame with stage 2 summary

References

Cochran, W. G. (1951). Improvement by means of selection. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 449-470.

Cunningham, E. P. (1975). Multi-stage index selection. Theoretical and Applied Genetics, 46(2), 55-61.

Young, S. S. Y. (1964). Multi-stage selection for genetic gain. Heredity, 19(1), 131-144.

Examples

## Not run: 
# Two-stage selection example
# Stage 1: Select based on 3 traits
# Stage 2: Select based on all 7 traits

# Compute variance-covariance matrices
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Stage 1 uses first 3 traits
P1 <- pmat[1:3, 1:3]
G1 <- gmat[1:3, 1:3]

# Stage 2 uses all 7 traits
P <- pmat
C <- gmat

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

# Run MLPSI (default: stage1_indices = 1:3)
result <- mlpsi(
  P1 = P1, P = P, G1 = G1, C = C, wmat = weights,
  selection_proportion = 0.1
)

# Or with non-contiguous traits (e.g., traits 1, 3, 5 at stage 1):
# P1 <- pmat[c(1,3,5), c(1,3,5)]
# G1 <- gmat[c(1,3,5), c(1,3,5)]
# result <- mlpsi(P1 = P1, P = P, G1 = G1, C = C, wmat = weights,
#                 stage1_indices = c(1, 3, 5))

print(result$summary_stage1)
print(result$summary_stage2)

## End(Not run)

selection.index documentation built on March 9, 2026, 1:06 a.m.