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Desired Gains Index (DG-LPSI)

Source: R/constrained_indices.R
dg_lpsi.Rd

Implements the Pesek & Baker (1969) Desired Gains Index where breeders specify target genetic gains instead of economic weights. This enhanced version includes calculation of implied economic weights and feasibility checking.

Usage

dg_lpsi(
  pmat,
  gmat,
  d,
  return_implied_weights = TRUE,
  check_feasibility = TRUE,
  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)

d

Vector of desired genetic gains (length n_traits). Example: d = c(1.5, 0.8, -0.2) means gain +1.5 in trait 1, +0.8 in trait 2, -0.2 in trait 3.

return_implied_weights

Logical - calculate implied economic weights? (default: TRUE)

check_feasibility

Logical - warn if desired gains are unrealistic? (default: TRUE)

selection_intensity

Selection intensity i (default: 2.063)

Value

List with:

  • summary - Data frame with coefficients, metrics, and implied weights

  • b - Vector of selection index coefficients

  • Delta_G - Named vector of achieved genetic gains per trait

  • desired_gains - Named vector of desired gains (input d)

  • gain_errors - Difference between desired and achieved gains

  • implied_weights - Economic weights that would achieve these gains in Smith-Hazel LPSI

  • implied_weights_normalized - Normalized implied weights (max absolute = 1)

  • feasibility - Data frame with feasibility analysis per trait

  • hI2 - Index heritability

  • rHI - Index accuracy

Details

Mathematical Formulation:

1. Index coefficients: \(\mathbf{b} = \mathbf{G}^{-1}\mathbf{d}\)

2. Expected response: \(\Delta \mathbf{G} = (i/\sigma_I) \mathbf{G}\mathbf{b}\)

CRITICAL: Scale Invariance Property

The achieved gains \(\Delta\mathbf{G}\) are determined by selection intensity (i), genetic variance (G), and phenotypic variance (P), NOT by scaling \(\mathbf{b}\). If you multiply \(\mathbf{b}\) by constant c, \(\sigma_I\) also scales by c, causing complete cancellation in \(\Delta\mathbf{G} = (i/(c\sigma_I))\mathbf{G}(c\mathbf{b}) = (i/\sigma_I)\mathbf{G}\mathbf{b}\).

What DG-LPSI Actually Achieves:

- Proportional gains matching the RATIOS in d (not absolute magnitudes) - Achieved magnitude depends on biological/genetic constraints - Use feasibility checking to verify if desired gains are realistic

3. Implied economic weights (Section 1.4 of Chapter 4): $$\hat{\mathbf{w}} = \mathbf{G}^{-1} \mathbf{P} \mathbf{b}$$

The implied weights represent the economic values that would have been needed in a Smith-Hazel index to achieve the desired gain PROPORTIONS. Large implied weights indicate traits that are "expensive" to improve (low heritability or unfavorable correlations), while small weights indicate traits that are "cheap" to improve.

Feasibility Checking:

The function estimates maximum possible gains as approximately 3.0 * sqrt(G_ii) (assuming very intense selection with i ~ 3.0) and warns if desired gains exceed 80% of these theoretical maxima.

References

Pesek, J., & Baker, R. J. (1969). Desired improvement in relation to selection indices. Canadian Journal of Plant Science, 49(6), 803-804.

Examples

# Load data
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Specify desired gains (e.g., increase each trait by 1 unit)
desired_gains <- rep(1, ncol(gmat))

# Calculate Desired Gains Index with all enhancements
result <- dg_lpsi(pmat, gmat, d = desired_gains)
#> Warning: Desired gains may be unrealistic for trait(s): rpp, ppr, spp, pw
#> Desired gains exceed 80% of theoretical maximum (i = 2.06).
#> Consider reducing targets or using higher selection intensity.

# View summary
print(result$summary)
#>       b.1    b.2     b.3      b.4    b.5       b.6     b.7 Delta_G     hI2 rHI
#> 1 11.4784 3.8899 -17.521 -17.3052 0.5096 -115.2781 71.4102 53.5871 -0.0931  NA
#>   implied_w.1 implied_w.2 implied_w.3 implied_w.4 implied_w.5 implied_w.6
#> 1    -46.6372     14.1965    178.2319     72.9888    -26.4785   -256.3287
#>   implied_w.7 implied_w_norm.1 implied_w_norm.2 implied_w_norm.3
#> 1    738.7446          -0.0631           0.0192           0.2413
#>   implied_w_norm.4 implied_w_norm.5 implied_w_norm.6 implied_w_norm.7
#> 1           0.0988          -0.0358           -0.347                1

# Extract implied weights to understand relative "cost" of gains
print(result$implied_weights_normalized)
#>        sypp         dtf         rpp         ppr         ppp         spp 
#> -0.06313034  0.01921701  0.24126326  0.09880111 -0.03584256 -0.34697876 
#>          pw 
#>  1.00000000 

# Check feasibility
print(result$feasibility)
#>      trait desired_gain achieved_gain genetic_sd max_possible_gain
#> sypp  sypp            1    0.07942154  1.1209826            2.3126
#> dtf    dtf            1    0.07942154  1.2490710            2.5768
#> rpp    rpp            1    0.07942154  0.3639840            0.7509
#> ppr    ppr            1    0.07942154  0.4931853            1.0174
#> ppp    ppp            1    0.07942154  0.9801869            2.0221
#> spp    spp            1    0.07942154  0.1319506            0.2722
#> pw      pw            1    0.07942154  0.1015121            0.2094
#>      feasibility_ratio is_realistic
#> sypp            0.4324         TRUE
#> dtf             0.3881         TRUE
#> rpp             1.3317        FALSE
#> ppr             0.9829        FALSE
#> ppp             0.4945         TRUE
#> spp             3.6736        FALSE
#> pw              4.7751        FALSE