In plant and animal breeding, quantitative traits (QTs) are expressions of genes distributed across the genome interacting with the environment. The phenotypic value of QTs (\(y\)) can be systematically partitioned into a genotypic component (\(g\)) and an environmental component (\(e\)):
\[ y = g + e \]
The primary goal in breeding is to maximize an individual’s net genetic merit. The net genetic merit (\(H\)) is a linear combination of the unobservable true breeding values (\(\mathbf{g}\)) weighted by their respective economic values (\(\mathbf{w}\)):
\[ H = {\mathbf{w}}^{\prime}\mathbf{g} \]
Because the net genetic merit is unobservable in field trials, breeders construct a Linear Phenotypic Selection Index (LPSI) to predict it. The LPSI (\(I\)) is a linear combination of the observable and optimally weighted phenotypic trait values (\(\mathbf{y}\)) adjusted by index coefficients (\(\mathbf{b}\)):
\[ I = {\mathbf{b}}^{\prime}\mathbf{y} \]
The objective of the LPSI is to predict the net genetic merit and maximize the multi-trait selection response.
To identify the optimal parents for the next selection cycle, the correlation between the net genetic merit (\(H\)) and the LPSI (\(I\)) must be maximized. The vector \(\mathbf{b}\) that simultaneously minimizes the mean squared difference between \(I\) and \(H\) and perfectly maximizes this correlation is mathematically derived as:
\[ \mathbf{b} = {\mathbf{P}}^{-1}\mathbf{Gw} \]
where: * \(\mathbf{P}\) is the phenotypic variance-covariance matrix. * \(\mathbf{G}\) is the genotypic variance-covariance matrix. * \(\mathbf{w}\) is the vector of economic weights defining relative trait importance.
Once these optimal coefficients are derived, we can evaluate two fundamental parameters:
The Maximized Selection Response (\(R_I\)): The expected mean improvement in the net genetic merit due to indirect selection on the index. \[ {R}_I = {k}_I\sqrt{{\mathbf{b}}^{\prime}\mathbf{Pb}} \]
The Expected Genetic Gain Per Trait (\(\mathbf{E}\)): The multi-trait selection response broken down per individual trait. \[ \mathbf{E} = {k}_I\frac{\mathbf{Gb}}{\sigma_I} \]
where \(k_I\) is the standardized selection intensity and \(\sigma_I\) is the standard deviation of the index score variance.
We can seamlessly translate this text theory into rigorous
statistical practice using the selection.index package. We
will utilize the built-in synthetic datasets: maize_pheno
(containing multi-environment phenotypic records for 100 genotypes) and
maize_geno (500 SNP markers).
First, we estimate the genotypic (\(\mathbf{G}\)) and phenotypic (\(\mathbf{P}\)) variance-covariance matrices from our raw phenotypic dataset.
library(selection.index)
# Load the synthetic phenotypic multi-environment dataset
data("maize_pheno")
# In maize_pheno: Traits are columns 4:6.
# Genotypes are in column 1, and Block/Replication is in column 3.
gmat <- gen_varcov(data = maize_pheno[, 4:6], genotypes = maize_pheno[, 1], replication = maize_pheno[, 3])
pmat <- phen_varcov(data = maize_pheno[, 4:6], genotypes = maize_pheno[, 1], replication = maize_pheno[, 3])Next, we establish the relative economic priority of each trait. Economic weights (\(\mathbf{w}\)) explicitly define our strategic breeding objectives.
With the covariance matrices and economic weights specified, we
integrate them into the primary lpsi() function, which
evaluates the combinatorial multi-trait selection indices
efficiently.
Finally, we evaluate the theoretical gains. The lpsi()
function returns a structured data frame containing the theoretical
selection response (\(R_I\)) and other
parameter estimates for all requested trait combinations.
# View the top combinatorial indices, including their selection response (R_A)
head(index_results)
#> ID b.1 b.2 b.3 GA PRE Delta_G rHI hI2
#> 1 1, 2, 3 2364.671 9453.933 1012156 6863519 686351898 6863519 42.0899 1771.794
#> Rank
#> 1 1
# Extract the phenotypic selection scores to strategically rank the parental candidates
# using the top evaluated combinatorial index
scores <- predict_selection_score(
index_results,
data = maize_pheno[, 4:6],
genotypes = maize_pheno[, 1]
)
# View the top performing candidates designated for the next breeding cycle
head(scores)
#> Genotypes I_1_2_3 I_1_2_3_Rank
#> 1 G001 138639943 78
#> 2 G002 138393284 83
#> 3 G003 138260546 88
#> 4 G004 139442740 46
#> 5 G005 139319528 51
#> 6 G006 139661606 36The classical linear selection index theories seamlessly extend to marker-assisted genomic selection. If you have genome-wide marker profiles for your genotypes, you can incorporate them to estimate the Linear Marker Selection Index (LMSI).
# Load the associated synthetic genomic dataset (500 SNPs for the 100 genotypes)
data("maize_geno")
# Calculate the marker-assisted index combining our matrices and raw SNP profiles
marker_index_results <- lmsi(
pmat = pmat,
gmat = gmat,
marker_scores = maize_geno,
wmat = weights
)
summary(marker_index_results)In scenarios where the phenotypic (\(\mathbf{P}\)) and genotypic (\(\mathbf{G}\)) matrices are poorly estimated (e.g., due to limited data), the true optimal coefficients (\(\mathbf{b}\)) can be systematically biased. The Base Index provides a robust, non-optimized alternative where coefficients are set strictly equal to the fixed economic weights (\(I_B = \mathbf{w}'\mathbf{y}\)).
# Calculate the Base Index and automatically compare its efficiency to the LPSI
base_results <- base_index(
pmat = pmat,
gmat = gmat,
wmat = weights,
compare_to_lpsi = TRUE
)
# Observe the expected genetic gains and efficiency comparison
base_results$summary
#> b.1 b.2 b.3 GA PRE Delta_G hI2 rHI
#> 1 10 -5 -5 1824121 182412094 14577.53 125.1324 11.1863The theory demonstrates that the correlation between the net genetic
merit (\(H\)) and the expected index
(\(I\)) differs from the traditional
index heritability mathematically (\(h^2_I
\neq \rho^2_{HI}\)). The lpsi() function
intrinsically estimates both of these fundamental statistics: