Calculate reliability using variance parameters — H2_Reliability

Compute the reliability ($r^2$, also known as the coefficient of determination) for each genotype from the genotypic variance-covariance matrix and the prediction error variance (PEV) matrix of the genotype BLUPs.

Usage

H2_Reliability_parameters(G_g, C22_g)

Arguments

G_g: Genotypic variance-covariance matrix.
C22_g: Prediction error variance matrix associated with the genotype effects.

Value

A named numeric vector of per-genotype reliabilities. Entries for genotypes with a non-positive genotypic variance are NA.

Details

The reliability of the $i$th genotype is

$$r^2_i = 1 - \frac{var(\hat{g}^{BLUP}_i)}{var(g_i)}$$

where:

$var(\hat{g}^{BLUP}_i)$ is the $i$th diagonal element of the PEV matrix $C_{22(g)}$
$var(g_i)$ is the $i$th diagonal element of the genotypic variance-covariance matrix $G_{(g)}$

As only the diagonal elements are used, $r^2_i$ does not account for the off-diagonal elements (i.e. covariances) of either matrix. The overall reliability is obtained as the mean across genotypes, $\bar{r}^2 = \frac{1}{n_g}\sum_{i=1}^{n_g} r^2_i$.

Genotypes whose genotypic variance (the corresponding diagonal element of $G_{(g)}$) is not strictly positive yield an undefined reliability and are returned as NA (with a warning) rather than Inf/NaN.

References

Schmidt, P., Hartung, J., Bennewitz, J., & Piepho, H.-P. (2019). Heritability in Plant Breeding on a Genotype-Difference Basis. Genetics, 212(4), 991–1008. https://doi.org/10.1534/genetics.119.302134
Mrode, R. A. (2014). Linear Models for the Prediction of Animal Breeding Values (3rd ed.). CABI.

Examples

G_g <- diag(0.15, 3, 3)
C22_g <- matrix(
  c(
    0.08, 0.01, 0.00,
    0.01, 0.07, 0.01,
    0.00, 0.01, 0.09
  ),
  nrow = 3, byrow = TRUE
)
H2_Reliability_parameters(G_g, C22_g)
#> [1] 0.4666667 0.5333333 0.4000000