Computes the Smith (1936) and Hazel (1943) index given economic weights and phenotypic and genotypic variance-covariance matrices. The Smith-Hazel index is computed as follows: $\bf{b = P^{-1}Aw}$

where $$\bf{P}$$ and $$\bf{G}$$ are phenotypic and genetic covariance matrices, respectively, and $$\bf{b}$$ and $$\bf{w}$$ are vectors of index coefficients and economic weightings, respectively.

The genetic worth $$I$$ of an individual genotype based on traits x, y, ..., n, is calculated as:

$I = b_xG_x + b_yG_y + ... + b_nG_n$

where b the index coefficient for the traits x, y, ..., n, respectively, and G is the individual genotype BLUPs for the traits x, y, ..., n, respectively.

## Usage

Smith_Hazel(
.data,
use_data = "blup",
pcov = NULL,
gcov = NULL,
SI = 15,
weights = NULL
)

## Arguments

.data

The input data. It can be either a two-way table with genotypes in rows and traits in columns, or an object fitted with the function gamem(). Please, see Details for more details.

use_data

Define which data to use If .data is an object of class gamem. Defaults to "blup" (the BLUPs for genotypes). Use "pheno" to use phenotypic means instead BLUPs for computing the index.

pcov, gcov

The phenotypic and genotypic variance-covariance matrix, respectively. Defaults to NULL. If a two-way table is informed in .data these matrices are mandatory.

SI

The selection intensity (percentage). Defaults to 20

weights

The vector of economic weights. Defaults to a vector of 1s with the same length of the number of traits.

## Value

An object of class hz containing:

• b: the vector of index coefficient.

• index: The genetic worth.

• sel_dif_trait: The selection differencial.

• sel_gen: The selected genotypes.

• gcov: The genotypic variance-covariance matrix

• pcov: The phenotypic variance-covariance matrix

## Details

When using the phenotypic means in .data, be sure the genotype's code are in rownames. If .data is an object of class gamem them the BLUPs for each genotype are used to compute the index. In this case, the genetic covariance components are estimated by mean cross products.

## References

Smith, H.F. 1936. A discriminant function for plant selection. Ann. Eugen. 7:240-250. doi:10.1111/j.1469-1809.1936.tb02143.x

Hazel, L.N. 1943. The genetic basis for constructing selection indexes. Genetics 28:476-90. https://www.genetics.org/content/28/6/476.short

mtsi(), mgidi(), fai_blup()

## Author

Tiago Olivoto tiagoolivoto@gmail.com

## Examples

# \donttest{
vcov <- covcor_design(data_g, GEN, REP, everything())
means <- as.matrix(vcov$means) pcov <- vcov$phen_cov
gcov <- vcov\$geno_cov

index <- Smith_Hazel(means, pcov = pcov, gcov = gcov, weights = rep(1, 15))
# }