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## Estimates of genetic correlation between two traits / one trait in two selection environments

For estimating selection efficiency, an estimate of the genetic correlation has to be given. This value can be estimated using different equations and/or models:

Based on the phenotypic correlation between two traits or the same trait estimated in different environments: (Falconer and Mackay, 1996):

$r_{g}=\frac{r_{p}-e_{1}e_{2}r_{e}}{h_{1}h_{2}}$
$r_{e}=\frac{\sigma_{e_{12}}^{2} }{\sigma_{e_{1}}^{2}\sigma_{e_{2}}^{2}}$

where $r_{p}$ is the phenotypic correlation between trait 1 and 2 and $h_{1}$ and $h_{2}$ are  the broad-sense heritability of trait 1 and 2, respectively. $e_{1}$ is the environmental variance of trait 1 and $r_{e}$ is the environmental correlation between trait 1 and 2. The environmental correlation is not 0. Thus, the genetic correlation is a function of the phenotypic and environmental correlation as well as their heritability

Assuming that, the environmental covariance is zero, the genetic correlation between two subregions is (Cooper et al., 1996):

$r_{g}=\frac{r_{p}} {h_{1}h_{2}}$

Estimates of the product moment correlation coefficient can exceed 1.

The genetic correlation can be estimated using heritability estimates based on the available number of locations. An estimation based on a standardized number would be only of interest if one would be interested to say something about the error of this estimation. As the genotypic correlation is a population parameter, its magnitude should not change considerably with increasing or decreasing testing effort.

Assuming that the environmental covariance is zero and the heritability of both traits is the same the genetic correlation can be estimated as:

$r_{g}=r_{p}$

$r_{p}$  usually underestimates $r_{g}$ because lack of correlation between traits or sets of variety means in two different environments which is explained by G×E interaction and lack of repeatability.

According to Atlin (2000b) the genetic correlation among target region and several subregions of the target region can be estimated as:

$r_{g}=\sqrt{\frac{\sigma_{g}^{2} }{\sigma_{g}^{2}\sigma_{ge}^{2}}}$

Genotypic correlation among target- and subregion based on two subregions (e) and homogeneous variances (compound symmetry variance-covariance structure, cor() in asrem-R, i.e. gen + gen: subregion). The variance components have to be evaluated in pairwise models. They  might be over-/underestimated due to heterogeneity of variances between subregions.

The homogeneity of variances can be tested using the Bartlett test which is a log-likelihood test. Homogeneity of variances is the most important assumption when applying an Anova!

Alternatively a diagonal, factor-analystic or unstructured variance-covariance structure can be used to estimate the variance within and among subregions. With a diagonal variance-covariance structure (diag() in asreml-R, i.e. gen + gen:diag(sub-region) the covariance is estimated for each paire of subregions, while the variance within subregions is assumed to be the same.

With a factor analytic variance-covariance structure (i.e. fa() in asreml-R, gen:fa(subregion,1)) the main effect of the genotype has to be excluded as the structure apllyed to g x subregion seperates into g and g x subregion.

$\begin{pmatrix}\lambda_{1}\\\lambda_{2}\end{pmatrix}\begin{pmatrix}\lambda_{1}&\lambda_{2}\end{pmatrix}+\begin{pmatrix}\delta_{1}^{2}&0\\0&\delta_{1}^{2}\end{pmatrix}=\begin{pmatrix}\lambda_{1}^{2}&\lambda_{1}\lambda_{2}\\\lambda_{1}\lambda_{2}&\lambda_{2}^{2}\end{pmatrix}+\left(\begin{matrix}\delta_{1}^{2}&0\\0&\delta_{1}^{2}\end{matrix}\right)=$

$\begin{pmatrix}\lambda_{1}^{2}+\delta_{1}^{2}&\lambda_{1}\lambda_{2}\\\lambda_{1}\lambda_{2}&\lambda_{2}^{2}+\delta_{2}^{2}\end{pmatrix}$

where $\delta_{1}^{2}$ and $\delta_{2}^{2}$ are the genotype-by-subregion variances estimated assuming a factor-analytic variance-covariance structure. If $\delta_{2}^{2}=0$ , then

$r_{g_{12}}=\frac{\lambda_{1}\lambda_{2}}{\lambda _{1}\left(\lambda_{2}^{2}+\delta ^{2}\right)}$

With an unstructured variance-covariance structure (i.e., us() in asreml-R, gen:us(subregion)) a variance component within and among subregions is estimated. On this basis the genetic correlation can be estimated as follows:

$\sigma \left ( \frac{n_{1}}{n_{2}}\right )=\left [\begin{matrix}\sigma_{g}^{2}+\sigma_{gs}^{2} &\sigma_{g}^{2}\\\sigma_{g}^{2}&\sigma_{g}^{2}+\sigma_{gs}^{2}\end{matrix}\right ]$

$r_{g}=\frac{\sigma _{g_{12}}^{2}}{\sigma _{g_{1}}^{2}\sigma _{g_{2}^{2}}}$

where  $\sigma _{g}$ is the genetic variance and $\sigma _{gs}$ the genotype-by-subregion interaction variance estimated across the target population of environments.

Genetic correlation between performance in the sub- and the target region for the same trait based on variance components assuming different weights and homogeneous variances

The compound symmetry variance-covariance structure for two subregions under consideration with different weights is (H.P. Piepho, personal communication):

$\sigma\left (\frac{n_{1}}{n_{2}} \right)=\left [\begin{matrix}\sigma_{g}^{2}+\sigma_{gs}^{2} &\sigma_{g}^{2}+p\sigma_{gs}^{2}\\\sigma_{g}^{2}+p\sigma_{gs}^{2}&\sigma_{g}^{2}+\left(p^{2}+q^{2}\right )\sigma_{gs}^{2}\end{matrix}\right]$

Genetic correlations and best linear unbiased predictors (BLUP) can be estimated using information from all subregions, regarding the surbegion factor as fixed, provided that the genotype-by-subregion variance is large relative to the genetic variance. The weights (p and q) are based on the proportion of environments in each subregion and the correlation between subregions, which can be estimated as follows:

$r_{g1r}=\frac{\sigma _{g}^{2}+p\sigma _{gs}^{2}}{\left ( \sigma _{g}^{2}+\sigma _{gs}^{2} \right )\left ( \sigma _{g}^{2}+\left ( \sum_{i=1}^{q} p_{i}^{2}\right )\sigma _{gs}^{2} \right )}$

The weight of the subregion of interest can be regarded as 1 and that of the complementary subregion as 0, if one does not want to involve different economic weights for each subregion.

If the main effect of the genotype is discarded from the model, than

$\sigma_{12}^{2}=0$
$\sigma _{T}^{2}=p\sigma _{g_{1}}^{2}+(1-p)\sigma _{g_{2}}^{2})$

Literature

Atlin, G.N., R.J. Baker, K.B. Mcrae, and X. Lu. 2000a. Selection Response in Subdivided Target Regions. Crop Sci. 40: 7–13.

Atlin, G.N., K.B. McRae, and X. Lu. 2000b. Genotype x region interaction for two-row barley yield in Canada. Crop Science 40(1): 1–6.

Cooper, M., I.. H. DeLacy, and K.E.E. Basford. 1994. Relationshis among analytical methods used to analyse genotypic adaptation in multi-environment trials. p. 193–224. In Cooper, M., Hammer, G.L. (eds.), Plant adaptation and crop improvement. CAB International, Wallingford, UK.

Falconer, D.S., and T.F.C. Mackay. 1996. Introduction to Quantitative Genetics. Pearson Education Limited, Essex, England.

Piepho, H.P., and J. Möhring. 2005. Best linear unbiased prediction of cultivar effects for subdivided target regions. Crop Science 45(3): 1151–1159.

February 10th, 2013
Topic: Plant breeding Tags: None