class: center, middle, inverse, title-slide # Correlated traits: Index selection ### Jinliang Yang ### Dec. 9th, 2019 --- # The correlation between two traits `\begin{align*} & r_P = r_Ah_Xh_Y + r_Ee_Xe_Y \\ \end{align*}` - __ `\(r_P\)`__: the phenotypic correlation between two traits X and Y - __ `\(r_A\)`__: the genetic correlation due to breeding values between X and Y - __ `\(r_E\)`__: the environmental correlation between X and Y, including non-additive genetic effects - __ `\(h^2\)`__: heritability - __ `\(e^2\)`__: `\(1-h^2\)` --- This proof generally shows that the genetic and environmental correlation come together to create the phenotypic correlation. - If both traits have __low heritabilites__: - then phenotypic correlation is determined mainly be the environmental correlations. - If they have __high heritabilities__: - genetic correlation is more important. --- # The correlation between two traits `\begin{align*} & r_P = r_Ah_Xh_Y + r_Ee_Xe_Y \\ \end{align*}` --- ### Estimates of genetic correlation - Rests on the resemblance between relatives in a similar manner to the estimation of `\(h^2\)` - But rather than perform an ANOVA, perform __an analysis of covariance__. - The interpretation of covariance is exactly the same. -- ### Some precautions with genetic correlations: - They have large sampling error. - Increasing the precision of genetic correlations can be attained through using the same techniques for increasing the precision of heritability estimates. - Subject to change with allele frequency differences. --- # Correlated response to selection If we select on trait X, how will trait Y change? -- This comes down to the relationship between the breeding values (BVs) for trait Y and breeding values for trait X. - It can be expressed as __"what is the expected BV of Y given a BV of X"__ - Or `\(E(A_Y | A_X)\)` -- .pull-left[ <img src="Ch19-c2_files/figure-html/unnamed-chunk-1-1.png" width="80%" style="display: block; margin: auto;" /> ] -- .pull-right[ `\begin{align*} & b_{A_{YX}} = \frac{Cov_A}{\sigma^2_{A_X}} \\ & Cov_A = r_A\sigma_{A_X}\sigma_{A_Y} \\ \end{align*}` Therefore, `\begin{align*} b_{A_{YX}} & = \frac{Cov_A}{\sigma^2_{A_X}} \\ & = \frac{r_A\sigma_{A_Y}}{\sigma_{A_X}} \\ \end{align*}` ] --- # Correlated response to selection `\begin{align*} b_{A_{YX}} = r_A\frac{\sigma_{A_Y}}{\sigma_{A_X}} \\ \end{align*}` -- Recall that `\(R_X = ih_X\sigma_{A_X}\)`, correlated response of trait Y is: -- `\begin{align*} CR_Y & = b_{A_{YX}}R_X \\ & = r_A \frac{\sigma_{A_Y}}{\sigma_{A_X}} ih_X\sigma_{A_X}\\ & = ih_Xr_A\sigma_{A_Y} \\ \end{align*}` -- Because `\(\sigma_{A_Y} = h_Y\sigma_{P_Y}\)`, therefore, `\begin{align*} & CR_Y = ih_Xh_Yr_A\sigma_{P_Y} \\ \end{align*}` -- In the formula: - `\(h_Xh_Yr_A\)` is referred to as the __coheritability__, as it takes the place of the heritability in the direct response equation. - If `\(h_Xh_Yr_A\)` is larger than `\(h^2\)` of trait Y, then selection on a correlated trait should be used. --- # Indirect selection The trade-off between selection on a correlated trait and direct selection on a trait can also be seen: `\begin{align*} \frac{CR_Y}{R_Y} & = \frac{i_Xh_Xr_A\sigma_{A_Y}}{i_Yh_Y\sigma_{A_Y}} \\ & = \frac{i_Xh_Xr_A}{i_Yh_Y}\\ \end{align*}` -- ### Assuming seletion intensity is the same - When __ `\(h_Xr_A > h_Y\)`__, a correlated response from selection on a secondary trait (X) is greater than response to direct selection on Y . -- - Note that this exact same property applies to usefulness of molecular markers for __marker-assisted selection__ --- # Indirect selection The trade-off between selection on a correlated trait and direct selection on a trait can also be seen: `\begin{align*} \frac{CR_Y}{R_Y} & = \frac{i_Xh_Xr_A\sigma_{A_Y}}{i_Yh_Y\sigma_{A_Y}} \\ & = \frac{i_Xh_Xr_A}{i_Yh_Y}\\ \end{align*}` ### Practical considerations - If trait Y is very __expensive__ and __difficult__ to measure, but trait X is very cheap and easy to measure. - e.g. high-throughput phenotyping techonologies -- - Or the desired traits is measurable __in one sex__ only, but the secondary traits is measurable in both. - e.g. milk yield and body weight in dairy cow --- # Indirect selection The trade-off between selection on a correlated trait and direct selection on a trait can also be seen: `\begin{align*} \frac{CR_Y}{R_Y} & = \frac{i_Xh_Xr_A\sigma_{A_Y}}{i_Yh_Y\sigma_{A_Y}} \\ & = \frac{i_Xh_Xr_A}{i_Yh_Y}\\ \end{align*}` ### Genotype-by-environment interaction - Performance in different environments can be regarded as __two separate, but correlated traits__. - Improvement in one environment by selection in another environment can be predicted by knowing the heritability in each environment and the genetic correlation between them. --- # Selection on multiple traits Total economic value is a composite of many traits. How does one __maximize response for many traits__ simultaneously? -- #### Options for multiple trait selection - __Tandom selection__: - Selection for one trait at a time until that trait is improved to a desired level. - After that, selection proceeds for another trait. -- - __Independent culling levels__: - Only individuals that meet the minimum standard for each trait are selected. -- - __Index selection__: - Select for multiple traits simultaneously by constructing an index value. - Index value is then treated as a single economic trait. --- # Genetic merit - Any individual has a genetic value for what we'll call as __genetic merit__, or simply, __merit__. -- - Merit is the summation of all traits contributing to an individual's worth, fitness, economic value, etc. -- #### The __true value of merit__ is represented as: `\begin{align*} T & = a_1G_1 + a_2G_2 + ... + a_mGm \\ & = \sum_{i=1}^ma_iG_i \\ \end{align*}` - `\(G_i\)` is the genetic value for trait `\(i\)` - `\(a_i\)` is the economic weight placed on trait `\(i\)` - The economic weights are set by the breeder according to production needs and value. --- # Index trait To accurately predict genentic merit `\(T\)`, we want to combine the values of multiple traits into one value, denoted `\(I\)`. -- `\begin{align*} I & = b_1P_1 + b_2P_2 + ... + b_mPm \\ & = \sum_{i=1}^mb_iP_i \\ \end{align*}` - `\(P_i\)` is the phenotypic value for trait `\(i\)` that goes into the index - `\(b_i\)` is the weighting factor on trait `\(i\)` -- ### Correlation between T and I The goal is to __find the values of the `\(b_i\)`s__ that could __maximize the correlations__ between `\(T\)` and `\(I\)`, or `\(r_{TI}\)`. `\begin{align*} r_{TI} = \frac{Cov(T, I)}{\sigma_T\sigma_I} \end{align*}` --- # Correlation between T and I `\begin{align*} r_{TI} = \frac{Cov(T, I)}{\sigma_T\sigma_I} \end{align*}` Obtaining the maximum value of the correlation involves taking the derivative and setting to zero. -- ### Optimum index The vector of weights, which is called the __Smith-Hazel index__, or __optimum index__: - It is the most widely used set of weights for a linear selection index - See [here](ch37_index-selection.pdf) Page 411 for the proof `\begin{align*} & \mathbf{b} = \mathbf{P^{-1}}\mathbf{G^Ta} \\ \end{align*}` - `\(\mathbf{P}\)` is the phenotypic variance-covariance matrix - `\(\mathbf{G}\)` is the genetic variance-covariance matrix - `\(\mathbf{a}\)` is the vector of __known__ economic weights - `\(\mathbf{b}\)` is the vector of __unknow__ weights, or the weights to be applied to the phenotypic values of the different traits composing the index, `\(I\)`. --- # Optimum index The vector of weights, which is called the __Smith-Hazel index__, or __optimum index__: `\begin{align*} & \mathbf{b} = \mathbf{P^{-1}}\mathbf{G^Ta} \\ \end{align*}` -- - The vector of `\(\mathbf{b}\)` can be estimated if we know the __phenotypic and genetic variances__ and the __phenotypic and genetic covariances__ -- - This assumes that we have good information on the economic weights, which can be hard to determine. -- - Then the resulting vector of weights will give a selection index that maximized genetic gain in `\(T\)`, the true genetic merit. -- - The drawback of this index is that the genetic variance and covariances are often estimated with large amounts error, which may reduce the correlation between I and T. ??? # A problem This litter size of mice could be increased: - 1) by selection of females for their little size - 2) by selection of both parents for body weight Which would be the better of these two simple procedures, given the following parameters? - `\(h^2\)` of litter size: 0.22 - `\(h^2\)` of body weight: 0.35 - Genetic correlation `\(r_A\)`: 0.43 - Proportion of selection: - Females: 25% ( `\(i_f\)` =1.271) - Males: 10% ( `\(i_m\)` =1.755)