class: center, middle, inverse, title-slide .title[ # Correlated traits ] .author[ ### Jinliang Yang ] .date[ ### April 24, 2024 ] --- # The correlation between two traits .pull-left[ <div align="center"> <img src="figure19.1_cor.png" height=250> </div> ] .pull-right[ - Heart girth with body weight - Body weight with sale price - Body weight with feed conversion ratio ] -- ### Genetic correlation - Pleiotropy - property of a locus whereby it affects two or more traits -- - Linkage disequilibrium - Two loci that each affect a trait may be in LD caused by physical linkage - transient cause of correlation. --- # The correlation between two traits .pull-left[ <div align="center"> <img src="htp.png" height=200> </div> ] .pull-right[ - Plant height vs. canopy coverage - Canopy coverage vs. yield - Vegetation index vs. N content in leaf ] -- ### Environmental correlation Caused by common response of two traits to shared environmental conditions. --- # The correlation between two traits Knowing the values of each component of the phenotype allows the calculation of the different types of correlations. ### Genetic correlation - This can be further broken down into __additive genetic correlation__ by correlating breeding values for two traits ### Environmental correlation - Different __environmental factors__ -- --- `\begin{align*} & Cov_P = Cov_A + Cov_E \\ \end{align*}` - Here, `\(Cov_P\)`, `\(Cov_A\)`, and `\(Cov_E\)` denotes the phenotypic, additive genetic and environmental covariance of the __two traits X and Y__. --- # The correlation between two traits The __phenotypic correlation__ between __two traits__ X and Y: `\begin{align*} & r_P = \frac{Cov_P}{\sigma_{P_X}\sigma_{P_Y}} \\ \end{align*}` -- `\begin{align*} & Cov_P = r_P\sigma_{P_X}\sigma_{P_Y} \\ \end{align*}` - __ `\(Cov_P\)`__: the phenotypic covariance between the two traits X and Y - __ `\(r_P\)`__: the phenotypic correlation coefficient between the two traits X and Y -- ----- `\begin{align*} & Cov_P = Cov_A + Cov_E \\ & r_P\sigma_{P_X}\sigma_{P_Y} = r_A\sigma_{A_X}\sigma_{A_Y} + r_E\sigma_{E_X}\sigma_{E_Y} \\ \end{align*}` - __ `\(r_A\)`__: 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. - __ `\(\sigma\)`__: standard deviation, with subscripts `\(P\)`, `\(A\)`, and `\(E\)`, and X or Y to the traits referred to. - e.g., `\(\sigma_{A_Y}\)`, standard deviation of the trait Y due to breeding value. --- # The correlation between two traits `\begin{align*} & h^2 = \frac{\sigma_A^2}{\sigma_P^2} \\ & 1 - h^2 = 1 - \frac{\sigma_A^2}{\sigma_P^2} \\ \end{align*}` -- `\begin{align*} e^2 = & 1 - h^2 = \frac{\sigma_P^2 - \sigma_A^2}{\sigma_P^2} = \frac{\sigma_E^2}{\sigma_P^2} \\ \end{align*}` Here, `\(\sigma_E^2\)` includes __non-additive genetic effects__ and __environmental effects__. -- `\begin{align*} & e^2 = \frac{\sigma_E^2}{\sigma_P^2} \\ & e = \frac{\sigma_E}{\sigma_P} \rightarrow e \sigma_P = \sigma_E \\ \end{align*}` --- # The correlation between two traits `\begin{align*} & Cov_P = Cov_A + Cov_E \\ & r_P\sigma_{P_X}\sigma_{P_Y} = r_A\sigma_{A_X}\sigma_{A_Y} + r_E\sigma_{E_X}\sigma_{E_Y} \\ \end{align*}` By subsitituing: `\begin{align*} & \sigma_E = e \sigma_P \\ & \sigma_A =h\sigma_P \\ \end{align*}` -- We get: `\begin{align*} & r_P\sigma_{P_X}\sigma_{P_Y} = r_Ah_X\sigma_{P_X}h_Y\sigma_{P_Y} + r_Ee_X\sigma_{P_X}e_Y\sigma_{P_Y} \\ \end{align*}` -- Divide through by `\(\sigma_{P_X}\sigma_{P_Y}\)`: `\begin{align*} & r_P = r_Ah_Xh_Y + r_Ee_Xe_Y \\ \end{align*}` --- # 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. --- # The correlation between two traits `\begin{align*} & r_P = r_Ah_Xh_Y + r_Ee_Xe_Y \\ \end{align*}` --- ### 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 conditional expectation: `\(E(A_Y | A_X)\)` -- .pull-left[ ``` ## `geom_smooth()` using formula = 'y ~ x' ``` <img src="w14_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 `\(R_X = ih_X^2 \sigma_{P_X} = ih_X\sigma_{A_X}\)`, 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.