class: center, middle, inverse, title-slide # Interactive deterministic formulas for genomic prediction ## ASCI431-831: Advanced Animal Breeding ### Gota Morota <br /><br /> <a href="http://morotalab.org/" class="uri">http://morotalab.org/</a> ### 2018/04/12 --- # Shiny - [https://shiny.rstudio.com/](https://shiny.rstudio.com/) - A web application framework for **interactive** visualization - Able to generate user friendly web interfaces - Built on a reactive programming model - Entirely extensible - custom inputs and outputs - CSS themes - JavaScript and D3.js - Example - [Collision Detection](https://bl.ocks.org/mbostock/raw/3231298/) --- # Shiny framework <img src="Shinyframework.png" height="300px" width="650px"/> **Template** ```r library(shiny) ui <- fluidPage() server <- function(input, output) {} shinyApp(ui = ui, server = server) ``` --- # Control widgets <img src="widgets.png" width=700 height=470> .left[[RStudio](https://shiny.rstudio.com/tutorial/written-tutorial/lesson3/)] .right[[Example](https://github.com/morota/asci431-2018/blob/gh-pages/day28/ex.R)] --- # Cross-validation for genomic prediction <div align="center"> <img src="Fig1CV.png" width=650 height=450> </div> .right[[doi:10.1093/jas/sky014](http://dx.doi.org/10.1093/jas/sky014)] --- # Deterministic genomic prediction formulas - highlight the relationships among prediction accuracy and potential factors influencing prediction accuracy - no computationally intensive cross-validation - prior to genotyping individuals --- class: inverse, left, middle # ShinyGPAS - Shiny Genomic Prediction Accuracy Simulator Can be used for - _interactive_ exploration of potential factors influencing prediction accuracy - simulation of achievable prediction accuracy - prior to genotyping individuals or performing CV - supporting in-class teaching - no knowlege of R, HTML, JavaScript, or CSS is required. R code encapsulated as a web-based Shiny application Available at [https://chikudaisei.shinyapps.io/shinygpas/](https://chikudaisei.shinyapps.io/shinygpas/) and [https://github.com/morota/ShinyGPAS](https://github.com/morota/ShinyGPAS) --- # Deterministic formula (1) Daetwyler et al. (2008; 2010) `\begin{align} r &= \sqrt{\frac{N h^2}{N h^2 + M_e} } \end{align}` where `\(N\)` is the number of individuals in the reference population, `\(h^2\)` is the heritability, and `\(M_e\)` is the number of independent chromosome segments. - treating genetic markers as fixed - independence of quantitative trait loci (QTL) - regression of phenotypes on genotype one locus at a time - `\(\sigma^2_{\epsilon}\)` and `\(\sigma^2_g + \sigma^2_{\epsilon}=1\)` - identical accuracy of QTL effect size estimates across QTL allele frequencies - perfect linkage disequilibrium (LD) between marker and QTL pairs --- # Deterministic formula (2) Goddard (2009) `\begin{align} r &= \sqrt{1 - \frac{\lambda}{2N\sqrt{\alpha}} \ln\left( \frac{1 + \alpha + 2\sqrt{\alpha}}{1 + \alpha - 2\sqrt{\alpha}}\right) } \end{align}` where `\(\lambda\)` is `\(M_e/(h^2\ln(2N_e))\)` and `\(\alpha\)` is `\(1 + 2(M_e/Nh^2\ln(2N_e))\)` - treating markers as random - assuming complete LD between marker and QTL pairs - QTL effects were assumed to be sampled from a normal distribution - assumes that QTL with high minor allele frequencies have more accurate effect size than QTL with low minor allele frequencies --- # Deterministic formula (3) Goddard et al. (2011) `\begin{align} r &= \sqrt{b \frac{Nbh^2/M_e}{1 + Nbh^2/M_e}} \end{align}` where `\(b = M/(M + M_e)\)` is the proportion of genetic variance explained by the markers and `\(M\)` is the is the number of markers. - accounts for incomplete LD between markers and QTL --- # Deterministic formula (4) Rabier et al. (2016) `\begin{align*} r &= \sqrt{\frac{h^2/(1-h^2)}{M_e/N + h^2/(1-h^2)}}. \end{align*}` - relaxing the assumption of `\(\sigma^2_{\epsilon}\)` and `\(\sigma^2_g + \sigma^2_{\epsilon}=1\)` - can be applied with any value of `\(\sigma^2_{\epsilon}\)` and `\(\sigma^2_g\)` --- # Deterministic formula (5) Rabier et al. (2016) `\begin{align*} r &= \sqrt{\frac{h^2/(1-h^2)}{\mathbb{E}(||\mathbf{x}'_{n_{\text{TRN} + 1}} \mathbf{X}'\mathbf{V}^{-1} ||^2) + h^2/(1-h^2)}} \end{align*}` `\(M_e/N\)` is equal to `\(\mathbb{E}(||\mathbf{x}'_{n_{\text{TRN} + 1}} \mathbf{X}'\mathbf{V}^{-1} ||^2)\)` under RRBLUP. - `\(\mathbf{x}'_{n_{\text{TRN} + 1}}\)` is the vector of markers belonging to the testing set individual - `\(\mathbf{X}\)` is the training set allele content matrix - `\(\mathbf{V} = \mathbf{X}\mathbf{X}' + \lambda \mathbf{I}\)` - `\(\lambda\)` is the regularization parameter - `\(||.||^2\)` is the squared norm Note that if we can assume individuals in training and testing sets were sampled from the same population, `\(\hat{\mathbb{E}}(||\mathbf{x}'_{n_{\text{TRN} + 1}} \mathbf{X}'\mathbf{V}^{-1} ||^2) \le 1\)` --- # Deterministic formula (6) de los Campos et al. (2013) Under the genomic best linear unbiased prediction framework `\begin{align} r &= \sqrt{ [1 - (1 - b)^2] h^2 } \end{align}` - assuming that the patterns of allele sharing between markers and causal loci are different - `\(b\)` is the average regression coefficient of the marker-based genomic relationships on genomic relationships at QTL --- # Deterministic formula (7) - Karaman et al. (2016) `\begin{align} r &= \sqrt{ h^2_M \left[ \frac{N h^2_M}{N h^2_M + M_e} \right] } \end{align}` - correlation between phenotypes and estimated breeding values (additive genetic values) - `\(h^2_M\)` is the genomic heritability, which is the proportion of phenotypic variance that is explained by regression on markers. --- # Deterministic formula (8) - Wientjes et al. (2016) `\begin{align} r = \sqrt{ \begin{bmatrix} b_{AC} r_{G_{AC}} \sqrt{\frac{h^2_A}{M_{e_{A,C}}} } & b_{BC} r_{G_{BC}} \sqrt{\frac{h^2_B}{M_{e_{B,C}}}} \end{bmatrix} \begin{bmatrix} \frac{h^2_A}{M_{e_{A,C}}} + \frac{1}{N_A} & r_{G_{AB}} \sqrt{\frac{h^2_A h^2_B}{M_{e_{A,C}} M_{e_{B,C}} } } \\ r_{G_{AB}} \sqrt{\frac{h^2_A h^2_B}{M_{e_{A,C}} M_{e_{B,C}} } } & \frac{h^2_B}{M_{e_{B,C}}} + \frac{1}{N_B} \end{bmatrix}^{-1} } \\ \times \sqrt{\begin{bmatrix} b_{AC} r_{G_{AC}} \sqrt{\frac{h^2_A}{M_{e_{A,C}}}} \\ b_{BC} r_{G_{BC}} \sqrt{\frac{h^2_B}{M_{e_{B,C}}}} \end{bmatrix}} \end{align}` Combines two populations A and B to predict prediction accuracy in population C. - `\(b_{AC}\)` is the square root of the proportion of the genetic variance in predicted population C explained by the markers in training population A - `\(r_{G_{AC}}\)` is the genetic correlation between populations A and C - `\(M_{e_{A,C}}\)` is the effective number of chromosome segments shared between populations A and C --- # Paper <img src="GSE.png" height="420px" width="710px"/> [doi:10.1186/s12711-017-0368-4](https://doi.org/10.1186/s12711-017-0368-4)