class: center, middle, inverse, title-slide # Further common genetic models ### Facundo Muñoz<br/><a href="mailto:facundo.munoz@cirad.fr">facundo.munoz@cirad.fr</a><br/><i class="fab fa-twitter"></i> <i class="fab fa-github"></i> famuvie ### Orléans, Sep. 18, 2018 --- class: inverse, center, middle # Competition ## a.k.a. Indirect Genetic Effects --- # Diagnosis of Competition 1. Plot of residuals vs average neighbouring residuals Negative correlation, after accounting for Direct Genetic Effects and Spatial Autocorrelation 2. Variogram assessment Peak at the first lag in the variogram of residuals, after accounting for direct genetic effects and spatial autocorrelation 3. Model comparison Compare (*e.g.* AIC) _competition_ vs. _animal_ models --- # Competition model .pull-left[] .pull-right[ - Each individual have **two** (unknown) Breeding Values (BV): - `direct` BV affects its **own** phenotype, - `competition` BV affects its **neghbours'** - The total effect of the neighbouring `competition` BVs is given by their **distance-weighted sum** ] --- # Weighted neighbour competition effect .pull-left[ <!-- --> ] .pull-right[ `$$\omega_i(\alpha) = \sum_{k \in \partial i} z_{ik}(\alpha) u_{c,k}$$` ] Where `\(\partial i\)` be the set of __neighbouring locations__ of tree `\(i\)`, `\(u_c = (u_{c,k})'\)` the vector of __competition BVs__ and `\(z_{ik}(\alpha) \propto 1/d_{ik}^\alpha\)`, such that `$$\sum_{k \in \partial i} z_{ik}(\alpha)^2 = 1.$$` This condition is **variance-estabilizer** ensuring `\(\forall i\)`: `$$\text{Var}(\omega_i) = \text{Var}(u_c) = \sigma_c^2$$` --- # The decay parameter The **decay parameter** `\(\alpha\)` controls the **relative intensity of competition** of the neighbours .pull-left[ <!-- --> ] .pull-right[ - The weights `\(z_{ik}\)` are **scale-invariant** - *e.g.* a tree twice as far is weighted `\(1/2^\alpha\)` as much - higher values of `\(\alpha\)` concentrate the weights on the closest trees ] --- # Random-effect representation `$$Z_d u_d + Z_c(\alpha) u_c, \quad \begin{pmatrix} u_d \\ u_c \end{pmatrix} \sim \mathcal{N}\big(0, \Sigma_a \otimes A\big), \quad \Sigma_a = \begin{pmatrix} \sigma_d^2 & \sigma_{dc} \\ \sigma_{dc} & \sigma_c^2 \end{pmatrix}$$` - Each set of BVs is modelled as a zero-mean **random effect** with structure matrix given by the **pedigree** and independent **variances** `\(\sigma^2_d\)` and `\(\sigma^2_c\)` - Both random effects are modelled jointly with **covariance** `\(\sigma_{dc}\)` - `\(Z_d\)` is an indicator matrix linking observations and individuals - `\(Z_c(\alpha)\)` weights the competition effect of the neighbours with (fixed) **decay** parameter `\(\alpha\)` --- # Permanent Environmental Competition Effect `$$Z_p u_p, \quad Z_p = Z_c, u \sim \mathcal{N}(0, \sigma_p^2 I)$$` - **Optional** companion effect with **environmental** (rather than genetic) basis - Modelled as an individual **independent** random effect that affects **neighbouring** trees in the same (weighted) way --- # Implementation in breedR ```r fm <- remlf90( fixed = ···, random = ···, * genetic = list( * model = 'competition', * pedigree = ···, * coord = ···, * id = ···, * pec = TRUE/FALSE), data = ···, * method = 'em' ) ``` --- # Selection under competition .center[ <!-- --> ] --- class: inverse, center, middle # GBLUP & Dominance effects --- # Using genomic markers `$$Z u, \quad u \sim \mathcal{N}\big(0, \sigma_G^2 G\big)$$` - Use markers to compute a **relationship matrix** `\(G\)` for individuals - Several methods available - *e.g.* VanRaden et al. 2009 `$$G = XX'/\sum 2p(1-p)$$` - **Replace** the `additive-genetic` model, which uses the pedigree-based relationship matrix `\(A\)` with a `generic` model with a genomic relationship matrix `\(G\)` - `\(Z\)` is an **indicator** matrix linking observations with individuals - Predicts genetic value of **individuals**, not markers - Improved **accuracy** wrt pedigree-based evaluation --- # Relationship matrices ## pedigree-based vs. genomic  Note the increased level of detail in the relationship structure --- # Implementation in breedR - breedR allows random effects with **arbitrary covariance** structures (`generic` terms, see `?remlf90`) - These additional components allow to introduce random effects with **arbitrary** incidence and covariance/precision matrices `\(Z\)` and `\(\Sigma\)` ```r fm <- remlf90( ···, * generic = list( * G = list(Zg, Gmat), * ··· * D = list(Zd, precision = Dmat)), data = ··· ) ``` --- # Applications ## include __additional not-predefined components__ *e.g.* Dominance, Hybrid populations, Genomic evaluation, etc. --- class: inverse, center, middle background-image: url(img/breedRhex.png) background-position: 50% 80% # Further common genetic models .left[ - Competition - GBLUP - Dominance ]