Genotype-Environment interaction

# Genotype-Environment interaction
## in multi-site trials
### Facundo Muñoz <a href="mailto:facundo.munoz@cirad.fr">facundo.munoz@cirad.fr</a> famuvie
### Orléans, Sep. 18, 2018

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# Excercise 1

## Fit a __reference__ model for the variable `C13` of the `douglas` dataset (included with `breedR` with fixed effects of provenance (`orig`) and site (`site`) and a random family (`mum`) effect

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## Douglas dataset

- Genetic material (provenances and families) planted in 3 sites

- Circumference in 2013 (`C13`) measured in the 3 sites

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# Excercise 2

## Fit a simple __interaction model__ with constant variance accross sites.

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# Multiple _environments_

- **Environments** may refer to sites, but also to years or climates

- Here we will develop the __multi-site__ case.

- The other cases might require different
approaches (e.g. continuous variation, shared spatial effects, etc.)

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# Basic interaction

## which are the __genotype__ and __environment__ variables in the `globulus` dataset?

- Remember from module 1 how to crate __interactions__

- Is this interaction a __fixed__ or __random__ effect?

- what did I mean by _constant variance across sites_?

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# Basic interaction model

`$$\begin{aligned}
  \text{C13} & \sim \text{orig} + \text{site} + \text{mum} + f_e + \varepsilon \\
  \text{mum} & \sim \mathcal N(0, \sigma_f^2) \\
  f_e & \sim \mathcal N(0, \sigma_{f:e}^2) \\
  \varepsilon & \sim \mathcal N(0, \sigma^2)
\end{aligned}$$`

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# Excercise 3

## Extend the previous model to account for __site-varying interaction__ variances.

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# Heterogeneous-variance interaction model

`$$\begin{aligned}
  \text{C13} & \sim \text{orig} + \text{site} + \text{mum} + \sum_{e=1}^3 f_e \mathbb{1}_{e} + \varepsilon \\
  \text{mum} & \sim \mathcal N(0, \sigma_f^2) \\
  f_1 & \sim \mathcal N(0, \sigma_{f:1}^2) \\
  f_2 & \sim \mathcal N(0, \sigma_{f:2}^2) \\
  f_3 & \sim \mathcal N(0, \sigma_{f:3}^2) \\
  \varepsilon & \sim \mathcal N(0, \sigma^2)
\end{aligned}$$`

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# Data preparation

- The previous model requires creating 3 new variables to represent
the interactions with each site

| site | family | f1 |  f2  |  f3  |
|:----:|:------:|:--:|:----:|:----:|
|  s1  |   1    | 1  |      |      |
|  s1  |   2    | 2  |      |      |
|  s2  |   1    |    |  1   |      |
|  s2  |   2    |    |  2   |      |
|  s3  |   1    |    |      |   1  |
|  s3  |   2    |    |      |   2  |

- When added, only one of them will affect the __likelihood__
of each observation, depending on its site

- `breedR` will automatically fit three __independent variances__

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class: middle, inverse

# Excercise 4

## Examine and compare the site-specific breeding values from each of the three models above

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# Extracting Breeding Values

- Remember from Module 1 how to extract PBVs safely (`model.matrix() %*% ranef()`)

- Here, the PBVs are the sum of the main effects and the interactions

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# Genetic correlations

- Ideally, we would like to fit the following model:

`$$\begin{aligned}
  \text{C13} & \sim \text{orig} + \text{site} + \text{mum} + \sum_{e=1}^3 f_e \mathbb{1}_{e} + \varepsilon \\
  \text{mum} & \sim \mathcal N(0, \sigma_f^2) \\
    (f_1, f_2, f_3)' & \sim
    \mathcal{N}\big(0, \Sigma_{G\times E} \otimes \mathrm I\big) \\
  \varepsilon & \sim \mathcal N(0, \sigma^2)
\end{aligned}$$`
  but unfortunately, `breedR` does not support it.

- Instead, we can always check the __empirical correlations__ of the
PBVs from the previous model.

- We can also compute the Type-B genetic correlation (as a function of variances)

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# Ecovalence

Similarly, we can derive the _Ecovalence_ and other measures of
interactivity from the PBVs

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# Site-specific spatial effects

We can leverage the `generic` model to fit three independent spatial effects

```r
## Use breedR internal functions to compute splines models on each site
sp1 <- breedR:::breedr_splines(douglas[douglas$site == 's1', c('x', 'y')])
sp2 <- ...; sp3 <- ...

## Manually build the full incidence matrices with 0
## and the values computed before
mm1 <- model.matrix(sp1)
inc.sp1 <- Matrix::Matrix(0, nrow = nrow(douglas), ncol = ncol(mm1))
inc.sp1[douglas$site == 's1', ] <- mm1
inc.sp2 <- ...; inc.sp2 <- ...

reml.spl <- remlf90(
 ...
 generic = list(sp1 = list(inc.sp1, breedR:::get_structure(sp1)),
 sp2 = ...
 sp3 = ...),
 ···,
 method = 'em'
)
```

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# Site-specific spatial effects

---
class: inverse, center, middle
background-image: url(img/breedRhex.png)
background-position: 50% 90%

# Genotype-Environment interaction

## In multi-site trials