Multi-trait models

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# Multi-trait models
### Facundo Muñoz <a href="mailto:facundo.munoz@cirad.fr">facundo.munoz@cirad.fr</a> famuvie
### Orléans, Sep. 18, 2018

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# Multivariate Linear Mixed Models
## 2-trait case

`$$\begin{aligned}
    Y_1 & = X\beta_1 + Zu_1 + \varepsilon_1 \\
    Y_2 & = X\beta_2 + Zu_2 + \varepsilon_2, \\
    (u_1, u_2)' & \sim N(0, \Sigma_u \otimes G) \\
    (\varepsilon_1, \varepsilon_2)' & \sim N(0, \Sigma \otimes I_n).
  \end{aligned}$$`

- `$\Sigma_u$` and `$\Sigma$` either **diagonal** or **fully-parameterized** `$2\times 2$` matrices

- Some of the fixed or random effects can affect only a **subset of the traits**
    - *e.g.* fixed effect of operator

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# Limitation
## of breedR's implementation

- All fixed and random effects are assumed to be **trait-specific**

- **transversal effects** not directly supported (ultimately by PROGSF90)

- Simpler covariance structures **not supported**

- *e.g.* independent effects with shared variance, exchangeable structure
    
- A workaround is to **reshape the dataset** to long-layout

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background-image: url(img/reshaping.png)
background-position: 80% 30%
background-size: 40%

# Multi-trait with reshaping
## wide to long-layout

- Reshaping operation:
    - Stack traits into a **single variable** `value`
    - Additional variable `trait`
    - Duplicate individual information and other variables

- Use single-trait models with MET syntax
    - `trait` instead of `site`
    
- This overcomes the limitations breedR's multi-trait implementation
    - more complex models like multi-trait **and** multi-site become cumbersome

---

# Implementation in `breedR`

Specify the different traits in the main formula using `cbind()`.

```r
## Filter site and select relevant variables
dat <- 
 droplevels(
 douglas[douglas$site == "s3",
 names(douglas)[!grepl("H0[^4]|AN|BR|site",
 names(douglas))]]
 )

res <- 
 remlf90(
 fixed = cbind(H04, C13) ~ orig,
 genetic = list(
 model = 'add_animal', 
 pedigree = dat[, 1:3],
 id = 'self'),
 data = dat
 )
```

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A full covariance matrix across traits is estimated for each random effect,
and all results, including heritabilities, are expressed effect-wise:

```
## Formula: cbind(H04, C13) ~ 0 + orig + pedigree 
##    Data: dat 
##    AIC   BIC logLik
##  30968 31010 -15476
## 
## Parameters of special components:
## 
## 
## Variance components:
##                                       Estimated variances   S.E.
## genetic.direct.H04                                  918.1  438.6
## genetic.direct.H04_genetic.direct.C13              1872.4  824.0
## genetic.direct.C13                                 5827.6 1829.6
## Residual.H04                                       8373.7  461.7
## Residual.H04_Residual.C13                         10922.0  755.3
## Residual.C13                                      18439.0 1484.2
## 
##                  Estimate    S.E.
## Heritability:H04   0.0990 0.04589
## Heritability:C13   0.2391 0.07036
## 
## Fixed effects:
##              value    s.e.
## orig.H04.pA 352.00  6.2389
## orig.H04.pB 370.90 10.7947
## orig.H04.pC 346.93 13.0788
## orig.H04.pF 339.66  6.2268
## orig.H04.pG 313.00 24.0430
## orig.H04.pH 305.39 19.9334
## orig.H04.pI 323.29 20.0946
## orig.H04.pJ 343.87 19.8567
## orig.H04.pK 335.48 19.6409
## orig.C13.pA 460.01 13.6444
## orig.C13.pB 494.58 19.8635
## orig.C13.pC 430.86 25.5477
## orig.C13.pF 429.48 12.5501
## orig.C13.pG 376.42 48.3133
## orig.C13.pH 376.98 43.4266
## orig.C13.pI 404.62 43.6194
## orig.C13.pJ 418.91 43.2856
## orig.C13.pK 441.99 43.0567
```

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

# Multi-trait models

.left[
- Basic multivariate syntax

- Long-shape with `trait` variable
]