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Multi-trait models

Facundo Muñoz
facundo.munoz@cirad.fr
famuvie

Orléans, Sep. 18, 2018

1 / 7

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
2 / 7

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

3 / 7

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
4 / 7

Implementation in breedR

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

## 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
)
5 / 7

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
6 / 7

Multi-trait models

  • Basic multivariate syntax

  • Long-shape with trait variable

7 / 7

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
2 / 7
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