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

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

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().

## 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
  )

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