I swear there was a point in writing an introduction to covariance structures: now we can start joining all sort of analyses using very similar notation. In a previous post I described simple (even simplistic) models for a single response variable (or ‘trait’ in quantitative geneticist speak). The R code in three R packages (asreml-R, lme4 and nlme) was quite similar and we were happy-clappy with the consistency of results across packages. The curse of the analyst/statistician/guy who dabbles in analyses is the idea that we can always fit a nicer model and—as both Bob the Builder and Obama like to say—yes, we can.

Let’s assume that we have a single trial where we have assessed our experimental units (trees in my case) for more than one response variable; for example, we measured the trees for acoustic velocity (related to stiffness) and basic density. If you don’t like trees, think of having height and body weight for people, or whatever picks your fancy. Anyway, we have our traditional randomized complete block design with random blocks, random families and that’s it. Rhetorical question: Can we simultaneously run an analysis for all responses?

```
setwd('~/Dropbox/quantumforest')
library(asreml)
canty <- read.csv('canty_trials.csv', header = TRUE)
summary(canty)
m1 <- asreml(bden ~ 1, random = ~ Block + Family, data = canty)
summary(m1)$varcomp
gamma component std.error z.ratio constraint
Block!Block.var 0.2980766 162.74383 78.49271 2.073362 Positive
Family!Family.var 0.1516591 82.80282 29.47153 2.809587 Positive
R!variance 1.0000000 545.97983 37.18323 14.683496 Positive
m2 <- asreml(veloc ~ 1, random = ~ Block + Family, data = canty)
summary(m2)$varcomp
gamma component std.error z.ratio constraint
Block!Block.var 0.1255846 0.002186295 0.0011774906 1.856741 Positive
Family!Family.var 0.1290489 0.002246605 0.0008311341 2.703059 Positive
R!variance 1.0000000 0.017408946 0.0012004136 14.502456 Positive
```

Up to this point we are using the same old code, and remember that we could fit the same model using lme4, so what's the point of this post? Well, we can now move to fit a multivariate model, where we have two responses at the same time (incidentally, below we have a plot of the two response variables, showing a correlation of ~0.2).

We can first refit the model as a multivariate analysis, assuming block-diagonal covariance matrices. The notation now includes:

- The use of
`cbind()`

to specify the response matrix, - the reserved keyword
`trait`

, which creates a vector to hold the overall mean for each response, `at(trait)`

, which asks ASReml-R to fit an effect (e.g. Block) at each trait, by default using a diagonal covariance matrix (σ^{2}**I**). We could also use`diag(trait)`

for the same effect,`rcov = ~ units:diag(trait)`

specifies a different diagonal matrix for the residuals (`units`

) of each trait.

```
m3 <- asreml(cbind(bden, veloc) ~ trait,
random = ~ at(trait):Block + at(trait):Family, data = canty,
rcov = ~ units:diag(trait))
summary(m3)$varcomp
gamma component std.error
at(trait, bden):Block!Block.var 1.627438e+02 1.627438e+02 78.492736507
at(trait, veloc):Block!Block.var 2.186295e-03 2.186295e-03 0.001177495
at(trait, bden):Family!Family.var 8.280282e+01 8.280282e+01 29.471507439
at(trait, veloc):Family!Family.var 2.246605e-03 2.246605e-03 0.000831134
R!variance 1.000000e+00 1.000000e+00 NA
R!trait.bden.var 5.459799e+02 5.459799e+02 37.183234014
R!trait.veloc.var 1.740894e-02 1.740894e-02 0.001200414
z.ratio constraint
at(trait, bden):Block!Block.var 2.073362 Positive
at(trait, veloc):Block!Block.var 1.856733 Positive
at(trait, bden):Family!Family.var 2.809589 Positive
at(trait, veloc):Family!Family.var 2.703059 Positive
R!variance NA Fixed
R!trait.bden.var 14.683496 Positive
R!trait.veloc.var 14.502455 Positive
```

Initially, you may not notice that the results are identical, as there is a distracting change to scientific notation for the variance components. A closer inspection shows that we have obtained the same results for both traits, but did we gain anything? Not really, as we took the defaults for covariance components (direct sum of diagonal matrices, which assumes uncorrelated traits); however, we can do better and actually tell ASReml-R to fit the correlation between traits for block and family effects as well as for residuals.

```
m4 <- asreml(cbind(bden, veloc) ~ trait,
random = ~ us(trait):Block + us(trait):Family, data = a,
rcov = ~ units:us(trait))
summary(m4)$varcomp
gamma component std.error
trait:Block!trait.bden:bden 1.628812e+02 1.628812e+02 7.854123e+01
trait:Block!trait.veloc:bden 1.960789e-01 1.960789e-01 2.273473e-01
trait:Block!trait.veloc:veloc 2.185595e-03 2.185595e-03 1.205128e-03
trait:Family!trait.bden:bden 8.248391e+01 8.248391e+01 2.932427e+01
trait:Family!trait.veloc:bden 1.594152e-01 1.594152e-01 1.138992e-01
trait:Family!trait.veloc:veloc 2.264225e-03 2.264225e-03 8.188618e-04
R!variance 1.000000e+00 1.000000e+00 NA
R!trait.bden:bden 5.460010e+02 5.460010e+02 3.712833e+01
R!trait.veloc:bden 6.028132e-01 6.028132e-01 1.387624e-01
R!trait.veloc:veloc 1.710482e-02 1.710482e-02 9.820673e-04
z.ratio constraint
trait:Block!trait.bden:bden 2.0738303 Positive
trait:Block!trait.veloc:bden 0.8624639 Positive
trait:Block!trait.veloc:veloc 1.8135789 Positive
trait:Family!trait.bden:bden 2.8128203 Positive
trait:Family!trait.veloc:bden 1.3996166 Positive
trait:Family!trait.veloc:veloc 2.7650886 Positive
R!variance NA Fixed
R!trait.bden:bden 14.7057812 Positive
R!trait.veloc:bden 4.3442117 Positive
R!trait.veloc:veloc 17.4171524 Positive
```

Moving from model 3 to model 4 we are adding three covariance components (one for each family, block and residuals) and improving log-likelihood by 8.5. A quick look at the output from m4 would indicate that most of that gain is coming from allowing for the covariance of residuals for the two traits, as the covariances for family and, particularly, block are more modest:

```
summary(m3)$loglik
[1] -1133.312
summary(m4)$loglik
[1] -1124.781
```

An alternative parameterization for model 4 would be to use a correlation matrix with heterogeneous variances (`corgh(trait)`

instead of `us(trait)`

), which would model the correlation and the variances instead of the covariance and the variances. This approach seems to be more numerically stable sometimes.

As an aside, we can estimate the between traits correlation for Blocks (probably not that interesting) and the one for Family (much more interesting, as it is an estimate of the genetic correlation between traits: `1.594152e-01 / sqrt(8.248391e01*2.264225e-03) = 0.37`

).