In a previous post I summarily described our options for (generalized to varying degrees) linear mixed models from a frequentist point of view: nlme, lme4 and ASReml-R^{†}, followed by a quick example for a split-plot experiment.

But who is really happy with a toy example? We can show a slightly more complicated example assuming that we have a simple situation in breeding: a number of half-sib trials (so we have progeny that share one parent in common), each of them established following a randomized complete block design, analyzed using a ‘family model’. That is, the response variable (dbh: tree stem diameter assessed at breast height—1.3m from ground level) can be expressed as a function of an overall mean, fixed site effects, random block effects (within each site), random family effects and a random site-family interaction. The latter provides an indication of genotype by environment interaction.

setwd('~/Dropbox/quantumforest') trees <- read.table('treedbh.txt', header=TRUE) # Removing all full-siblings and dropping Trial levels # that are not used anymore trees <- subset(trees, Father == 0) trees$Trial <- trees$Trial[drop = TRUE] # Which let me with 189,976 trees in 31 trials xtabs(~Trial, data = trees) # First run the analysis with lme4 library(lme4) m1 <- lmer(dbh ~ Trial + (1|Trial:Block) + (1|Mother) + (1|Trial:Mother), data = trees) # Now run the analysis with ASReml-R library(asreml) m2 <- asreml(dbh ~ Trial, random = ~Trial:Block + Mother + Trial:Mother, data = trees)

First I should make clear that this model has several drawbacks:

- It assumes that we have homogeneous variances for all trials, as well as identical correlation between trials. That is, that we are assuming compound symmetry, which is very rarely the case in multi-environment progeny trials.
- It also assumes that all families are unrelated to each other, which in this case I know it is not true, as a quick look at the pedigree will show that several mothers are related.
- We assume only one type of families (half-sibs), which is true just because I got rid of all the full-sib families and clonal material, to keep things simple in this example.

Just to make the point, a quick look at the data using ggplot2—with some jittering and alpha transparency to better display a large number of points—shows differences in both mean and variance among sites.

library(ggplot2) ggplot(aes(x=jitter(as.numeric(Trial)), y=dbh), data = trees) + geom_point(colour=alpha('black', 0.05)) + scale_x_continuous('Trial')

Anyway, let's keep on going with this simplistic model. In my computer, a two year old iMac, ASReml-R takes roughly 9 seconds, while lme4 takes around 65 seconds, obtaining similar results.

# First lme4 (and rounding off the variances) summary(m1) Random effects: Groups Name Variance Std.Dev. Trial:Mother (Intercept) 26.457 5.1436 Mother (Intercept) 32.817 5.7286 Trial:Block (Intercept) 77.390 8.7972 Residual 892.391 29.8729 # And now ASReml-R # Round off part of the variance components table round(summary(m2)$varcomp[,1:4], 3) gamma component std.error z.ratio Trial:Block!Trial.var 0.087 77.399 4.598 16.832 Mother!Mother.var 0.037 32.819 1.641 20.002 Trial:Mother!Trial.var 0.030 26.459 1.241 21.328 R!variance 1.000 892.389 2.958 301.672

The residual matrix of this model is a, fairly large, diagonal matrix (residual variance times an identity matrix). At this point we can relax this assumption, adding a bit more complexity to the model so we can highlight some syntax. Residuals in one trial should have nothing to do with residuals in another trial, which could be hundreds of kilometers away. I will then allow for a new matrix of residuals, which is the direct sum of trial-specific diagonal matrices. In ASReml-R we can do so by specifying a diagonal matrix at each trial with `rcov = ~ at(Trial):units`

:

m2b <- asreml(dbh ~ Trial, random = ~Trial:Block + Mother + Trial:Mother, data = trees, rcov = ~ at(Trial):units) # Again, extracting and rounding variance components round(summary(m2b)$varcomp[,1:4], 3) gamma component std.error z.ratio Trial:Block!Trial.var 77.650 77.650 4.602 16.874 Mother!Mother.var 30.241 30.241 1.512 20.006 Trial:Mother!Trial.var 22.435 22.435 1.118 20.065 Trial_1!variance 1176.893 1176.893 18.798 62.606 Trial_2!variance 1093.409 1093.409 13.946 78.403 Trial_3!variance 983.924 983.924 12.061 81.581 ... Trial_29!variance 2104.867 2104.867 55.821 37.708 Trial_30!variance 520.932 520.932 16.372 31.819 Trial_31!variance 936.785 936.785 31.211 30.015

There is a big improvement of log-Likelihood from m2 (-744452.1) to m2b (-738011.6) for 30 additional variances. At this stage, we can also start thinking of heterogeneous variances for blocks, with a small change to the code:

m2c <- asreml(dbh ~ Trial, random = ~at(Trial):Block + Mother + Trial:Mother, data = wood, rcov = ~ at(Trial):units) round(summary(m2c)$varcomp[,1:4], 3) # Which adds another 30 variances (one for each Trial) gamma component std.error z.ratio at(Trial, 1):Block!Block.var 2.473 2.473 2.268 1.091 at(Trial, 2):Block!Block.var 95.911 95.911 68.124 1.408 at(Trial, 3):Block!Block.var 1.147 1.147 1.064 1.079 ... Mother!Mother.var 30.243 30.243 1.512 20.008 Trial:Mother!Trial.var 22.428 22.428 1.118 20.062 Trial_1!variance 1176.891 1176.891 18.798 62.607 Trial_2!variance 1093.415 1093.415 13.946 78.403 Trial_3!variance 983.926 983.926 12.061 81.581 ...

At this point we could do extra modeling at the site level by including any other experimental design features, allowing for spatially correlated residuals, etc. I will cover some of these issues in future posts, as this one is getting a bit too long. However, we could gain even more by expressing the problem in a multivariate fashion, where the performance of Mothers in each trial would be considered a different trait. This would push us towards some large problems, which require modeling covariance structures so we have a change of achieving convergence during our lifetime.

**Disclaimer:** I am emphasizing the use of ASReml-R because 1.- I am most familiar with it than with nlme or lme4 (although I tend to use nlme for small projects), and 2.- There are plenty of examples for the other packages but not many for ASReml in the R world. I know some of the people involved in the development of ASReml-R but otherwise I have no commercial links with VSN (the guys that market ASReml and Genstat).

^{†} There are other packages that I have not had the chance to investigate yet, like hglm.