A few years ago we had this really cool idea: we had to establish a trial to understand wood quality *in context*. Sort of following the saying “we don’t know who discovered water, but we are sure that it wasn’t a fish” (attributed to Marshall McLuhan). By now you are thinking WTF is this guy talking about? But the idea was simple; let’s put a trial that had the species we wanted to study (*Pinus radiata*, a gymnosperm) **and** an angiosperm (*Eucalyptus nitens* if you wish to know) to provide the contrast, as they are supposed to have vastly different types of wood. From space the trial looked like this:

The reason you can clearly see the pines but not the eucalypts is because the latter were dying like crazy over a summer drought (45% mortality in one month). And here we get to the analytical part: we will have a look only at the eucalypts where the response variable can’t get any clearer, trees were either totally dead or alive. The experiment followed a randomized complete block design, with 50 open-pollinated families in 48 blocks. The original idea was to harvest 12 blocks each year but—for obvious reasons—we canned this part of the experiment after the first year.

The following code shows the analysis in asreml-R, lme4 and MCMCglmm:

```
load('~/Dropbox/euc.Rdata')
library(asreml)
sasreml <- asreml(surv ~ 1, random = ~ Fami + Block,
data = euc,
family = asreml.binomial(link = 'logit'))
summary(sasreml)$varcomp
# gamma component std.error z.ratio
#Fami!Fami.var 0.5704205 0.5704205 0.14348068 3.975591
#Block!Block.var 0.1298339 0.1298339 0.04893254 2.653324
#R!variance 1.0000000 1.0000000 NA NA
# constraint
#Fami!Fami.var Positive
#Block!Block.var Positive
#R!variance Fixed
# Quick look at heritability
varFami <- summary(sasreml)$varcomp[1, 2]
varRep <- summary(sasreml)$varcomp[2, 2]
h2 <- 4*varFami/(varFami + varRep + 3.29)
h2
#[1] 0.5718137
library(lme4)
slme4 <- lmer(surv ~ 1 + (1|Fami) + (1|Block),
data = euc,
family = binomial(link = 'logit'))
summary(slme4)
#Generalized linear mixed model fit by the Laplace approximation
#Formula: surv ~ 1 + (1 | Fami) + (1 | Block)
# Data: euc
# AIC BIC logLik deviance
# 2725 2742 -1360 2719
#Random effects:
# Groups Name Variance Std.Dev.
# Fami (Intercept) 0.60941 0.78065
# Block (Intercept) 0.13796 0.37143
#Number of obs: 2090, groups: Fami, 51; Block, 48
#
#Fixed effects:
# Estimate Std. Error z value Pr(>|z|)
#(Intercept) 0.2970 0.1315 2.259 0.0239 *
# Quick look at heritability
varFami <- VarCorr(slme4)$Fami[1]
varRep <- VarCorr(slme4)$Block[1]
h2 <- 4*varFami/(varFami + varRep + 3.29)
h2
#[1] 0.6037697
# And let's play to be Bayesians!
library(MCMCglmm)
pr <- list(R = list(V = 1, n = 0, fix = 1),
G = list(G1 = list(V = 1, n = 0.002),
G2 = list(V = 1, n = 0.002)))
sb <- MCMCglmm(surv ~ 1,
random = ~ Fami + Block,
family = 'categorical',
data = euc,
prior = pr,
verbose = FALSE,
pr = TRUE,
burnin = 10000,
nitt = 100000,
thin = 10)
plot(sb$VCV)
```

You may be wondering Where does the 3.29 in the heritability formula comes from? Well, that's the variance of the link function that, in the case of the logit link is pi*pi/3. In the case of MCMCglmm we can estimate the degree of genetic control quite easily, remembering that we have half-siblings (open-pollinated plants):

```
# Heritability
h2 <- 4*sb$VCV[, 'Fami']/(sb$VCV[, 'Fami'] +
sb$VCV[, 'Block'] + 3.29 + 1)
posterior.mode(h2)
# var1
#0.6476185
HPDinterval(h2)
# lower upper
#var1 0.4056492 0.9698148
#attr(,"Probability")
#[1] 0.95
plot(h2)
```

By the way, it is good to remember that we need to back-transform the estimated effects to probabilities, with very simple code:

```
# Getting mode and credible interval for solutions
inv.logit(posterior.mode(sb$Sol))
inv.logit(HPDinterval(sb$Sol, 0.95))
```

Even if one of your trials is trashed there is a silver lining: it is possible to have a look at survival.

This is extremely interesting! I would tremendously appreciate a bit more of clarification, though… I am just moving into statistical genetics with my coral studies, to understand their adaptive responses to local environment and climate change. Working with lmer now, thinking of mcmc.. But to business: Your treatment of variance to derive heritability is very neat but could you please clarify a bit more, how did you derive the formula? Another question: how would you derive evolvability, which is additive genetic variance scaled by the square of the mean for the trait?

Hi Misha,

Many programs for the estimation of variance components use a unit variance (1) for the residuals; because of this, it is necessary to use the variance of the link function (logit in this case, 3.29) to scale the calculation (see, for example, this paper). If you use a family model (assuming half-siblings), you would need to construct evol using 4*sb$VCV[, 'Fami'] in the numerator and extract the mean from sp$Sol[1] (I think it Sol[1].

You can also use the pedigree for the analysis and use an 'animal model', which is what I use most of the time. I will cover that in a near(ish) future; probably in January.

In my opinion, if you are going to do quite a bit of work in quantitative genetics (which is what I do very often) your best bets are ASReml-R (for restricted maximum likelihood, I think it is way more flexible than lmer) or MCMCglmm (For Bayesian Analysis). ASReml-R is a commercial package, but it is free for academic use.

P.S. I just noticed that the denominator for the MCMCglmm estimation of h2 has an additional 1(it should finish with 3.29). I'm traveling right now but as soon as I have a time to rerun that part I will update the post.

Hello Luis,

Please could you explain me how do you get that 3.29 value you are using in the denominator for the heritability estimation

It is the variance of the logistic distribution when s=1 (pi^2/3), which is what many software packages fit.

Hey Luis – I’m sure this article and your comments are still useful to other people (not just me!), so thought I’d post the updated link for the paper you mention as an example for scaling the calculation of the variance: http://journalofanimalscience.org/content/81/6/1399.full.pdf

I guess it moved once again. The article is now found here: https://doi.org/10.2527/2003.8161399x

Hi Luis- Very interesting. Is there a typo in the ASREML and lmer case in the heritability estimates? I guess one should include in the denominator the value 1 (fixed value of the residuals) as you are doing in the MCMCglmm case. Also do you have by any chance a reference for the above equations?