ASR001. LMM for a randomized complete block design - Wheat varieties

The complete script for this example can be downloaded here:

Dataset

In this example we will use the D001 dataset, and the first few rows are presented below:

yield	gen	col	row
3.37917	G01	1	1
3.58748	G02	1	2
4.32812	G03	1	3
4.53642	G04	1	4
2.59224	G05	1	5
2.40708	G06	1	6

Model

The basic LMM for RCBD that we will fit in this example is:

\[ y = \mu + col + gen + e\ \] where,

\(y\) is the grain yield,

\(\mu\) is the overall mean,

\(col\) is the fixed effect of column (block),

\(gen\) is the random effect of treatment (genotype), with \(gen \sim \mathcal{N}(0,\,\sigma^{2}_{g})\),

\(e\) is the random residual effect, with \(e \sim \mathcal{N}(0,\,\sigma^{2}_{e})\).

Now, let’s take a look at how to write the model with ASReml-R. Note that before fitting the model, gen and col need to be set as factors.

asr001 <- asreml(
  fixed = yield ~ col,
  random = ~gen,
  residual = ~units,
  data = d001
)

Exploring output

Evaluate the statistical significance of fixed effect of block:

wald(asr001, denDF = 'numeric')$Wald

            Df denDF   F.inc        Pr
(Intercept)  1    49 967.900 0.0000000
col          2    98   1.135 0.3257146

Lets see the variance components:

summary(asr001)$varcomp

        component  std.error  z.ratio bound %ch
gen     0.1388994 0.06738716 2.061215     P   0
units!R 0.5153082 0.07361595 6.999953     P   0

Estimate the heritability:

vpredict(asr001, h2 ~ V1/(V1+V2))

    Estimate         SE
h2 0.2123171 0.09254986

The value of 0.212 for the heritability seems reasonable for this experiment, and the approximated standard error is also relatively small.

Extract the random additive genetic effects (BLUP/breeding values):

BLUP <- summary(asr001, coef = TRUE)$coef.random
head(BLUP)

           solution std.error    z.ratio
gen_G01  0.42473388 0.2793489  1.5204422
gen_G02  0.28676626 0.2793489  1.0265523
gen_G03  0.26607112 0.2793489  0.9524688
gen_G04  0.49716613 0.2793489  1.7797317
gen_G05 -0.06850108 0.2793489 -0.2452169
gen_G06  0.03497463 0.2793489  0.1252005

More information

Note that extensions of this model are available here: ASR002 and ASR009. The former uses covariates and the later incorporates spatial analysis with the ar1() function.