ASR002. LMM for a randomized complete block design with covariates - 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 LMM that we will fit in this example is an extension of the ASR001 model by including covariates. This model’s equation is:

\[ y = \mu + col + row + gen + e\ \]

where,

\(y\) is the grain yield,

\(\mu\) is the overall mean,

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

\(row\) is the fixed covariate effect of row,

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

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

Now, let’s take a look at how to write the model with ASReml-R. Note that before fitting the model, gen needs to be defined as a factor, and col and row need to be set as (co)variates. This is shwown in the next few lines of code, followed by the model as defined in ASReml-R.

d001$gen <- as.factor(d001$gen)
d001$row <- as.numeric(d001$row)
d001$col <- as.numeric(d001$col)

asr002 <- asreml(
  fixed = yield ~ col + row,
  random = ~gen,
  residual = ~units,
  data = d001
)

Exploring output

Evaluate the statistical significance of fixed effects:

wald(asr002, denDF = 'numeric')$Wald

            Df denDF    F.inc         Pr
(Intercept)  1  48.8 1318.000 0.00000000
col          1  98.1    4.855 0.02990027
row          1 127.4  129.600 0.00000000

We report below the fixed effects (BLUEs):

summary(asr002, coef = TRUE)$coef.fixed

               solution   std error    z.ratio
(Intercept)  3.57713516 0.143371942  24.950036
col         -0.10808700 0.049053064  -2.203471
row         -0.03563415 0.003129811 -11.385402

From the table above, negative slopes are associated with increases in both column and row numbers.

And the variance component estimates are:

summary(asr002)$varcomp

        component  std.error  z.ratio bound %ch
gen     0.1478455 0.04754391 3.109662     P   0
units!R 0.2406203 0.03434901 7.005160     P   0

Finaly, we can estimate the heritability as:

vpredict(asr002, h2 ~ V1/(V1+V2))

    Estimate         SE
h2 0.3805882 0.09004703