ASR009. LMM for a randomized complete block design with spatial analysis - Wheat varieties

The complete script for this example can be downloaded here:

Dataset

The model that we will fit in this example is based on 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 model we will fit here is an extension of the model ASR001 that will now incorporate spatial analysis with the use of the ar1() function. The model is specified as:

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

\(y\) grain yield,

\(\mu\) is the overall mean,

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

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

\(e\) is the random residual effect, with \(e \sim \mathcal{N}(0,\, \mathbf{R})\).

Also, we assume for the \(\mathbf{R}\) matrix:

\(\mathbf{R} = AR1{_r} \otimes AR1{_c}\), which specifies an autocorrelation of order \(1\) for rows and columns, respectively.

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

asr009 <- asreml(
  fixed = yield ~ 1 + col,
  random = ~gen,
  residual = ~ar1v(col):ar1(row),
  data = d001
)

We have included the fixed effect of col to describe the blocks of this experiment.

It is important to indicate that we are using ar1v(col) and ar1(row) to specify the variance structure. The first includes explicitly a variance in front of the error structure. Alternatively, we could have used ar1(col) and ar1v(row) providing with the same results, but now the variance will be included in the second term.

Exploring output

This model has some BLUE effect (the intercept + blocks) that are shown below:

summary(asr009, coef = TRUE)$coef.fixed

              solution std error    z.ratio
(Intercept)  2.5826405 0.2438254 10.5921703
col_1        0.0000000        NA         NA
col_2       -0.1103087 0.2916326 -0.3782453
col_3       -0.2449970 0.3253569 -0.7530100

And, the variance components estimated from this model are:

summary(asr009)$varcomp

                component  std.error   z.ratio bound %ch
gen             0.1583183 0.04245445  3.729133     P 0.0
col:row!R       1.0000000         NA        NA     F 0.0
col:row!col!cor 0.2448299 0.12924554  1.894300     U 0.2
col:row!col!var 0.4738903 0.10441160  4.538675     P 0.1
col:row!row!cor 0.7368368 0.06126281 12.027473     U 0.1

Note that the spatial correlation are 0.25 and 0.74 for columns and rows, respectively.

The heritability (repeatability) is:

vpredict(asr009, h2 ~ V1/(V1+V4))

    Estimate         SE
h2 0.2504209 0.06648334

Some of the random effects (BLUPs) for the factor gen are:

BLUP <- summary(asr009, coef = TRUE)$coef.random
head(BLUP)

          solution std.error   z.ratio
gen_G01  0.4920799 0.2116153  2.325351
gen_G02  0.2514980 0.1981158  1.269449
gen_G03  0.2757732 0.1985501  1.388935
gen_G04  0.6693526 0.1956572  3.421048
gen_G05 -0.1219122 0.2099151 -0.580769
gen_G06 -0.3127731 0.1991610 -1.570454