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 |

# 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:

### 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.

```
<- asreml(
asr009 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:

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