rat | time | resp |
---|---|---|

1 | After | 9.4 |

1 | Before | 8.7 |

2 | After | 9.8 |

2 | Before | 7.9 |

3 | After | 9.9 |

3 | Before | 8.3 |

4 | After | 10.3 |

4 | Before | 8.4 |

# ASR023. LMM specifying a generalized unstructured correlation matrix - Effect of ozone on lung capacity

The complete script for this example can be downloaded here:

### Dataset

The model that we will fit here is based on the D019 dataset, and the first few rows are presented below:

### Model

In this example we will fit a LMM specifying a generalized unstructured correlation matrix (`corgh()`

), and using initial values for the estimation of its variance components. The specification of the model is: \[
y = \mu + time + e\
\]

\(y\) is the lung capacity in milliliters,

\(\mu\) is the overall mean,

\(time\) is the fixed effect of the measurement time (*i.e.*, Before and After),

In the above model, the residual matrix \(\mathbf{R = U \otimes I_m}\) represents a direct product between a 2 \(\times\) 2 matrix (\(\mathbf{U}\)) describing the structure of each individual over their two measurements. This implies a variance for each time point and a correlation between residuals. In addition, the second matrix (\(\mathbf{I_m}\)) is describing the independence between the \(m\) individuals.

Before fitting our model, we need to obtain the initial values that we will use in the `corgh()`

structure. For this, we will fit two simpler models to obtain the variance components associated with the `time`

levels of “Before” and “After”. We will be using the ASReml-R option `subset`

to select each measurement separately. For the correlation between the two measurement, we will use a rough estimate of 0.5. Further more, the data have to be re-ordered to match the order specified by the residual formula, which in this case is by `time`

level. Note that before fitting the models, `rat`

, and `time`

need to be set as factors.

```
<- asreml(
asr023_before fixed = resp ~ 1,
subset = time == 'Before',
data = d019
)
```

`summary(asr023_before)$varcomp`

```
component std.error z.ratio bound %ch
units!R 0.2663636 0.1135778 2.345208 P 0
```

```
<- asreml(
asr023_after fixed = resp ~ 1,
subset = time == 'After',
data = d019
)
```

`summary(asr023_after)$varcomp`

```
component std.error z.ratio bound %ch
units!R 0.9753788 0.4159029 2.345208 P 0
```

So, now using the results from the above models, we can set up the initial values for our next model. This corresponds to three terms. The first is a guess of the correlation between the residual effects, followed by our previous estimates of the residual variance by `time`

level.

`<- c(0.5, 0.266, 0.975) initR `

Now, let’s take a look at how to write our final model with ASReml-R, but note that we are first sorting the data:

```
<- d019[order(d019$time),]
d019 <- asreml(
asr023 fixed = resp ~ time,
residual = ~corgh(time, init = initR):id(rat),
data = d019
)
```

### Exploring output

The variance components estimated from this model are:

`summary(asr023)$varcomp`

```
component std.error z.ratio bound %ch
time:rat!R 1.00000000 NA NA F 0.0
time:rat!time!Before:!time!After.cor 0.07936761 0.2994146 0.265076 U 0.1
time:rat!time_After 0.97537227 0.4142945 2.354297 P 0.3
time:rat!time_Before 0.26636364 0.1135774 2.345216 P 0.0
```

Here, V2 corresponds to the correlation of the residuals between “Before” and “After” with a relatively low value of 0.08. The components V3 and V4 correspond to their variance components.

Let’s take a look at the ANOVA table:

`wald(asr023, denDF = 'numeric')$Wald`

```
Df denDF F.inc Pr
(Intercept) 1 11 4070.00 1.776357e-15
time 1 11 15.09 2.541295e-03
```

This indicates that there are statistically significant differences between time points.

Finally, we can request the predictions for the factor `time’:

`predict(asr023, classify = 'time')$pvals`

```
time predicted.value std.error status
1 After 9.658333 0.2850992 Estimable
2 Before 8.450000 0.1489865 Estimable
```

These results indicate that after 30 days of exposure to ozone, the rats have increased, on average, their lung capacity.