designAmeasures {dae} | R Documentation |

## Calculates the average variance of pairwise differences from the variance matrix for predictions

### Description

Calculates the average variance of pairwise differences between, or of
elementary contrasts of, predictions using the variance matrix for the
predictions. The weighted average variance of pairwise differences can be
computed from a vector of `replications`

, as described by Williams and
Piepho (2015). It is possible to compute either
A-optimality measure for different subgroups of the predictions. If groups
are specified then the A-optimality measures are calculated for the differences
between predictions within each group and for those between predictions from
different groups. If groupsizes are specified, but groups are not, the
predictions will be sequentially broken into groups of the size specified by
the elements of groupsizes. The groups can be named.

### Usage

`designAmeasures(Vpred, replications = NULL, groupsizes = NULL, groups = NULL)`

### Arguments

`Vpred` |
The variance |

`replications` |
A |

`groupsizes` |
A |

`groups` |
A |

### Details

The variance matrix of pairwise differences is calculated as
`v_{ii} + v_{jj} - 2 v_{ij}`

,
where `v_{ij}`

is the element from the ith row and jth column of
`Vpred`

. if `replication`

is not `NULL`

then weights are computed as
`r_{i} * r_{j} / \mathrm{mean}(\mathbf{r})`

,
where `\mathbf{r}`

is the `replication`

vector and `r_{i}`

and `r_{j}`

are elements of `\mathbf{r}`

. The `(i,j)`

element of the variance matrix of pairwise differences is multiplied by the
`(i,j)`

th weight. Then the mean of the variances of the pairwise
differences is computed for the nominated `groups`

.

### Value

A `matrix`

containing the within and between group A-optimality measures.

### Author(s)

Chris Brien

### References

Smith, A. B., D. G. Butler, C. R. Cavanagh and B. R. Cullis (2015).
Multi-phase variety trials using both composite and individual replicate
samples: a model-based design approach.
*Journal of Agricultural Science*, **153**, 1017-1029.

Williams, E. R., and Piepho, H.-P. (2015). Optimality and contrasts in
block designs with unequal treatment replication.
*Australian & New Zealand Journal of Statistics*, **57**, 203-209.

### See Also

### Examples

```
## Reduced example from Smith et al. (2015)
## Generate two-phase design
mill.fac <- fac.gen(list(Mrep = 2, Mday = 2, Mord = 3))
field.lay <- fac.gen(list(Frep = 2, Fplot = 4))
field.lay$Variety <- factor(c("D","E","Y","W","G","D","E","M"),
levels = c("Y","W","G","M","D","E"))
start.design <- cbind(mill.fac, field.lay[c(3,4,5,8,1,7,3,4,5,8,6,2),])
rownames(start.design) <- NULL
## Set up matrices
n <- nrow(start.design)
W <- model.matrix(~ -1+ Variety, start.design)
ng <- ncol(W)
Gg<- diag(1, ng)
Vu <- with(start.design, fac.vcmat(Mrep, 0.3) +
fac.vcmat(fac.combine(list(Mrep, Mday)), 0.2) +
fac.vcmat(Frep, 0.1) +
fac.vcmat(fac.combine(list(Frep, Fplot)), 0.2))
R <- diag(1, n)
## Calculate the variance matrix of the predicted random Variety effects
Vp <- mat.Vpred(W = W, Gg = Gg, Vu = Vu, R = R)
## Calculate A-optimality measure
designAmeasures(Vp)
designAmeasures(Vp, groups=list(fldUndup = c(1:4), fldDup = c(5,6)))
grpsizes <- c(4,2)
names(grpsizes) <- c("fldUndup", "fldDup")
designAmeasures(Vp, groupsizes = grpsizes)
designAmeasures(Vp, groupsizes = c(4))
designAmeasures(Vp, groups=list(c(1,4),c(5,6)))
## Calculate the variance matrix of the predicted fixed Variety effects, elminating the grand mean
Vp.reduc <- mat.Vpred(W = W, Gg = 0, Vu = Vu, R = R,
eliminate = projector(matrix(1, nrow = n, ncol = n)/n))
## Calculate A-optimality measure
designAmeasures(Vp.reduc)
```

*dae*version 3.2.28 Index]