designAmeasures {dae}

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 matrix of the predictions. It can be obtained using mat.Vpredicts.

replications

A numeric vector whose length is equal to the number of rows (columns) in Vpred and whose elements are to be used to calculate weights for each pair of differences.

groupsizes

A numeric containing group sizes. The sum of the elements of groupsizes must be less than or equal to the order of Vpred. If groupsizes is a named vector, the names are used to label the groups. If NULL, either groups is used or the average for all pairwise differences is obtained.

groups

A list, each element of which is a numeric, vector with integers that specify the subroup of the predictions over whose pairwise differences the variances are to be averaged. If there is more than one group, the variances of all between and within group pairwise differences are averaged. If the elements of groups are named , the names are used to label the groups. If groups is NULL, either groupsizes is used or the average for all pairwise differences is obtained.

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

mat.Vpred, designAnatomy.

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)