# Calculates the average variance of pairwise differences from the variance matrix for predictions 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. ```r 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. ## Returns A `matrix` containing the within and between group A-optimality measures. ## 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 $r_{i} * r_{j} / mean(r)$, where $r$ is the `replication` vector and $r_{i}$ and $r_{j}$ are elements of $r$. 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`. ## Author(s) Chris Brien ## See Also `mat.Vpred`, `designAnatomy`. ## 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. ## Examples ```r ## 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) ```

designAmeasures function