We are now in a position to define the design optimality_criteria that have been implemented.
This is the log of the product of the canonical variances, called the D-criterion (for ``determinant''). The product is proportional to the volume of the confidence ellipsoid for joint estimation of the canonical contrasts.
This is the arithmetic mean of the canonical variances, called the A-criterion (for ``average''). It is also proportional to the average of the pairwise variances .
This is the mean of the squared canonical variances. For any fixed value of this is minimized when the are as close as possible in the square error sense. Thus it is a measure of balance of the design. A design is said to be variance balanced when all normalized treatment contrasts are estimated with the same variance. This occurs if and only if all the are equal, which gives the smallest conceivable (and often unattainable) value for for fixed . Among binary, equiblocksize designs, only balanced incomplete block designs achieve equality of the .
The largest pairwise variance ( ), called the MV-criterion (for ``maximum variance''). This is a minimax criterion: minimize the maximum loss (as measured by variance) for estimating the elementary contrasts.
The sum of the largest canonical variances, called the criterion. is usually called ``the'' E-criterion; minimization of is minimization of the worst variance over all possible normalized treatment contrasts. is the counterpart of maximum_pairwise_variances for the set of all contrasts. More generally, minimization of is minimization of the sum of the worst variances over all possible sets of normalized treatment contrasts whose estimators are uncorrelated. Thus the are a family of minimax criteria. is equivalent to . A design which minimizes all of the for is Schur-optimal (it minimizes all Schur-convex functions of the canonical variances).