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### Optimality criteria

We are now in a position to define the design optimality_criteria that have been implemented.

phi_0

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.

phi_1

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 .

phi_2

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 .

maximum_pairwise_variances

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.

E_criteria

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).

Next: Other ordering criteria Up: Statistical Properties Previous: Pairwise variances   Contents
Peter Dobcsanyi 2003-12-15