Optimality criteria

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

phi_0

$\Phi_0 = \sum log(z_{di})$
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

$\Phi_1 = \sum z_{di} / (v-1)$
This is the arithmetic mean of the canonical variances, called the A-criterion (for ``average''). It is also proportional to the average of the $v(v-1)/2$ pairwise variances $v_{dii'}$.

phi_2

$\Phi_2 = \sum z_{di}^2 / (v-1)$
This is the mean of the squared canonical variances. For any fixed value of $\Phi_1$ this is minimized when the $z_{di}$ 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 $z_{di}$ are equal, which gives the smallest conceivable (and often unattainable) value for $\Phi_2$ for fixed $\Phi_1$ . Among binary, equiblocksize designs, only balanced incomplete block designs achieve equality of the $z_{di}$.

maximum_pairwise_variances

The largest pairwise variance ( $\max(v_{dii'})$), 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

$z_{d1} + z_{d2} + \ldots + z_{di}$
The sum of the $i$ largest canonical variances, called the $E_i$ criterion. $E_1$ is usually called ``the'' E-criterion; minimization of $E_1$ is minimization of the worst variance over all possible normalized treatment contrasts. $E_1$ is the counterpart of maximum_pairwise_variances for the set of all contrasts. More generally, minimization of $E_i$ is minimization of the sum of the $i$ worst variances over all possible sets of $i$ normalized treatment contrasts whose estimators are uncorrelated. Thus the $E_i$ are a family of minimax criteria. $E_{v-1}$ is equivalent to $\Phi_1$. A design which minimizes all of the $E_i$ for $i = 1, \ldots, v-1$ is Schur-optimal (it minimizes all Schur-convex functions of the canonical variances).