The external representation implements optimality criteria and other ordering criteria as aids in judging statistical properties of of members of a class of block designs. Definitions and motivating principles for these two classes of criteria are given here.
Denote by the class of information matrices for the
class of designs under consideration, that is,
While it is trivial to define ordering functions , what does it mean for a function to be an optimality criterion? Any ordering of information matrices could be allowed, but not all orderings reflect a reasonable statistical concept of optimality. We work here towards appropriate definitions.
The first fundamental consideration is that of relative interest
in the members of the treatment set. Let be the
class of permutation matrices. If treatments are of
equal interest, then order should satisfy the symmetry
Another fundamental principle arises from the nonnegative definite
ordering on information matrices:
Fact: In the reference universe of all binary block designs with treatments and fixed block size distribution,
Proof: The trace is fixed for all in the reference universe. Consequently says that is a nonnegative definite matrix with zero trace, that is, it is the zero matrix.
Thus the nonnegative definite ordering does not distinguish among ordering functions for the reference universe. While the external representation does not currently include nonbinary designs, we take as part of our definition that an optimality criterion must respect the nonnegative definite ordering; effectively, it must be able to make this fundamental distinction in the larger class of all designs with the same and block size distribution. A criterion that cannot do this has little (if any) capacity to detect inflated variances.
Typically one wishes to consider not arbitrary functions on the matrices , but functions of some characteristic(s) of those matrices. Of particular interest are the lists of canonical variances and pairwise variances. A criterion which is a function of a list of values should respect orderings of lists, as follows. A list of real values calculated from may be thought of as the uniform probability distribution for each . Probability distributions may be stochastically ordered: the distribution of is stochastically larger than that of , written , if Pr( Pr() for every . Thus define to be stochastically larger than , written , if for every . Criterion respects the stochastic ordering with respect to list if
The nnd order on information matrices (or their M-P inverses) implies the stochastic order on both the lists of canonical variances and the lists of pairwise variances.
Fact: In the reference universe of all binary block designs with treatments and fixed block size distribution, if is the list of canonical variances, then .
Proof: This follows from fixed trace of the information matrix in the reference universe, and that element-wise inversion of nonnegative lists reverses the stochastic ordering.
Thus every ordering criterion that is a function of the list of canonical variances trivially respects the stochastic order over the binary class. This may not be so for a criterion based on the list of pairwise variances.
A weaker ordering of lists than stochastic ordering, which is of
some interest and which is not trivially respected in the binary
class, is the weak majorization ordering. Let be
the largest member of list . Define to
weakly majorize , written
. If also equality of the two sums holds at ,
then is said simply to majorize .
Criterion respects the weak majorization ordering with
respect to list if
For any connected design , the inverses of the canonical variances are the eigenvalues of the information matrix . Now the list of eigenvalues has constant sum for all in the reference universe; for these lists, majorization and weak majorization are equivalent. Moreover, if two lists of eigenvalues are ordered by majorization, then the corresponding lists of canonical variances are ordered by weak majorization. Consequently, weak majorization can sometimes be determined for canonical variances over the reference universe via the corresponding eigenvalues of information matrices.
Relationships among the three ordering principles discussed are
We call a symmetric
ordering criterion an optimality criterion if (1) it
preserves the nnd definite ordering of information matrices over
the generalized universe of all designs for given and block
size distribution, and (2) it admits direct interpretation as a
summary measure of magnitude of variances of one or more
treatment contrast estimators. Each of the functions in
optimality_criteria possesses these two properties.
Ordering criteria can fall outside this scope yet still be of
interest, such as those provided in the element
other_ordering_criteria. These functions, discussed next,
typically fail on both requirements for an optimality criterion,
but may preserve orderings in restricted classes.
The -criterion ( ) is typically employed as the second step in a so-called -optimality argument: first maximize (that is, restrict to the binary class - our reference universe), then minimize . Within the binary class, preserves the weak majorization order on the canonical variances; outside of that class, it is possible to find considerably smaller values of , though inevitably at considerable cost on one or more optimality criteria. Thus may be viewed as an ordering criterion suitable for use in restricted classes, and/or in a subsidiary role to one or more optimality criteria in a multi-criterion design screening.
the weak majorization order over the binary class (indeed within
max_min_ratio_pairwise_variances preserves the
majorization order over that class. Both suffer the same defects
as outside the reference universe. Each of these three
criteria is a summary measure of scatter of variances, not of
magnitude; minimizing over too large a class will reduce scatter
at the cost of increasing magnitude.
Two additional ordering criteria implemented are the support
sizes of the distributions of canonical variances and pairwise
variances. These, too, can be informative as subsidiary criteria
in a multi-criterion design search, but because they do not
employ the values in the corresponding distributions,
no_distinct_pairwise_variances cannot be guaranteed to
preserve (outside of the reference universe) any of the list
orderings discussed. Like and the variance ratios, these
measures give information on scatter in a list of variances, and
thus are fairly called balance criteria.
absolute_comparisons" and \verb"calculated_comparisons.
These serve the same role, and are computed with the same rules,
calculated_efficiencies" for \verb"optimality_criteria.
other_ordering_criteria typically do not measure
magnitude of variance, we do not consider it correct
terminological usage to call their relative values ``