Design orderings based on the information matrix

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 $\mathcal{C}$ the class of information matrices for the class of designs $\mathcal{D}$ under consideration, that is,

$$\mathcal{C}=\{C_d:d\in\mathcal{D}\}.$$

If $g$ map elements of $\mathcal{C}$ to a subset of the reals plus $\infty$, then $g$ provides an ordering on $d$:

$$d_1\geq_g d_2 \ \iff \ g(C_{d1})\leq g(C_{d2})$$

Usually $\mathcal{D}$ is our reference universe, but need not be so. In any case $\mathcal{D}$ is finite and $g(C_d)=\infty$ if and only if $d$ is disconnected.

While it is trivial to define ordering functions $g$, what does it mean for a function $g:\mathcal{C}\rightarrow\mathcal{R}$ 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 $v$ members of the treatment set. Let $\cal{P}$ be the class of $v \times v$ permutation matrices. If treatments are of equal interest, then order $g$ should satisfy the symmetry condition

$$g(C_d)=g(PC_dP') \hbox{ for every $P\in\cal{P}$.}$$

Only $g$ satisfying this condition are considered here.

Another fundamental principle arises from the nonnegative definite ordering on information matrices:

$$C_{d_1} \mbox{\ $\geq_{nnd}\ $}C_{d_2} \ \iff \ C_{d_1}-C_{d_2}\hbox{ is nonegative definite}$$

Now $C_{d_1}\mbox{\ $\geq_{nnd}\ $}C_{d_2} \ \iff \ C_{d_2}^+\mbox{\ $\geq_{nnd}\ $} C_{d_1}^+$ so this ordering says $\hbox{var}_{d_1}(\widehat{l'\tau})\leq \hbox{var}_{d_2}(\widehat{l'\tau})$ for every contrast $l'\tau$. A reasonable restriction to place on an optimality criterion $g$ is that it respect the nonnegative definite ordering:

$$C_{d_2}^+\mbox{\ $\geq_{nnd}\ $}C_{d_1}^+ \Rightarrow g(C_{d_1})\leq g(C_{d_2})$$

Fact: In the reference universe of all binary block designs with $v$ treatments and fixed block size distribution, $C_{d_2}^+\mbox{\ $\geq_{nnd}\ $}C_{d_1}^+ \ \iff \ C_{d_1}=C_{d2}$

Proof: The trace $\hbox{tr}(C_{d})$ is fixed for all $d$ in the reference universe. Consequently $C_{d_1}\mbox{\ $\geq_{nnd}\ $} C_{d_2}$ says that $C_{d_1}-C_{d_2}$ 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 $g$ 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 $g$ 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 $v$ 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 $C_d$, 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 $L_d$ of $s$ real values calculated from $C_d$ may be thought of as the uniform probability distribution $p(l)=\frac{1}{s}$ for each $l\in L_d$. Probability distributions may be stochastically ordered: the distribution of $X$ is stochastically larger than that of $Y$, written $X\mbox{\ $\geq_s\ $}Y$, if Pr($X\le a)\leq$ Pr($Y\leq a$) for every $a$. Thus define $L_{d_2}$ to be stochastically larger than $L_{d_1}$, written $L_{d_2}\mbox{\ $\geq_s\ $}L_{d_1}$, if $\vert L_{d_2}\leq a\vert\leq\vert L_{d1}\leq a\vert$ for every $a$. Criterion $g$ respects the stochastic ordering with respect to list $L$ if

$$L_{d_2}\mbox{\ $\geq_s\ $}L_{d_1} \Rightarrow g(C_{d_1})\leq g(C_{d_2})$$

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 $v$ treatments and fixed block size distribution, if $L$ is the list of canonical variances, then $L_{d_2}\mbox{\ $\geq_s\ $}L_{d_1}\iff \ L_{d_2}=L_{d1}$.

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 $L_{d[i]}$ be the $i^{th}$ largest member of list $L_d$. Define $L_{d_2}$ to weakly majorize $L_{d_1}$, written $L_{d_2}\mbox{\ $\geq_m\ $}L_{d_1}$, if $\sum_{i=1}^t L_{d_2[i]}\geq \sum_{i=1}^t L_{d_1[i]}$ for every $t=1,2,\ldots,s$. If also equality of the two sums holds at $t=s$, then $L_{d_2}$ is said simply to majorize $L_{d_1}$. Criterion $g$ respects the weak majorization ordering with respect to list $L$ if

$$L_{d_2}\mbox{\ $\geq_m\ $}L_{d_1} \Rightarrow g(C_{d_1})\leq g(C_{d_2}).$$

The weak majorization ordering is respected by every function of the form $g(C_d)=\sum_{i=1}^{s}h(L_{di})$ for continuous, increasing, convex $h$.

For any connected design $d$, the inverses of the canonical variances are the eigenvalues of the information matrix $C_d$. Now the list of eigenvalues has constant sum for all $d$ 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

$$\hbox{nnd ordering }\Rightarrow\hbox{ stochastic ordering }\Rightarrow\hbox{ weak majorization ordering}$$

the latter two for either the pairwise variances or the canonical variances. None of the implications can in general be reversed.

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 $v$ 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 $S$-criterion ( $\hbox{tr}(C_d^2)$) is typically employed as the second step in a so-called $(M,S)$-optimality argument: first maximize $\hbox{tr}(C_d)$ (that is, restrict to the binary class - our reference universe), then minimize $S$. Within the binary class, $S$ preserves the weak majorization order on the canonical variances; outside of that class, it is possible to find considerably smaller values of $S$, though inevitably at considerable cost on one or more optimality criteria. Thus $S$ 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 function max_min_ratio_canonical_variances preserves the weak majorization order over the binary class (indeed within any fixed $\hbox{tr}(C_d)$ class), and max_min_ratio_pairwise_variances preserves the majorization order over that class. Both suffer the same defects as $S$ 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_canonical_variances and no_distinct_pairwise_variances cannot be guaranteed to preserve (outside of the reference universe) any of the list orderings discussed. Like $S$ and the variance ratios, these measures give information on scatter in a list of variances, and thus are fairly called balance criteria.

Included with other_ordering_criteria are absolute_comparisons" and \verb"calculated_comparisons. These serve the same role, and are computed with the same rules, as absolute_efficiencies and calculated_efficiencies" for \verb"optimality_criteria. Because other_ordering_criteria typically do not measure magnitude of variance, we do not consider it correct terminological usage to call their relative values `` efficiencies."