|Details of the OPTEX Procedure
An optimality criterion is a single number that summarizes how good a
design is, and it is maximized or minimized by an optimal design.
This section discusses in detail the optimality criteria available in
the OPTEX procedure.
Types of Criteria
Two general types of criteria are available: information-based
criteria and distance-based criteria.
criteria that are directly available are D- and A-optimality; they are
both related to the information matrix X'X for the design. This matrix
is important because it is proportional to the inverse of the
variance-covariance matrix for the least-squares estimates of the linear
parameters of the model. Roughly, a good design should "minimize" the
variance (X'X)-1, which is the same as "maximizing" the information
X'X. D- and A-efficiency are different ways of saying how large
(X'X) or (X'X)-1 are.
For the distance-based criteria, the candidates are viewed as comprising a
point cloud in p-dimensional Euclidean space, where p is the number
of terms in the model. The goal is to choose a subset of this cloud
that "covers" the whole cloud as uniformly as possible (in the
case of U-optimality) or that is as broadly "spread" as possible (in the
case of S-optimality). These ideas of coverage and spread are defined
at "Distance-based Criteria"
. The distance-based criteria
thus correspond to the
intuitive idea of filling the candidate space as well as possible.
The rest of this section discusses different optimality criterion in
D-optimality is based on the determinant of the information matrix
for the design, which is the same as the reciprocal of the determinant
of the variance-covariance matrix for the least-squares estimates of the
linear parameters of the model.
The determinant is thus a general measure of the size of (X'X)-1.
D-optimality is the most common criterion for computer-generated optimal
designs, which is why it is the default criterion for the OPTEX procedure.
The D-optimality criterion has the following characteristics:
- D-optimality is the most computationally efficient criterion to
optimize for the low-rank update algorithms of the OPTEX
procedure, since each update depends only on the variance of
prediction for the current design; see
"Useful Matrix Formulas"
- |X'X| is inversely proportional to the size of a confidence ellipsoid for the least-squares estimates of
the linear parameters of the model.
- |X'X|1/p is equal to the geometric mean of the eigenvalues
- The D-optimal design is invariant to non-singular recoding
of the design matrix.
A-optimality is based on the sum of the variances of the estimated
parameters for the model, which is the same as the sum of the diagonal
elements, or trace, of (X'X)-1. Like the determinant, the A-optimality
criterion is a general measure of the size of (X'X)-1. A-optimality
is less commonly used than D-optimality as a criterion for computer
optimal design. This is partly because it is more computationally
difficult to update; see "Useful Matrix Formulas" . Also, A-optimality is not invariant
to non-singular recoding of the design matrix; different designs will
be optimal with different codings.
Both G-efficiency and the average prediction variance are well-known
criteria for optimal design. Both are based on the variance of
prediction of the candidate points, which is proportional to
x'(X'X)-1x. As this formula shows, these two
criteria are also related to the information matrix X'X.
Minimizing the average prediction variance has also been called
I-optimality, the "I" denoting integration over the
It is possible to apply the search techniques available in the OPTEX
procedure to these two criteria, but this turns out to be a poor
way to find G- and I-optimal designs. One reason for this is that there
are no efficient low-rank update rules (see
"Useful Matrix Formulas"
), so that the searches can take a very long
time. More seriously, for G-optimality such a search often does not
converge on a design with good G-efficiency. G-efficiency is
simply too "rough" a criterion to be optimized by the relatively
short steps of the search algorithms available in the OPTEX procedure.
However, the OPTEX procedure does offer an approach for finding G-efficient
Begin by searching for designs according to the default
D-optimality criterion. Then, from the various designs found on the
different tries, you can save the one that has the best G-efficiency by
specifying the NUMBER=GBEST option in the OUTPUT statement. Since D- and
G-efficiency are highly correlated over the space of all designs, this
method usually results in adequately G-efficient designs, especially when
the number of tries is large. See the
ITER= option for
details on specifying the number of tries.
To find I-optimal designs, note that if the design is orthogonally coded
I-optimality is equivalent to the A-optimality, since the sum of
the prediction variances of all points x in the candidate space
where NC is the number of candidate points and XC is the design
matrix for the candidate points. Thus, you can use the option CODING=ORTH
in the PROC OPTEX statement together with the option CRITERION=A in the
GENERATE statement to search for I-optimal designs.
Note that both G- and I-optimality are invariant to non-singular recoding
of the design matrix, since the coding does not affect how well a point
The distance-based criteria are based on the distance
d(x,A) from a point x in the p-dimensional
Euclidean space Rp to a set .This distance is defined as follows:
where is the usual p-dimensional Euclidean
U-optimality seeks to minimize the sum of the distances from each candidate
point to the design.
where C is the set of candidate points and D is the set
of design points.
You can visualize the U criterion by associating with any design point
those candidates to which it is closest. Thus, the design defines a
clustering of the candidate set, and indeed cluster analysis has
been used in this context. Johnson, Moore, and Ylvisaker (1990) consider
a similar measure of design efficiency, but over infinite rather than
finite candidate spaces. Computationally, the U-optimality criterion can
be very difficult to optimize, especially if the matrix of all
pairwise distances between candidate points does not fit in memory. In
this case, the OPTEX procedure recomputes each distance as needed.
When searching for a U-optimal design, you should start with a small
version of the problem to get an idea of the computing resources required.
S-optimality seeks to maximize the harmonic mean distance from each
design point to all the other points in the design.
For an S-optimal design, the distances d(y,D-y)
are large, so the points are as spread out as possible. Since the
S-optimality criterion depends only
on the distances between design points, it is usually computationally
easier to compute and optimize than the U-optimality criterion, which
depends on the distances between all pairs of candidate points.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.