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Details of the OPTEX Procedure |

The information-based
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
in detail
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 detail.

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 of*X*'*X*. - The D-optimal design is invariant to non-singular recoding
of the design matrix.

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 designs. 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
then
I-optimality is equivalent to the A-optimality, since the sum of
the prediction variances of all points **x** in the candidate space
*C* is

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 is predicted.

U-optimality seeks to minimize the sum of the distances from each candidate point to the design.

S-optimality seeks to maximize the harmonic mean distance from each design point to all the other points in the design.

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