*Details of the OPTEX Procedure* |

## Design Efficiency Measures

The output from the OPTEX procedure includes efficiency measures for the
resulting designs according to various criteria. This section
gives the precise definitions for these measures.
By default, the OPTEX procedure calculates the following efficiency measures
for each design found in its search for an optimum design:

where *p* is the number of parameters in the linear model, *N*_{D} is the
number of design points, and *C* is the set of candidate
points. The D- and A-efficiencies are the relative number of runs
(expressed as percents) required by a hypothetical orthogonal design
to achieve the same |*X*'*X*| and trace(*X*'*X*)^{-1}, respectively;
refer to Mitchell (1974b).
When you specify a BLOCKS statement, the D- and A-efficiencies for
the treatment part of the model are calculated. These are calculated
similarly to the preceding efficiencies, except that they are based on the
information matrix after correcting for covariate effects. This matrix
can be written as *X*'*AX* for a symmetric, positive definite matrix *A*
that depends on the model for the covariate effect. If you specify a
block structure or a covariate model, then *A*=*I* - *Z*(*Z*'*Z*)^{-1}*Z*', where
*Z* is the design matrix for the block or covariate effect. Alternatively,
you can use the COVAR= option to specify the matrix *A* directly. Given
*A*, the efficiencies in the presence of covariates are defined as follows:

where are the *p* largest eigenvalues of *A*.
If you use the STRUCTURE= block model specification and there is only one
class variable in the treatment model, then the design fits into the
traditional block design framework. In this case, the D-efficiency
relative to a balanced incomplete block design is also listed.
Because these efficiencies measure the goodness of the design relative to
theoretical designs that may be far from possible in many cases, they are
typically not useful as absolute measures of design goodness. Instead,
efficiency measures should be used relatively, to compare one design
to another for the same situation.

For the distance-based criteria, there are no simple measures of
design efficiency that can be scaled from 0 to 100. See
the "Output" section
for a definition of the design measures
tabulated for these criteria.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.