*Details of the OPTEX Procedure* |

## Optimal Blocking

Building on the work of Harville (1974), Cook and Nachtsheim (1989)
give an algorithm for finding D-optimal designs in the presence of
fixed block effects. In this case, the design for the original candidate
points is called the *treatment design*; the information matrix for
the treatment design has the form *X*'*AX* for a certain symmetric,
nonnegative-definite matrix *A* that depends on the blocks. The
algorithm is based on two kinds of low-rank changes to the treatment
design matrix *X*: *exchanging* a point in the design with a
potential treatment point, and *interchanging* two points in the design.
Cook and Nachtsheim (1989) give formulas for computing the resulting
change in *X*'*AX* and |*X*'*AX*|. These update formulas can be generalized
to apply whenever the information matrix for the treatment design has
the form *X*'*AX*, not just when *A* is derived from fixed blocks. This
is the basis for the optimal blocking algorithm in the OPTEX procedure.
Notice that you can combine several options to use the OPTEX procedure to
*evaluate* a design with respect to the fixed covariates. Assume
the design you want to evaluate is in a data set named EDESIGN. Then
first specify

generate initdesign=edesign method=sequential;

This makes the data set EDESIGN the treatment design. Then specify the
following BLOCKS statement options:
blocks {block-specification} init=chain iter=0;

The INIT=CHAIN option ensures that the starting ordering for the
treatment points is the same as in the EDESIGN data set, and the ITER=0
specification causes the procedure simply to output the efficiencies
for the initial design, without trying to optimize it.

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