*Theory of Orthogonal Designs* |

# Searching for Confounding Rules

The goal in constructing a design, then, is to find confounding
rules that do not confound with zero any of the effects in the set
*M* defined previously. This section describes the sequential
search performed by the FACTEX procedure to accomplish this goal.
First, construct
the set *C*_{1} of candidates for the first confounding rule, taking
into account the set *M* of effects not to be confounded with
zero. If *C*_{1} is empty, then no design is possible; otherwise, choose
one of the candidates for the first confounding rule and
construct the set *C*_{2} of candidates for the second confounding rule,
taking both *M* and *r*_{1} into account. If *C*_{2} is empty,
choose another candidate from *C*_{1}; otherwise, choose one of the
candidates rules and go on to the third rule. The search
continues either until it succeeds in finding a rule for every
non-run-indexing factor or the search fails because the set *C*_{1} is
exhausted.

The algorithm used by the FACTEX procedure to select confounding rules
is essentially a depth-first tree search. Imagine a tree
structure in which the branches connected to the root node correspond
to the candidates *C*_{1}. Traversing one of these branches corresponds to
choosing the corresponding rule *r*_{1} from *C*_{1}. The branches attached
to the node at the next level correspond to the candidates for the
second rule given *r*_{1}. In general, each node at level *i* of the tree
corresponds to a set of feasible choices for rules *r*_{1}, ... , *r*_{i},
and the rest of the tree above this node corresponds to the set of all
possible feasible choices for the rest of the rules.

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