Searching for Confounding Rules

### 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 defined previously. This section describes the sequential search performed by the FACTEX procedure to accomplish this goal.

First, construct the set of candidates for the first confounding rule, taking into account the set of effects not to be confounded with zero. If is empty, then no design is possible; otherwise, choose one of the candidates for the first confounding rule and construct the set of candidates for the second confounding rule, taking both and into account. If is empty, choose another candidate from ; 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 until the search fails because the set 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 . Traversing one of these branches corresponds to choosing the corresponding rule from . The branches attached to the node at the next level correspond to the candidates for the second rule given . In general, each node at level i of the tree corresponds to a set of feasible choices for rules , and the rest of the tree above this node corresponds to the set of all possible feasible choices for the rest of the rules.

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