PROC FACTEX: 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 $\text{[math]}$ defined previously. This section describes the sequential search performed by the FACTEX procedure to accomplish this goal.

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