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 $\mc {M}$ 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 $\mc {M}$ of effects not to be confounded with zero. If $C_1$ is empty, then no design is possible; otherwise, choose one of the candidates $r_1 \in C_1$ for the first confounding rule and construct the set $C_2$ of candidates for the second confounding rule, taking both $\mc {M}$ and $r_1$ into account. If $C_2$ is empty, choose another candidate from $C_1$; otherwise, choose one of the candidates rules $r_2 \in C_2$ 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 $C_1$ 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 $r_1, \ldots , r_ i$, and the rest of the tree above this node corresponds to the set of all possible feasible choices for the rest of the rules.