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 C₁ of candidates for the first confounding rule, taking into account the set M of effects not to be confounded with zero. If C₁ 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 C₂ of candidates for the second confounding rule, taking both M and r₁ into account. If C₂ is empty, choose another candidate from C₁; 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 the search fails because the set C₁ 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 C₁. Traversing one of these branches corresponds to choosing the corresponding rule r₁ from C₁. The branches attached to the node at the next level correspond to the candidates for the second rule given r₁. In general, each node at level i of the tree corresponds to a set of feasible choices for rules r₁, ... , 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.

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