Triads

We have studied a design with seven treatments, whose blocks are the 35 triples on the treatments. The blocks are arranged in seven rounds, such that:

This is called a triad design on 7 points.

(In the original application, it was required that the blocks be ordered so that no treatment occurs twice in the same position in any round. However, this condition can obviously be met by any triad design.)

Each round of such a design contains five blocks; one treatment occurs in three of the blocks and the others in two each. If we refer to the treatment of frequency 3 as the focus of that round, then each treatment is the focus of exactly one round. It is convenient to label the treatments as 1, 2, 3, 4, 5, 6, 7, and to order the rounds so that treatment i is the focus of round i.

If we delete from each round the two triples that do not contain the focus, the remaining triples are like

123    145    167,

and we associate this with the near-one-factor

1    23    45    67.

A set of seven factors of the form appropriate to be the near-one-factors associated with a 7-point triad design is called a compatible factorization.

In our paper we found all compatible factorizations and then tested to see which can be completed to form a triad design. In this way we constructed a complete census of triad designs on seven points.