Rosemary and I have just put on the arXiv a paper prompted by a question by Valery Fedorov. Suppose that we are testing a number v1 of drugs on a number v2 of cancer types. To simplify the protocol, at each centre involved in the trial, only a limited number k1 (less than v1) of drugs will be used, and only a limited number k2 (less than v2) of cancer types will be treated. Thus, at each centre, the number of combinations of drug and cancer type is k1k2, and these combinations have the structure of a rectangle. Other desirable “balance” properties are that each pair of distinct drugs are used together at a constant number of centres, each pair of distinct cancer types are treated together at a constant number of centres, and each drug is used on each cancer type at a constant number of centres.
These designs have some familiar aspects. If you know about triple arrays, you might think that they are just triple arrays, but this is not so: if we lay out the plan with drugs and cancer types as rows and columns, then each cell contains a fixed number (not necessarily 1) of centres; moreover, the last condition says just this, whereas the last condition for triple arrays asks for the intersection of the set of letters in a row and the set of letters in a column.
Of course it is interesting to know how few centres are required if all these conditions are met, for reasons of reducing the cost of the trial. One of the things we prove is that the number b of centres is at least v1+v2−1. Despite my claim above, this is exactly the same inequality that is known for triple arrays, and our proof works almost without change for triple arrays. Moreover, it also generalises the well-known inequalities of Fisher and Bose for balanced designs and balanced resolvable designs respectively (these two types of structure being “degenerate” examples of our new designs).
We give many different constructions.
The reason we have put it on the arXiv now is that it appears that these designs are actually useful, and being used, in practice; moreover, they might be useful even in a more general setting where combinations of drugs can be given to a patient.
But, inevitably, the effect of preparing it for the arXiv has led us to further insights, solutions to some questions, and new lines of possible inquiry. So the published version, if one is ever produced, will contain more …