# Combinatorics Seminar 2018

Date: Thu, Oct 04, 2018

Time: 14:00 - 15:00

Venue: Large Meeting Room

Title: Multi-part balanced incomplete-block designs

Abstract:

In order to keep the protocol for a cancer clinical trial simple for each medical centre involved, it is proposed to limit each medical centre to only a few of the cancer types and only a few of the drugs.
Let $v_1$ be the total number of cancer types, and $v_2$ the total number of drugs. At the workshop on Design and Analysis of Experiments in Healthcare at the Isaac Newton Institute, Cambridge, UK in 2015, Valerii Fedorov listed the following desirable properties.

(a) All medical centres involve the same number, say $k_1$, of cancer types, where $k_1<v_1$.

(b) All medical centres use the same number, say $k_2$, of drugs, where $k_2<v_2$.

(c) Each pair of distinct cancer types are involved together at the same non-zero number, say $\lambda_{11}$, of medical centres.

(d) Each pair of distinct drugs are used together at the same non-zero number, say $\lambda_{22}$, of medical centres.

(e) Each drug is used on each type of cancer at the same number, say $\lambda_{12}$, of medical centres.

The first four conditions state that, considered separately, the designs for cancer types and drugs are balanced incomplete-block designs (a.k.a.\ $2$-designs) with the medical centres as blocks. We propose calling a design that satisfies all five properties a $2$-part $2$-design.

The parameters of a $2$-part $2$-design satsify some equations, and also an inequality that generalizes both Fisher's inequality and Bose's inequality.

I shall give several constructions of $2$-part $2$-designs, then generalize them to $m$-part $2$-designs.

Slides: View slides

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