On a Minimax Property of a Balanced Incomplete Block Design

Abstract

It is shown that for a given set of parameters ($b$ blocks, $k$ plots per block and $v$ treatments), among the class of connected incomplete block designs, a balanced incomplete block design (if it exists) is the design which maximizes the minimum efficiency, efficiency being defined as $$\frac{\text {Variance of an estimated treatment contrast in a randomized block}{Variance of the estimated treatment contrast in the incomplete block}}.$$ The proof will be preceded by a lemma.