On Some Desirable Patterns in Block Designs

Abstract

SUMMARY A general iterative formula for the covariance matrix of the adjusted treatment means in block designs is given and its relation to design patterns discussed. It is shown that the general formula becomes very simple if the design has some degree of balance. Particularly interesting are those designs that combine features of balance and orthogonal designs. A matrix derived in a simple way from the incidence matrix proves useful in describing some properties of block designs. The simplification of the general formula depends entirely on the pattern of this matrix which also determines the efficiency of the design. Furthermore, its particular relation to treatment contrasts is helpful in designing block experiments of desirable properties. The construction of some useful designs that are simple in analysis and practical in application is illustrated by several examples.