The efficiency of block designs in general

Abstract

SUMMARY By the efficiency of a design is meant its precision relative to that of an orthogonal design with the same treatment replications. The general case of orthogonal block designs is first studied, because it provides the basis of comparison. An expression is then given for the efficiency of a general block design with respect to the estimation of treatment means. It leads to a relationship between the mean efficiency of a design and its dual. However, efficiency depends upon the equations of constraint that are adopted. For the estimation of differences of treatment means no design is ever more efficient than for the estimation of the means themselves. Also, for this purpose, the efficiency depends upon a set of quantities, which are here named ‘quasi-replications’, rather than upon the actual replications.