When a factorial experiment involves many factors each of which is tested at two levels, economy of space and material may be effected by using only a fraction of a complete replicate of all possible combinations of levels of the factors. The consequence is that each main effect and each interaction has one or more aliases, and the experiment cannot distinguish which member of a set of aliases is responsible for an observed difference in yields. For less than five factors, this feature prevents fractional replicate designs from being much use; for five or more factors, the fraction can be so chosen that main effects and two-factor interactions have only higher order interactions as their aliases, and the possibility that these are important may often be ignored. Within the fraction of a replicate chosen for an experiment, confounding of interactions may be introduced in order to reduce the block size.

Designs likely to be useful in field experiments are those for six factors in four blocks of eight, for seven factors in eight blocks of eight, and for eight factors in eight blocks of sixteen or in sixteen blocks of eight; in the first of these a two-factor interaction is amongst those confounded, but elsewhere all main effects and two-factor interactions are unconfounded. The method of construction of the blocks, for these designs and those involving more factors, has been described, and an example of the statistical analysis of experimental results (which presents no special difficulties) has been given.