The Construction of Balanced Designs for Experiments Involving Sequences of Treatments

This paper is concerned with the type of experimental design in which treatments are applied to the experimental material in a number of successive periods, each experimental unit receiving a different treatment in each period. Although some of the simpler examples of these designs have been in practical use in at least one field of research for several years, a systematic discussion of their general properties and the enumeration of possible designs does not appear to have been previously attempted. In order that estimates of the effects of treatments and the errors can be estimated by a reasonably simple statistical analysis certain elements of balance, set out below, are required. The requirements depend largely upon which residual effects are to be estimated. For example, any Latin square arrangement, in which the rows represent the different periods so that a column of symbols refers to a sequence of treatments, is suitable for the case in which residual effects are negligible. When first residual effects (i.e. the effects of treatments in the period after application) are to be estimated the class of available designs is more restricted. A simple example has been described by Cochran, Autrey & Cannon (1 94 1) in connexion with feeding experiments on dairy cows. This arrangement, shown in Fig. 1, consists of two particular Latin squares.