Experiments with Two Treatments per Experimental Unit in the Presence of an Individual Covariate

Abstract

Temporal or spatial trends in experimental material can sometimes be largely eliminated from treatment comparisons by the stratification involved in, for example, Latin-square designs. As an alternative, covariance analysis on the trend variable is available and a recent example on this has been given by Federer and Schlottfeldt [1954] and further discussed by Outhwaite and Rutherford [1955]. Both of these methods eliminate the mean curvature over a set of experimental units or of replicates and while this may suffice in, for example, field experiments with neighbouring experimental units, it is less likely to be adequate when the latter are selected at random from some large population, or when treatments are compared a biological individual, for example, within a lactating cow, as shown by Cox [1956] or a tree-Rasmussen [1956]. The switchback trial with two treatments per block, for example, Lucas [1956], allows for individual linear trends by confounding treatment comparisons with the second-order curvatures. These, however, cannot always be safely neglected. An analysis eliminating individual curvatures of order up to one less than the number of periods, using sets of orthogonal squares, has been given by Patterson [1951] and this design also allows after-effects to be estimated. By the use of intra-period observations, the author [1956: 3, 7] has shown that individual curvatures of order limited only by the number of independent observations per individual may be removed to isolate the treatment comparisons in Latin-square designs. This particular design restriction has now been relaxed and a general solution giving within-individual treatment comnparisons free from general curvature effects, has b-een obtained. Population treatment