Crossover designs for comparing treatments with a control

Abstract

SUMMARY We introduce a class of crossover designs with residual effects for experimental situations in which primary interest centres on the comparison of several test treatments to a control treatment rather than on all pairwise comparisons among the treatments. We shall call these designs 'control balanced residual effects designs'. Their structure is studied in detail, some methods of their construction are given and their relative efficien- In crossover designs, each experimental subject receives some or all of the treatments in an appropriate sequence over a number of successive periods. These designs have proved attractive because of their economical use of experimental subjects and because of the more sensitive treatment comparisons that result from the elimination of intersubject variation. However, the application of a sequence of treatments in this manner has the potential of producing a direct effect of the treatment in the period of its application as well as a residual or carryover effect in the periods following the application of the treatment. A residual effect which persists in the ith period after its application is called a residual effect of the ith order. It is common to assume that second- and higher-order residual effects are negligible and consequently we restrict our attention to crossover designs having only first-order residual effects. The most commonly used crossover designs are the balanced residual effects designs of Williams (1949) which consist of either one or two Latin squares such that every ordered pair of treatments occurs in successive periods an equal number of times. Patterson (1950, 1951, 1952) developed balanced residual effects designs for which the number of periods is less than the number of treatments. These designs are variance balanced for the estimation of direct effects contrasts as well as for residual effects contrasts. However, in many experimental situations, interest is focused on the simultaneous comparison of several test treatments to a control treatment rather than on all pairwise comparisons. In this paper, we formally define a class of residual effects designs which will be called control balanced residual effects designs since they are balanced with respect to the comparison of test treatments with a control treatment. After developing the definition of a control balanced residual effects design, some methods of their construction are given and their relative efficiencies are evaluated in terms of the variances