Some efficient incomplete block sequences

Abstract

SUMMARY Serially balanced sequences of types 1 and 2 are used for designing experiments where one experimental unit receives several treatments over successive periods. Both the direct effect of a treatment applied in a certain period, as well as the residual effect of the treatment applied in the previous period, are of interest. Optimality properties of such sequences have been studied in the literature. However, these sequences have block sizes equal to the number of treatments v and so they pose a problem to the practitioner when v is large. In this paper, a class of sequences with incomplete blocks is proposed and their construction and analysis given. These sequences can be used to estimate direct and residual effects of treatments with high efficiencies. The sequences may be replicated over any number of units when the experiment is run with more than one unit. In many situations, experiments are to be designed where each experimental unit receives some or all of the treatments, one at a time, over successive periods of time. In such designs, each treatment has a 'direct effect' in the period in which it is applied and also a 'residual effect' in the following period. These experiments are called cross-over experiments or, following Hedayat & Afsarinejad (1975), repeated measurements experiments. The corresponding designs are called cross-over or repeated measurements designs. Such experiments are widely used in clinical trials, agricultural experiments, pharmaceutical experiments and in many other fields. One form of such experiments is one in which the number of experimental units must be very small, sometimes only one, but a far greater number of periods may be used. For such experiments, Finney & Outhwaite (1956) introduced serially balanced sequences of types 1 and 2, primarily for use in bioassay, where all the trials are made on one subject. A typical situation for using these sequences is one where many tests can be made in fairly rapid succession, so that one or more treatments may be assigned to the one subject. One of the earliest such examples is in Finney (1955, p. 132), where in the assay of histamine, the experimental unit is an isolated strip of guinea- pig's gut and a sequence of doses is applied to it. Serially balanced sequences are particularly useful here since, with repeated use of one strip of gut, trends in responsiveness may occur and sets of successive doses can be grouped into blocks so as to permit the elimination of the trend. For details of use of such sequences and for discussion and construction of these sequences we refer to Sampford (1957), Finney (1955), Finney & Outhwaite (1955), Finney (1960) and Street & Street (1987). Moreover, if more than one subject is to be used, then these sequences may be easily replicated over the available units. Optimality properties of repeated measurements designs have been investigated by, among others, Hedayat & Afsarinejad (1975, 1978), Cheng & Wu (1980), Magda (1980), Kunert (1983, 1984, 1985), Dey, Gupta & Singh (1983) and Bose (1993). Optimality properties of type 1 and type 2