Optimal and efficient crossover designs under different assumptions about the carryover effects

In certain studies it is desirable or necessary that a subject, such as a patient in a medical trial, receive a treatment in each period. This facilitates a within-subject comparison of the treatments. Designs for studies of this type are called crossover designs or repeated measurements designs. If there are s subjects in p periods, the design should specify which of the t treatments is assigned to subject j in period i, i = 1,... ,p,j = 1,..., s. Equivalently we may think of a design as assigning each subject to one of the t(p) possible treatment sequences. The choice of a design will clearly depend on the values of p, s, and t, to which we will refer as the design parameters. But for any set of design parameters, we will typically still have many design choices. To distinguish between different designs for the same design parameters, we will compare the designs under criteria that are related to the objective of the study. Often the objective is a comparison of the treatments, and we would choose a design that, in some sense, provides good estimates of the treatment differences. For these criteria, a design that is optimal under one statistical model may not be optimal under another. It is therefore also of interest to identify designs that are efficient (relative to an optimal design) for more than one model. The main difference in the models that we will consider is in how the possible first-order carryover effects are modeled. This is a controversial issue, and it is by no means our intent to resolve this here. But a design that is efficient under a variety of plausible models is preferable to one that performs well under one model but poorly under another. Our main focus will be on two models. One of these models has been considered extensively in the literature, while the other is relatively new. For selected design parameters, we will compare selected designs under these models.