Modelling and design of cross-over trials

There are many diseases and conditions that can be studied using a cross-over clinical trial, where the subjects receive sequences of treatments. The treatments are then compared using the repeated measurements taken 'within' subjects. The actual plan or design of the trial is usually obtained by consulting a published table of designs or by applying relatively simple rules such as using all possible permutations of the treatments. However, there is a danger is this approach because the model assumed for the data when the tables or rules were constructed may not be appropriate for the new trial being planned. Also, there may be restrictions in the new trial on the number of treatment sequences that can be used or on the number of periods of treatment particular subjects can be given. Such restrictions may mean that a published design of the ideal size cannot be found unless compromises are made. A better approach is to make the design satisfy the objectives of the trial rather than vice versa. In this paper we describe an approach to constructing such tailor-made designs which we hope will lead to ill-fitting 'off the peg' designs being a thing of the past. We use a computer algorithm to search for optimal designs and illustrate it using a number of examples. The criterion of optimality used in this paper is A-optimality but our approach is not restricted to one particular criterion. The model used in the search for the optimal design is chosen to suit the nature of the trial at hand and as an example a variety of models for three treatments are considered. We also illustrate the construction of designs for the comparison of two active treatments and a placebo where it can be assumed that the carry-over effects of the active treatments are similar. Finally, we illustrate an augmentation of a design that could arise when the objectives of a trial change.