The two-period cross-over clinical trial.

A cross-over clinical trial is one in which the effects of different treatments are compared on the same subject during different treatment periods. Such trials are useful when the treatments are intended to alleviate a condition, rather than effect a cure, so that after the first treatment is withdrawn the subject is in a position to receive a second. Two common examples are the comparison of anti-inflammatory drugs in arthritis and the comparison of hypotensive agents in essential hypertension. In both of these the symptom treated returns after the treatment is stopped, although the return may not be immediate or to the same degree. It would be most unwise to give the treatments in the same order for all the subjects, so order should always be chosen at random. Sometimes the choice is completely random but more commonly a restriction is introduced to ensure a balance between the treatments and orders. The main advantages and disadvantages of the cross-over design in clinical trials are well known. On the positive side a comparison of treatments on the same subject is expected to be more precise than a comparison between subjects and therefore to require fewer subjects for the same precision. On the negative side the task of disentangling treatment effects from both time and carry over effects from the previous treatment period can be difficult or even impossible. Although these points are essentially statistical they are rarely presented in precise statistical terms. In fact the published results of cross-over trials suggest that the appropriate statistical methods are less well known, or less -well understood, than those for parallel group trials. The substance of this paper is an explicit account of the models and methods for analysing a two-period cross-over design. It is not suggested that the results of all cross-over trials should be submitted to the whole range of statistical tests discussed. On the other hand any analysis should be clearly based on one of these models if it is to carry conviction. For a more mathematical treatment of the problems considered in this paper the reader should consult Grizzle (1965) and Gart (1969). A very careful discussion of the principles on which the design of cross-over trials are based is given in Cox (1958). The last reference should also be consulted for details of designs suitable for more than two periods.