Optimal Cross-Over Designs

Investigator: Howard Kushner, Ph.D.

In a repeated measurements (crossover, changeover) design whose purpose is the evaluation of treatment effects, a subject is sequentially exposed to study treatments, some of which may be repeats. The advantage of such a (crossover) design is that subjects become their own controls, thereby reducing the error variance.  In many cases, however, the observations are biased by carryover effects -- the persistence of a treatment effect from one administration to the next. The simplest crossover design contrasts two treatment interventions, A and B, with half of the subjects receiving A and then B (denoted AB) and the other half receiving BA. This design may produce biased estimates of intervention effects, but unbiased estimates may be obtained, for example, from the four sequence design AA, BB, AB, BA with one quarter of the patients on each sequence.  More complicated designs based on multiple interventions have also been considered. The problem is to find an optimal design, i.e., one that produces unbiased estimates, even in the presence of carryover, with the greatest possible precision.

Through a set of linear equations, Dr. Kushner (1997b) has given necessary and sufficient conditions for a repeated measurements design to be optimal. The variables in the equations are the optimal proportions of the total number of subjects in the trial to assign to the treatment sequences. The number of periods, the number of treatments, and the covariance matrix of the normal response model, is arbitrary. The equations were applied in two special cases. (a) the two-treatment case (Kushner, 1997a) and (b) the case of uncorrelated responses (Kushner, 1998).

(a) The two treatment case. Dr. Kushner gave the optimality equations for the simple case of two treatments. Methods for computing the solution to the equations were derived and illustrated. For dual-balanced designs, a single equation suffices to produce all optimal designs. Also a simple formula for the efficiency of non-optimal designs was derived. The theory was further applied to the case of an auto-regressive covariance matrix.

(b) The uncorrelated observations case. The case of uncorrelated responses is, in fact, fairly general. Kushner (1997b) showed that if the covariance matrix has a certain common structure, such as completely symmetric (as is frequently assumed in applications) or "type H" matrix, then the results are the same as in the uncorrelated case. The set of treatment sequences which comprise an optimal design were characterized and the optimality equations for this case given. A number of solutions, i.e., optimal designs, were found. Other results from the literature were reconsidered and were extended by using the new methods. Finally, the paper proves the inefficiency of two-period designs compared to higher-period designs. The marked advantage of three-period designs over two -period designs is demonstrated.

The specification of an optimal design requires that the covariance matrix of subject responses be known - a situation which is rarely true. To deal with this problem, an adaptive design approach was developed in Kushner (2000). The paper proposes an allocation rule that assigns new experimental subjects to treatment sequences. The allocation rule aims to maximize the treatment information that is to be obtained from these subjects. The efficacy of the allocation rule is demonstrated by simulated experiments. 

Cross-over designs are highly efficient and optimal designs even more so because they reduce the sample size requirement and enable unbiased estimates of intervention effects even in the presence of carry-over. For randomized controlled trials, such designs should be carefully considered in any application. This was not previously the case because of the danger of carryover and the absence of optimal designs for many circumstances.

Kushner, H.B. (2000). Allocation rules for adaptive repeated measurements design. Submitted for publication.

Kushner, H.B. (1999). H-symmetric optimal repeated measurements designs. Journal of Statistical Planning and Inference, 76:  pp. 235-61. 

Kushner, H.B. (1998). Optimal and efficient repeated-measurements designs for uncorrelated observations. Journal of American Statistical Association, 93 (443), Sept 1998: pp. 1176-1187

Kushner, H.B.(1997a). Optimality and efficiency of two-treatment repeated measurement designs. Biometrica, 84: pp. 455-468.

Kushner, H.B. (1997b). Optimal repeated measurements designs: the linear optimality equations. Annals of Statistics, 25: 2328-2344

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