The Desirability of Adjusting for Residual Effects in a Crossover Design

SUMMARY Crossover experiments, in which treatments are applied in sequence to the same experimental unit, may give misleading results if there are carryover effects from previous treatments in each sequence and these effects are not allowed for in the analysis. This paper is concerned with how to decide whether or not to allow for carryover effects. The general conclusion is that, despite the increased variance resulting from adjustment, the safest policy is always to allow for these effects. In a crossover design different treatments are applied in turn to the same experimental unit (Kershner and Federer, 1981; Patterson, 1951). The commonest applications are in nutritional trials on animals and in clinical trials of drugs. However, one difficulty associated with this design is that, from the second period onwards, a residual or carryover effect may be present due to the treatment applied in the previous period. This problem can be mitigated by ensuring that each treatment is preceded equally often by every other treatment. Williams (1949) has shown how Latin squares can be chosen to achieve this balance. Even with this balance, the straightforward comparison of any two treatment means will be biased if their residual effects differ, because no treatment is preceded by itself. The repetition of one of the periods in order to achieve a complete balance of the residual effects, and hence remove the bias, is often impossible because of the time that would be required for the extended experiment. When crossover designs are used for treatment comparisons, the usual procedure is to test for residual effects and, only if they are significantly different from zero, adjust for them in the estimation of treatment effects. The decision whether to adjust or not is, therefore, based on a significance test applied at an arbitrarily specified significance level. If, however, the precise estimation of treatment effects is of prime interest, procedures which lead to estimators that have smallest mean square errors (MSEs) are more appropriate. The purpose of this paper is to investigate the effectiveness of such a procedure; attention is restricted to the case in which it is reasonable to suppose that the duration of residual effects is limited to the period immediately following the direct application of the treatment. We shall assume that the major interest is in the effect of the treatments during the period of their application. Some combination of direct and residual effects would merit consideration if, in practice, the treatment is to be applied for a longer period.