13.1 - Introduction to Cross-over Designs

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Cross-over designs begin with a simple concept and quickly becomes one of the more difficult methods in ANOVA. The simplest cross-over design is a 2-treatment level, 2-period design. If we use A and B to represent the two levels of an experimental treatment, then we can build the following table:

Sequence Period 1 Period 2
1 A B
2 B A

Experimental units are randomly assigned to receive one of the two different sequences. For example, if this were a clinical trial, patients assigned to sequence 2 would be given treatment B first, then after assessment of their condition, given treatment A and their condition re-assessed.

The complicated part of the cross-over design is the potential for carry-over effects. A carry-over effect is when the response to a particular treatment level has been influenced by the previous application of a different treatment level. The presence of carry-over effects are dealt with differently by various authors and research institutions. Avoiding carry-over effects is one solution to this potential problem. This is usually accomplished with a sufficient wash-out period. A washout period is a gap in time between the application of the treatment levels such that any residual effect of a previous treatment level has dissipated and there is no detectable carry-over effect.

timeline with carry over and wash out periods

One philosophy is that if there is a significant carryover effect, then the experiment has been compromised and unless a sufficient washout period can be found for a future experiment, that’s the end of it. Another philosophy is to recognize that carry-over effects exist, even if not significant, and we need to provide for an adjustment for carry-over effects when assessing treatment effects.

In the 2 treatment level case, it is sufficient to simply include a ‘sequence’ categorical variable in the model to assess the presence of a carry-over effect. If the sequence variable is significant, then a detectable carry-over effect exists.

With more than two treatment levels, the complexity of the analysis rises sharply. For 3 levels of a treatment, 3 periods will be needed, and now we have 3! = 6 sequences to consider. What is needed in this case, in addition to a sequence variable, is a way to adjust the assessment of treatment effects for the presence of carry-over effects. This can be accomplished with a set of coded covariates in a repeated measures ANCOVA.