Cross-over trials period, meaning

In this context, each patient would be receiving each of the multiple treatments. In the cross-over trial with three treatments this would likely be a three-period, three-treatment design and patients would be randomised to one of the six sequences ABC, ACB, BAC, BCA, CAB or CBA. Although there are again ways of asking a simultaneous question relating to the equality of the three treatment means through an analysis of variance approach this is unlikely to be of particular relevance questions of real interest will concern pairwise comparisons. 

If we now look at the ability of various types of clinical trial to resolve total variation, then the situation is as illustrated in Table 25.5, which is also taken from Senn (2001). We can see that, as discussed above, a single cross-over trial is not capable of distinguishing between sources of variation C and D. This is the situation discussed in Section 25.2.1. A slight caveat must be entered here. If more than two treatments are being studied in a cross-over trial, then some partial identification of treatment-by-patient interaction is possible. This is because in a cross-over trial in n and t treatments in t periods the nt —1 degrees of freedom resolve into n — for patients and t— for each of treatments and periods. This leaves nt — n — lt + liot error. If r = 2 the degrees of freedom reduce ton-2, which is fewer than the number of patients. However, if r > 3 then the residual degrees of freedom for even moderately sized trials exceed the number of patients and this means that some partial identification is possible (Senn, 2002 Senn and Hildebrand, 1991). Nevertheless, full identification requires full replication of patient-treatment combinations. 

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