**5. Universally optimal N-of-1 designs for more than two treatments**

Oftentimes, N-of-1 trials deal with comparing *t* >2 treatments and we briefly discuss selected universal optimal N-of-1 trials for such a situation. In N-of-1 trials with *t*>2 treatments, we can consider a sequence consisting of treatments in blocks of a size *t*. Every block within the sequence contains each of the *t* treatments exactly once. It follows that N-of-1 designs constructed in this way can ensure treatments are compared fairly, and poor balance can be prevented when the study is terminated prematurely. For example, in a 3-treatment N-of-1 trial, a six-period design could be *ABC*∣*BCA*, where the sign ∣ divides them into blocks. Li [31] denoted such a class of Nof-1 trial designs by No1(t,t), where the first *t* in the notation represents the number of treatments in the study and the second *t* denotes that the treatments are to be administered to be in blocks of size *t*. Therefore, a six-period design in the above example is No1(3,3).

Li [31] showed that Kiefer's conditions could not be satisfied with designs in the class No1(t,t). However, if we consider a slightly different class of designs, then universally optimal designs can be obtained. Li [31] used No1(t,t+) to denote the new class of designs, which consist of designs with one extra treatment to the last period in No1(t,t). For instance, a design in No1(3,3+) could be *ABC*∣*BCA*∣*A*, *ABC*∣*BCA*∣*B*, or *ABC*∣*BCA*∣*C*. Similarly, some examples from the class No1(2,2+) are *AB*∣*BA*∣*A*, *AB*∣*BA*∣*B*, etc.

One disadvantage of the universally optimal designs for *t*> 2 treatments is that the length of the sequence can be unmanageable, leading to drop-outs and noncompliances before the end of the trials. As discussed in [3], a universally optimal design in No1(t,t+) requires the length of the sequence equal to ð Þ *t* � 1 !*t* <sup>2</sup> <sup>þ</sup> 1, which is 5 for *t* ¼ 2, 19 for *t* ¼ 3, and 97 for *t* ¼ 4. It may be infeasible in practice because the longer the period of the experiment, the more expensive the experiment and the

*Optimal N-of-1 Clinical Trials for Individualized Patient Care and Aggregated N-of-1… DOI: http://dx.doi.org/10.5772/intechopen.106352*

higher the risk of drop-outs. To shorten the length of the experiment without losing the balance in the comparison of treatments, Li [31] introduced a class of designs in No1(t,s) or No1(t,s+) for some *s* <*t*, especially when *s* ¼ 2. To do so, the restriction that the block size must be equal to the number of the treatments can be relaxed [31]. By allowing the block size to be smaller than *t*, universally optimal designs can be manageable in practice, thereby reducing the risk of early dropouts and the burden of treatment administration.

Li [31] showed some practical universally optimal designs for three-, four- and five-treatment in blocks of size 2. In each block, two different treatments are assigned such as a crossover pair. For *t*-treatment designs, there are *t t*ð Þ � 1 different kinds of crossover pairs. To construct the universally optimal design, the crossover pairs are selected such that each subsequence of *AiAj*, 1≤*i*, *j*≤ *t*, appears only once. Therefore, for universally optimal designs, the number of periods is *p* ¼ *t* <sup>2</sup> <sup>þ</sup> 1. For example, *<sup>p</sup>* is 10 for three-treatment designs, 17 for four-treatment designs and 26 for five-treatment designs. We close by giving examples of universally optimal designs in selected situations. Omitting details, which are available in [31], they are:

• No1(3,2) with *t* ¼ 3:

f g *ABBCCAACBA or BCABBAACCB* f g,

• No1(4,2+) for *t* ¼ 4:

f g *ABBCCDDACBDCADBAA or BCCDDAABBDCADBACB* f g

and

$$\text{• No1(5,2) for } t = 5:$$

f g *ABBCCDDEEAACBDCEDAEBADBECA or* f g *CDDEEAABBCCEBDACBEDBAECADC :*
