**4. Optimal aggregated N-of-1 trial designs with** *N* **> 1**

In addition to the interest in the patient-based evidence of a treatment contrast, it may also be desirable to obtain a population average effect of treatments.

Aggregating the series of N-of-1 trials can give such an estimate of the average effect [7]. Using the one sequence that was found optimal for N-of-1 trial to all patients seems to be an obvious choice. However, it might not optimize the trial for estimating the effects on the average patient and therefore, using the one sequence that is optimal for a single individual patient to all patients might not serve this purpose.

The optimal designs for aggregated N-of-1 trials can also be derived from the information matrices we obtained, similarly as for N-of-1 trials for one patient, by allowing *j* ¼ 1, … , *N* with *N* >1. We approached the problem in two steps; first, we optimize single N-of-1 trials, as the primary goal is to optimize estimating the effects for each patient. Next, we optimize the overall N-of-1 trials in aggregation.

To find the optimal design, we typically choose *Nk* for *k* ¼ 1, … , *s* to allocate patients to a sequence *s*. The sufficient condition on *Nk* was given by [20] for a design to be optimal. The condition is called a duality in the design matrices, as defined earlier. Among other things, it permits simplification of the search for the optimal choice for *Nk* (see also [16]).

As noted earlier for **Table 1**, designs with the same value of *h* perform equally in estimation precision. Although all or only one of those with an equally optimal *h* can be used in a trial, practical consideration will lead to using the least necessary number of sequences for ease of treatment administration. Further, we found that there is a unique N-of-1 trial sequence in all *p*-period experiments. Since the designation of *A* and *B* is arbitrary, the optimal N-of-1 trial can be obtained by reversing the order of treatment administration. For example, the optimal 6-period N-of-1 trial is *ABBAAB* under the traditional model for *N* ¼ 1. Its dual, *BAABBA* also has the same value of *h* ¼ �1 and is optimal. Hence, when *N* ¼ 1, either of these two sequences will provide the maximum amount of information. When *N* > 1 and a multiple of 2, we can adopt both of these sequences, as they maximize the information, and this approach also simplifies the search for the optimal design for estimating the treatment effect for the average patients, satisfying the duality condition in [20]. Based on this rationale, we make the following two propositions.

**Proposition 1:** The optimal design for aggregated N-of-1 trials under the traditional model is to allocate the same number of patients to the optimal sequence with *AB* and *BA* alternating and its dual.

For example, the optimal design for aggregated six-period N-of-1 trials is the twosequence design using sequences *ABBAAB* and *BAABBA*, allocating the same number of patients to each. For a balanced design, *N* must be a multiple of 2.

**Proposition 2:** The optimal design for aggregated N-of-1 trials under the self and mixed model is to allocate the same number of patients to the optimal sequence with no alternation between *AB* and *BA* pairs and its dual. However, under the autocorrelation errors, the optimal design is to allocate the same number of patients to the optimal sequence that alternates between *AB* and *BA* pairs subsequently and its dual.

For example, the optimal aggregated 6-period N-of-1 trials for multi-clinic setting is to use the two-sequence design *ABABAB* and *BABABA* under the equal or uncorrelated errors, and to use the two-sequence design *ABBAAB* and *BAABBA* under the autocorrelated errors, allocating the same number of patients to each sequence.

From each sequence, we obtain individual patient specific treatment effects and by aggregating these one sequence of N-of-1 trials, we can quantify the average treatment effects.

### **4.1 Numerical comparisons**

To appreciate the practical performance of the optimal N-of-1 trials we constructed, we compare the efficiencies of selected designs for estimating the treatment and carryover effects under the two models. We also investigate their performances in some aggregation for estimating the average treatment effect. We limit the comparison to the models with independent and equi-correlation errors. In our comparisons, we also reference many designs, labeled with an A or S at the beginning, like A65 and S83, that were investigated in [31].

Recall that the optimal N-of-1 trials are either to alternate between *AB* and *BA* pairs or simply to repeat the *AB* pair in a sequence. Under the traditional model, the optimal N-of-1 trial uses *ABBAAB* and *ABBAABBA* for 6 and 8 period experiments, respectively. We refer them to S63 and S83. Under the self and mixed effects model, the optimal N-of-1 trial is to use *ABABAB* and *ABABABAB* for 6 and 8 period experiments, respectively, which we refer to S61 and S81. **Table 3** considers other mixtures and shows that the optimal individual-based N-of-1 trials are S63 and S81 under the respective models, as expected. We also observe from the table that (i) there are no real practical differences among various N-of-1 trials under the self and mixed model, and (ii) designs S61 and S81 cannot estimate self carryover effects, making S63 and S83 preferable. Therefore, we recommend using a sequence that alternates between *AB* and *BA* pairs, such as S63 and S83, as robust and optimal N-of-1 trials for all models.

Based on these single sequence trials, we also consider aggregated N-of-1 trials to numerically justify Propositions 1 and 2. We constructed 5 aggregated N-of-1 trials for *p* ¼ 6 and *p* ¼ 8 with *N* ¼ 32 and compare their efficiencies for estimating the average treatment effects as follows.


The design A63 uses the optimal sequence S63 under the traditional model; the design A61 uses the optimal sequence S61 under the self and mixed model although the self carryover effect is not estimable; the design A62 is a slight rearrangement of


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

*Note: NE means "Not Estimable." For N* ¼ 1*, a six-period N-of-1 trial may consider any one of S61,*⋯*, S64. For N* >1*, aggregated six-period N-of-1 trials may use a combination of these, A61,*⋯*, A65. Similarly, an eight-period N-of-1 trial may consider any one of S81,*⋯*, S84. For N* > *1, aggregated six-period N-of-1 trials may use a combination of these, A81,*⋯*, A85. The variances reported are divided by σ*<sup>2</sup> *<sup>ε</sup>=N (under an independence error) or σ*<sup>2</sup> *<sup>ε</sup>* ð Þ 1 � *ρ =N (under an equi-correlated error).*

#### **Table 3.**

*Variances of the estimators of treatment and carryover effects in six- and eight-period designs.*

designs A61 and A63; the design A64 is a combination of designs A62 and A63; the design A65 contains all 8 possible sequences of a 6-period design. Designs A81–A85 are also similarly constructed from various N-of-1 trials. We compare these designs under the traditional model and the self and mixed model. **Table 3** displays the comparison results under the two models and reports the variances of the estimated *τ*, *γ*, *γ<sup>s</sup>* and *γ<sup>m</sup>* after they are divided by their leading constants *σ*<sup>2</sup>*=N* (when errors are independent) or by *<sup>σ</sup>*<sup>2</sup>ð Þ <sup>1</sup> � *<sup>ρ</sup> <sup>=</sup><sup>N</sup>* (when errors equi-correlated).

**Table 3** shows that under the traditional model, the design A63 with the optimal sequence *ABBAAB* and its dual provides the best precision for estimating both the direct treatment effect and the carryover effect for the average patients. Each of the sequences optimally estimates the individual-based treatment effect. The least efficient choice would be the design A61. Design A65, which has been used in a recent multi-clinical trial [7], is rather inefficient as well, not to mention the unnecessarily lengthy administration time and cost required to manage many treatment groups, which requires the number of patients to be a multiple of 8.

When using the self and mixed effects model, Design A61 provides the best precision for estimating the direct treatment effect and the mixed carryover effect. However, the self carryover effect is not estimable. Overall, A63 is the optimal choice even in this case. However, all designs performed rather similar with over 95% relative efficiency under the self and mixed effects model, as observed earlier for single N-of-1 trials.

A similar observation is possible for 8-period designs and their sequences. In summary, it appears that there is no discernable advantage to distinguish among the two models and various error structures.

Overall, S63 and S83 for single N-of-1 trials or designs A63 and A83 in aggregation of S63, S83 and their duals seem to be the best under both models. They are optimal for estimating direct treatment and mixed carryover effects. Further, they are optimal for estimating both the treatment and carryover effects under the traditional model. Hence, we conclude that the optimal six-period aggregated N-of-1 trials is *ABBAAB* and its dual *BAABBA*, while the optimal eight-period aggregated N-of-1 trials is *ABBAABBA* and its dual *BAABBAAB*. For an N-of-1 trial, using one of these sequences will optimize the treatment for an individual patient.

We close this section with a summary note. Our numerical work suggests that alternating *AB* and *BA* pairs in sequence is likely to result in an optimal or nearly optimal *p*�period design for all the models we have considered for estimating both individual effects in N-of-1 trials and average effects in aggregated N-of-1 trials.
