**3. Practical and nearly optimal designs**

We apply the allocation method to construct some popular practical designs in clnical trials, two-treatment two-period designs and two-treatment three-period designs based on the parameter settings from Li [13], which are shown in **Table 1** with a slight modification on the values to incorporate the GEE modeling approach. Initially, one subject is assigned to each treatment sequence. Afterward, new subjects are introduced sequentially and are assigned to the treatment sequence with the highest Eq. (10). When all subjects are assigned, the variance of the estimated treatment effects, varð Þ ^*τ<sup>N</sup>* , is computed and compared with the variance obtained from the optimal fixed designs suggested by Mukhopadhyay [20]. Mukhopadhyay [20] conducted simulation study for the optimal fixed crossover design with binary outcomes using the GEE method and showed that *AA=AB=BB=BA* is optimal for *p* = 2 and


#### **Table 1.**

*Parameter values for simulation in construction of multiple-objective response-adaptive crossover design with binary outcomes.*

*ABB=AAB=BAA=BBA* is optimal for *p* = 3 under the compound symmetric covariance structure with binary outcomes.

#### **3.1 Two-period design**

There are four possible treatment sequences for two-treatment two-period crossover trials. Carriere and Reinsel [21] showed that an equal allocation on all sequences AA*=*BB*=*AB*=*BA, denoted as *d*opt,*<sup>p</sup>*2, is universally optimal for a continuous response, and Mukhopadhyay [20] confirmed that it is also numerically optimal even when responses are binary. We assign a subject to each of the four sequences and allocate the rest based on the objective function in Eq. (10). The following tables show the allocations of the adaptive designs, their efficiency compared with the fixed optimal design, and their success outcome ratio for various values of *λ* and *N*.

When *λ* ¼ 0, the resulting allocation focuses on the treatment sequence AA with very few assigned to the rest of the sequences due to randomness during the initial stage of the trial. We can see that the allocation to the sequence AA decreases as *λ* increases. The allocations move toward a dual balanced design *d*opt,*<sup>p</sup>*2, which assigns equal allocations to all four sequences. The relative efficiency, which is defined as the ratio of variance of estimated treatment effects of *d*opt,*p*<sup>2</sup> over the proposed multiple objective adaptive design, is low for *λ* ¼ 0 and approaches 1 as *λ* increases to 1. The success ratio is close to the expected success shown in **Table 1** when *λ* ¼ 0 and decreases as *λ* increases. Therefore, we must find a reasonable compromise between efficiency and a success ratio. For *n* ¼ 40, *λ*∈ ð Þ 0*:*85, 0*:*9 would construct an efficient design (efficiency > 0.8) with a sufficiently higher success ratio (5–8% increased) than *λ* ¼ 1. For *n* ¼ 80, *λ*∈ ð Þ 0*:*9, 0*:*95 would construct a similar design (efficiency > 0.8 and success ratio improved by 5–8%). For *n* ¼ 100, we note a drastic result around *λ*∈ð Þ 0*:*9, 0*:*95 where efficiency changes from 0.8957 to 0.7096, while the success ratio changes from 0.5168 to 0.5638, showing that the choice of suitable *λ* may vary significantly by the sample size *n*.

The consistent estimates for the above terms can be obtained by replacing the parameters with their GEE estimates. Also, the variance of the estimated *β*'s can easily be computed using the sandwich covariance matrix from GEE. The treatment sequences with a smaller variance do not necessarily improve efficiency in this case, and the efficiency depends on the covariance matrix of the estimates of parameters. This covariance matrix, in turn, does not have a closed form, unlike in the continuous response case.

#### **3.2 Three-period design**

Three-period two-treatment crossover designs constructed from the multiple objective response-adaptive approach behave similarly as the two-period two-treatment designs. When *λ* ¼ 0, the majority of the subjects are allocated to the treatment sequence *AAA*, which has the highest success ratio per period. For small sample size, *n* ¼ 40, the efficiencies remain high and the success ratios are improved for any values of *λ*<1. This is largely due to the conditions of the design, where 3�8 = 24 subjects out of 40 are assigned evenly to all eight sequences and thus only 16 subjects are allocated based on the multiple objective response-adaptive schemes. Therefore, the relative efficiency, which is computed based on the optimal design [20], remains high and the success ratio is improved only to a degree.

## *Practical and Optimal Crossover Designs for Clinical Trials DOI: http://dx.doi.org/10.5772/intechopen.104694*

However, in the case of *n* ¼ 80, the success ratio increases from 0*:*4323 to 0*:*5647 and the efficiency decreases from 1*:*0370 to 0*:*5793 as *λ* changes from 1 to 0. It is notable that the relative efficiencies of multiple objective response-adaptive designs for *λ* ¼ 1 are greater than 1, indicating that these designs are slightly better than the optimal design [20] for the given set of parameters. The design with *λ* ¼ 0*:*95 is as efficient as the optimal design, with a relative efficiency of 1*:*0055, and yet shows a higher success ratio (0*:*4708 compared with 0*:*4323), with an expected success ratio of 0*:*4696 (compared with 0*:*4375). In the case of *λ* ¼ 0*:*9, the relative efficiency decreases to 0*:*9220 while the success ratio increases to 0*:*5050 from 0*:*4323. Looking at the design with *λ* ¼ 0*:*85, we see that the relative efficiency decreases to 0*:*8133 while the success ratio increases to 0*:*5290. These two designs with *λ* ¼ 0*:*9 and *λ* ¼ 0*:*85 indicate that we could improve the success ratio of the design by 7–10% at the cost of relative efficiency between 0*:*1 and 0*:*2.

When *n* ¼ 100, the designs show a similar performance to the case of *n* ¼ 80 with respect to efficiency and the success ratio, except that efficiencies drop sharply, as we give attention to beneficial treatment effects with *λ*< 1.

In summary, the above tables show that adaptive schemes could benefit more subjects without much loss of efficiency for the given set of parameters. But it is important to find an appropriate *λ* to improve the success ratios while maintaining a sufficient level of statistical efficiency. In this case, *λ* ∈ ð Þ 0*:*85, 0*:*9 is recommended for both *n* ¼ 80 and *n* ¼ 100. However, we can see that the decrease in efficiency is more evident for *n* ¼ 100 than that of *n* ¼ 80, indicating that sample size *N* is another player determining the balance parameter *λ*. The resulting designs would have success ratios increased by 9–12% when compared with the optimal fixed design (*λ* ¼ 1). Taking a smaller value of *λ* can benefit further, but the gain in success ratio decreases marginally as the *λ* decreases.
