**5. Implementing the adaptive allocations**

In **Tables 5** and **6**, we observed that the decrease in efficiency following the decrease in *λ* is not consistent for differing sample sizes. That is, if we wish to maintain some level of relative efficiency with respect to a known fixed optimal design while applying the multiple objective adaptive allocation scheme, we must fully understand the behaviors of this adaptive allocation scheme and find the suitable *λ*, which is determined by the true parameters as well as the sample size. The simulations on this scheme may help suggest some *λ*'s, but is limited to the specific scenarios being studied. Therefore, we implement a sensible strategy of the multiple-objectivebased allocation scheme without having to precisely know which *λ* to use.

The multiple-objective function as in Eq. (10) is now split into two objective functions:

$$H\_{1,j,k} = \frac{\Delta\left(\hat{I}\_{j+1}^k(\boldsymbol{\beta})\right)}{\Delta\left(\hat{I}\_{j+1}^{k'}(\boldsymbol{\beta})\right)},\tag{11}$$


**Table 5.**

*Allocation, efficiency, and success ratio for two-period designs using the multiple objective criteria in Eq. (10).*

$$H\_{2,j,k} = \frac{f\_{j,k}}{f\_{j,k'}},\tag{12}$$

which are the first and second terms of the Eq. (10). The allocation scheme takes the following steps.



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

**Table 6.**

*Allocation, efficiency, and success ratio for three-period design using the multiple objective criteria in Eq. (10).*

To illustrate, we apply the above strategy to the parameters in **Table 1** with the aim of constructing a response-adaptive design with a relative efficiency around *r* <sup>∗</sup> >0*:*8. First, we construct two-period two-treatment response-adaptive designs with *n* ¼ 40, 80, and 100. We present the results for three-period two-treatment designs with *n* ¼ 80 and 100. The case for *n* ¼ 40 was excluded as all adaptive designs constructed using Eq. (10) with any *λ* have relative efficiencies > 0*:*9.

From **Table 4**, we can see that the designs constructed using the adaptive allocation method by Kim [23], denoted as *d*Adaptive, have relative efficiencies close to 0.8 or slightly larger than that while the success ratios are increased by 9% compared with the designs for *λ* ¼ 1. For *n* ¼ 40, the adaptive design follows the pattern of changes in the allocations, efficiency, and success ratio so that we can find one between *d*ð Þ <sup>0</sup>*:*<sup>8</sup> and *d*ð Þ <sup>0</sup>*:*<sup>9</sup> . For example, the allocation to the treatment sequence *AA* is 21.75 (*d*Adaptive), which is between 16.63 (*d*ð Þ <sup>0</sup>*:*<sup>8</sup> ) and 22.22 (*d*ð Þ <sup>0</sup>*:*<sup>9</sup> ). This pattern is also the case for all other columns in the table for *n* ¼ 80 and 100. Our adaptive designs appear to be constructed in a similar manner as the multiple objective responseadaptive designs as if they were constructed with the *λ* in the suggested range of


**Table 7.** *Comparison of our new revised response-adaptive three-period design with the results from Table 6.*

ð Þ 0*:*8, 0*:*9 . Similarly, the *d*Adaptive designs for *n* ¼ 80 and *n* ¼ 100 fall right in between *d*ð Þ <sup>0</sup>*:*<sup>9</sup> and *d*ð Þ <sup>0</sup>*:*<sup>95</sup> .

From **Table 7**, the relative efficiencies of our adaptive three-period designs are 0.7999 and 0.7854 for *n* ¼ 80 and *n* ¼ 100, respectively. These efficiencies are very close to our target *<sup>r</sup>* <sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:*8 while the success ratios are improved by approximately 9%. We can see that the allocation for treatment sequence *AAA*, relative efficiency, and the success ratio for the new adaptive designs *d*Adaptive follow the same pattern as the multiple objective response-adaptive designs. The allocations to the other sequences are relatively small and do not seem to affect the efficiency much as long as the allocation to *AAA* is well controlled. The above strategy successfully leads us to obtain desired success ratios and maintain efficiency to a prespecified level without having to determine what the ideal *λ* is.
