**4. Comparison with other approaches**

Bandyopadhyay [22] utilized an example of a three-period crossover trial of two treatments for hypertension. In this trial, 68 subjects were equally assigned to the treatment sequences *ABB=BAA=ABA=BAB*. Li [13] used the last two periods of this trial to obtain a crossover design with *AA=BB=AB=BA*. The response variable was continuous measurements of systolic blood pressure. Binary response variables were computed by dichotomizing the blood pressures at "135 or more" and "140 or more" and denoting the responses as failures. Two corresponding sets of success probabilities were estimated from this data. ð^*vA*1, ^*vA*2, ^*vB*1, ^*vB*2Þ ¼ ð Þ 0*:*24, 0*:*24, 0*:*24, 0*:*35 and ð Þ ^*vA*1, ^*vA*2, ^*vB*1, ^*vB*<sup>2</sup> = 0ð *:*35, 0*:*5, 0*:*35, 0*:*53Þ where *v* is the probability of success with the letters denoting treatments and numbers denoting periods.

These estimated probabilities were considered as actual success probabilities, and the multiple objective response-adaptive technique was applied with *λ* ¼ 1 and *λ* ¼ 0*:*9. A comparison of allocations, efficiencies, and success ratios of the three methods (B, L, K proposed by [13, 20, 23], respectively) is provided below. We included fixed group effects, *βk*, to the model in Eq. (2) to incorporate success probabilities. The parameters and other settings are provided in **Table 2**, and the results of simulations are found in **Tables 3** and **4**.

The efficiencies in **Table 3** were computed against the equal allocation design, which are nonadaptive but optimal for two-period and two-treatment designs. First, we examine multiple objective response-adaptive designs with *λ* ¼ 1. We see that when the


**Table 2.**

*Parameter values and expected success probabilities based on the crossover trial of [22].*


#### **Table 3.**

*Allocation, efficiency, and success ratio for two-period designs.*

difference of the expected success probabilities between the sequences is small (0.425 vs. 0.44, second example in **Table 2**), [13]'s strategy allocates an extensive number of the subjects to the treatment sequences *AB=BB* and results in a substantial loss of efficiency. Moreover, the gain in the expected success over an equal allocation design is minimal (0.4352 vs. 0.4325). The simulations confirm this observation, and *d*<sup>8</sup> has relative efficiency of 0.8376 without much gain as a result. On the other hand, *d*<sup>10</sup> adapts to the small differences in the sequences in a careful manner, and it assigns about three more subjects to better treatment sequences *AB=BB* without losing efficiency (0*:*9970). *d*<sup>10</sup> allocates fewer subjects to *AB=BB* compared with *d*7, *d*8, and *d*9.

It is noticeable that the pattern is not the same when there is some difference in the expected success probabilities between the treatment sequences (0.24 vs. 0.295). Design *d*<sup>2</sup> allocates 41.83 subjects to better sequences *AB=BB*, whereas *d*<sup>4</sup> allocates 42.7

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


#### **Table 4.**

*Comparison of new revised response-adaptive two-period design with the results from Table 5.*

subjects. The designs allocate more subjects to better treatment sequences than *d*<sup>1</sup> while maintaining a high level of efficiency.

The designs constructed using the multiple objective response-adaptive method with GEE are more responsive to the differences in treatments better than Bandyopadhyay [22] and Li [13], while maintaining a high level of efficiency when there is a large difference in the treatment effects. The method by Kim [23] assigns more subjects to the better treatment sequence when the treatment differences are large. Moreover, the resulting designs are close to the optimal design with an equal allocations on all four sequences, when the treatment differences are negligible. This assures that even if the treatment difference is not as large as expected, the multiple objective response-adaptive method is robust and creates an efficient design.
