**1. Introduction**

Crossover designs have enjoyed advantages over parallel designs, such as completely randomized design in terms of statistical efficiencies. Equal or balanced allocations play an important role in the construction of optimal designs under various model assumptions. However, equal allocations may pose ethical dilemma when researchers start to suspect that one treatment may be superior to the other. All trials start with the null hypothesis that the effects of a new treatment being tested are the same as comparators before we could prove its superiority. At some point in the trial, one may find an evidence indicating that the effects of treatments are notably different. Then, one may wonder whether to equally allocate remaining subjects to the treatments as per the protocol or to adapt to the findings and alter the allocation scheme to reflect the trial phenomena. Connor et al. [1] studied HIV treatment drug

named AZT. Among 477 pregnant mothers with HIV, 239 were assigned to a placebo, and 238 were assigned to the AZT. The trial resulted in 60 infants diagnosed with HIV from the placebo group and 20 infants diagnosed with HIV from the AZT group. A decade later, Tymofyeyev et al. [2] suggested that use of 50–50 allocation was ethically improper given the seriousness of the outcome of the study and recommended to use a response-adaptive allocation. Tymofyeyev et al. [2] utilized the Play the Winner Rule (PWR) allocation [3, 4] and simulated the trial in a way that 360 and 117 pregnant mothers were adaptively allocated to the AZT or the placebo, respectively. The results of simulation showed that 60 infants were expected to be diagnosed with HIV in two groups combined as opposed to 80 infants in 1994, which revealed some of the benefits of the adaptive allocations.

Response-adaptive designs may have several other goals. Many authors [3–6] aimed at allocating more subjects to a better treatment. Armitage [7] aimed at reducing the sample size, and Wang [8] aimed at increasing the sample size based on the prespecified statistical power and the data acquired. Furthermore, Bandyopadhyay and Biswas [9] introduced covariates in response-adaptive designs. Sorkness et al. [10] proposed designs that were adaptive to the prevalence of events, in which the sample size recalculation was done to remedy the loss of statistical power arising from the imbalance of the prevalence. However, these studies utilized the acquired information using only a single objective. Many authors proposed a multiple objective adaptive design for continuous responses where they defined an objective function with two components, controlled by a weight parameter [11–13].

Binary responses are modeled differently from continuous responses in a way that the information is a function of the outcome. Standard logistic regression assumes that the responses are independent although crossover trial data are dependent on each subject. We use the Generalized Estimating Equations (GEEs) method, which can incorporate a desired covariance structure of responses. Liang and Zeger [14] proposed the GEE, which takes into account for the time dependencies of the data by allowing correlations. The GEE method estimates parameters by solving the system of equations based on the Quasi-Likelihood function. The advantage of Quasi-Likelihood method is that it does not need to provide joint distribution of the data and only requires the marginal distribution and its mean and variance. GEE estimates are proven consistent under a mild regularity conditions [14]. Valois [15] utilized GEE in the analysis of crossover designs.

This chapter demonstrates how to construct multiple objective response-adaptive designs for two treatments with binary outcomes using the GEE. We first review the theoretical grounds for crossover designs with binary outcome and the GEE method. Adaptive designs are constructed using simulations, and some two- and three-period practical designs will be built for various weights of multiple objective functions. We also compare the GEE methods to the other approaches done by Li [13]. Lastly, we develop a new strategy for maximizing the success outcome, while maintaining certain level of prefixed desired statistical efficiency.
