**2. Multiple objective response-adaptive designs with GEE**

#### **2.1 Model and information matrix**

Agresti [16] discussed the Generalized Linear Model (GLM) for an exponential family of distributions. Suppose *Y* follows a distribution in an exponential family with parameters ð Þ *θ*, *ϕ :* Then the pdf of Y can be written as:

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

$$f(\boldsymbol{y}|\boldsymbol{\theta},\boldsymbol{\phi}) = \exp\left(\boldsymbol{y}\boldsymbol{\theta} - \boldsymbol{b}(\boldsymbol{\theta})\right) / a(\boldsymbol{\phi}) + c(\boldsymbol{y},\boldsymbol{\phi}).\tag{1}$$

Consider that the *Yijk* denotes the binary response of *i*th period of *j*th subject in *k*th treatment sequence, distributed as Bernoulli (*pijk*), and *X* is a design matrix for an overall mean effect (*μ*), period effects (*αi*), direct treatment effects (*τd i*ð Þ , *<sup>j</sup>*,*<sup>k</sup>* ), and carryover effects (*γd i*ð Þ �1, *<sup>j</sup>*,*<sup>k</sup>* ) with the corresponding vector of parameters *β*. By defining the relation *θ* ¼ *h*ð Þ*η* , *η* ¼ *x*<sup>0</sup> *β* and with a logit link function *g*ðÞ, we can entertain the following model:

$$\eta\_{ijk} = \lg\left(E\left(Y\_{ijk}\right)\right) = \lg\left(P\left(Y\_{ijk} = \mathbf{1}\right) = \text{logit}\left(P\left(Y\_{ijk} = \mathbf{1}\right)\right)\right) \tag{2}$$

$$\hat{\mu} = \log \left( \frac{P(Y\_{ijk} = 1)}{1 - P(Y\_{ijk} = 1)} \right) = \mu + a\_i + \tau\_{d(i,j,k)} + \gamma\_{d(i-1,j,k)} = X'\_{ijk} \beta\_{ijk}.\tag{3}$$

It is easy to see that the mean and variance of *Yijk* are defined as

$$E(Y\_{ijk}) = \mu\_{ijk} = b' \left(\beta\_{ijk}\right) = \frac{\exp\left(X\_{ijk}' \beta\_{ijk}\right)}{\mathbf{1} - \exp\left(X\_{ijk}' \beta\_{ijk}\right)},\tag{4}$$

$$\text{Var}\left(Y\_{ijk}\right) = \sigma\_{ijk} = b'' \left(\beta\_{ijk}\right) = \frac{\exp\left(X\_{ijk}' \beta\_{ijk}\right)}{\left(1 - \exp\left(X\_{ijk}' \beta\_{ijk}\right)\right)^2}.\tag{5}$$

#### **2.2 Generalized estimating equations**

We use Generalized Estimating Equations to estimate the parameters of GLM with unknown correlation structure using the mean *μijk* and unknown variance structure *V*�<sup>1</sup> *<sup>j</sup>* . The estimating equations can be shown as

$$\sum\_{j=1}^{n} \frac{\partial \mu\_j'}{\partial \beta} V\_j^{-1} \left( Y\_j - \mu\_j \right) = 0,\tag{6}$$

where *μ<sup>j</sup>* and *Y <sup>j</sup>* are vector of means and responses for periods 1 to *p*.

The above estimating equation resembles that of GLM but does not require an exponential distribution assumption for *Y*, which is the strength of GEE. McCullaugh [17] showed that under the correct specification of mean and variance functions, the quasi-likelihood estimators demonstrate characteristics similar to MLE. The covariance matrix then can be written as:

$$\text{Var}(\beta) = \left[ \sum\_{j=1}^{n} \frac{\partial \mu\_j'}{\partial \beta} V\_j^{-1} \frac{\partial \mu\_j}{\partial \beta} \right]^{-1} \tag{7}$$

Then, Bose and Dey [18] showed that the covariance matrix for parameters *β* can be defined with respect to *k* treatment sequences as follows:

$$\operatorname{Var}(\hat{\boldsymbol{\beta}}) = \left(\sum\_{k \in \mathcal{Q}} n\_k \frac{\partial \mu\_k^{'}}{\partial \boldsymbol{\beta}} \boldsymbol{V}\_k^{-1} \frac{\partial \mu\_k}{\partial \boldsymbol{\beta}}\right)^{-1},\tag{8}$$

where *nk* denotes number of subjects allocated to *k*th sequence and the design matrices being identical for subjects in the same treatment sequence. However, when the specified covariance matrix *V* is not identical to the observed covariance matrix Var(*Y*), then the sandwich variance estimator is suggested:

$$\mathrm{Var}(\beta) = A \left( \sum\_{k \in \mathcal{Q}} n\_k \frac{\partial \mu\_j'}{\partial \beta} V\_k^{-1} \mathrm{Var}(Y\_k) V\_k^{-1} \frac{\partial \mu\_j}{\partial \beta} \right) A,\tag{9}$$

where *A* is the variance in Eq.(8). This sandwich variance estimator is shown to be consistent [14].

#### **2.3 Multiple objective function**

Liang and Carriere [11] proposed the following multiple objective function for the continuous responses:

$$\boldsymbol{\Phi}\_{j,k} = \lambda \frac{\Delta \left( \boldsymbol{\hat{I}}\_{j+1}^{k}(\boldsymbol{\beta}) \right)}{\Delta \left( \boldsymbol{\hat{I}}\_{j+1}^{k'}(\boldsymbol{\beta}) \right)} + (1 - \lambda) \frac{\boldsymbol{f}\_{j,k}}{\boldsymbol{f}\_{j,k'}},\tag{10}$$

where ^*I k <sup>j</sup>*þ<sup>1</sup>ð Þ *<sup>β</sup>* is the Fisher's Information matrix for subject *<sup>j</sup>* <sup>þ</sup> 1 allocated to treatment sequence *k* with Δ being an optimality criterion of choice and *f <sup>j</sup>*,*<sup>k</sup>* is an evaluation function for treatment sequence *k* based on the first *j* subjects in the trial. In this function, treatment sequence *k*<sup>0</sup> refers to the sequence with maximum ^*I k <sup>j</sup>*þ<sup>1</sup>ð Þ *<sup>β</sup>* , and *k*00 refers to the sequence with maximum *f <sup>j</sup>*,*<sup>k</sup>*, which may not necessarily be identical. Among the two terms in the objective function, the first term of the function investigates the efficiency of design with respect to the Fisher's information matrix given that subject ð Þ *j* þ 1 is allocated to treatment sequence *k*. This is represented as a ratio over the sequence with maximum information so that the component may take value in 0, 1 ½ �. The second term of the function is called the evaluation function that evaluates the total efficacy of treatment sequences based on the estimated treatment effects. When *λ* ¼ 0, the objective function considers only the efficiency of the design and ignores any superiority/inferiority of the treatments being tested. On the other hand, the objective function with *λ* ¼ 1 would construct adaptive designs based solely on the positive effects of treatments being tested.

Liang et al. [12] and Li [13] extended their multiple objective function to binary responses and derived the information matrix for estimated success probabilities for binary responses. The observed number of successes for each treatment sequence was used for the evaluation function *f*. As the analysis of crossover trials mainly focuses on direct treatment effects, we choose the inverse of the variance of estimated treatment effects, 1*=*varð Þ^*τ* , as the criterion for comparing the efficiency of various treatment sequences. McCullagh [19] showed that quasi-likelihood estimates are invariant under a linear transformation. That is, *μ*^*<sup>k</sup>* maximizes the quasi-likelihood function.

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

Throughout this chapter, we will refer to the Eq. (10) as the multiple objective function and choose the first term Δ ^*I k <sup>j</sup>*þ1ð Þ *<sup>β</sup>* as the variance of the estimated treatment effects, var ^*τ <sup>j</sup>*þ1,*<sup>k</sup>* . The data acquired from the first *j* subjects are modeled using the GEE approach, and predictions for subject *j* þ 1 are made for all of the *K* treatment sequences. Then, we include the predicted responses of subject *j* þ 1 into the model and obtain the variance of an estimated treatment effects of each treatment sequence. Then, we evaluate the efficacy of each treatment sequence by using Σ*<sup>p</sup> <sup>i</sup>*¼1^*ηi*, *<sup>j</sup>*,*k*. The *<sup>η</sup>*<sup>0</sup> *s* take any values in I*R* where large values correspond to a better treatment sequence. We transform these values to positive numbers so that a larger value indicates a better sequence and the ratios could be easily implemented. For this reason, we choose *<sup>f</sup> <sup>j</sup>*,*<sup>k</sup>* <sup>¼</sup> logit <sup>Σ</sup>*<sup>p</sup> <sup>i</sup>*¼<sup>1</sup>^*η<sup>i</sup>*, *<sup>j</sup>*,*<sup>k</sup>* , which falls in 0, 1 ð Þ over all *<sup>p</sup>* periods.
