**2. Models and information matrix**

The traditional crossover design model assumes that the carryover effects last for only one period. The patient effects are considered fixed in the model. The traditional model assumes no carryover effects for the observations in the first period. An alternative model which has carryover effects in the first period as well is built by giving patients a pre-period or baseline period [23, 24]. More complex models have also been considered. Some models incorporate higher-order carryover effects [25]; some consider carryover effects are proportional to the treatment effects [26], some include the interaction effects between the treatment effects and carryover effects [27], and others have random patient effects [28–30].

We first focus on the traditional model, frequently used to analyze repeated measures crossover data:

$$Y\_{i\bar{\jmath}} = \mu + a\_i + \beta\_{\bar{\jmath}} + \tau\_{d(i,\bar{\jmath})} + \gamma\_{d(i-1,\bar{\jmath})} + \varepsilon\_{i\bar{\jmath}},\tag{1}$$

*i* ¼ 1, ⋯, *p* and *j* ¼ 1, ⋯, *N*. Here *Yij* is the outcome in the *i th* period from the *j th* patient; *α<sup>i</sup>* is the *i th* period effect and *β<sup>j</sup>* is the *j th* patient effect. Further, *d i*ð Þ , *<sup>j</sup>* represents the treatment assigned to the patient in period *i* of patient *j*, and *τd i*ð Þ,*<sup>j</sup>* and *γd i*ð Þ �1,*<sup>j</sup>* are, respectively, the treatment effect of the treatment in the *i th* period and the carryover effect of the treatment in the ð Þ *<sup>i</sup>* � <sup>1</sup> *th* period.

The model assumes that the carryover effects only depend on the treatment assigned in the previous period but not on the treatment in the current period, which may be unrealistic. Taking the interaction into account without introducing too many parameters, Kunert and Stufken [17] presented a model with self and mixed carryover effects. The self carryover effect occurs when the treatments administered in the current and the previous periods are the same; otherwise, we have a mixed carryover effect. The model with the self and mixed carryover effects is given by

$$Y\_{\vec{\eta}} = \begin{cases} \mu + \alpha\_i + \beta\_j + \tau\_{d(i,j)} + \gamma\_{s, d(i-1,j)} + \varepsilon\_{\vec{\eta}}, & \text{if } \mathbf{d}(\mathbf{i}, \mathbf{j}) = \mathbf{d}(\mathbf{i-1}, \mathbf{j}) \\\\ \mu + \alpha\_i + \beta\_j + \tau\_{d(i,j)} + \gamma\_{m, d(i-1,j)} + \varepsilon\_{\vec{\eta}}, & \text{if } \mathbf{d}(\mathbf{i}, \mathbf{j}) \neq \mathbf{d}(\mathbf{i-1}, \mathbf{j}) \end{cases}, \tag{2}$$

where *αi*, *β<sup>j</sup>* , *d i*ð Þ , *j* and *τd i*ð Þ,*<sup>j</sup>* , are defined as in model (1). The parameters *γ<sup>s</sup>*,*d i*ð Þ �1,*<sup>j</sup>* and *γ<sup>m</sup>*,*d i*ð Þ �1,*<sup>j</sup>* represent the self and mixed carryover effects of the treatment assigned in the ð Þ *<sup>i</sup>* � <sup>1</sup> *th* period, respectively.

In an N-of-1 trial with *N* ¼ 1, the *j* index can be omitted. Further, with one patient and *p* responses in total, the period effects and patient effects cannot be accommodated. Therefore, we need to reduce the models for the case when *N* ¼ 1.

For models (1) and (2), we define the contrast of the direct treatment effects by *τ* ¼ ð Þ *τ<sup>A</sup>* � *τ<sup>B</sup> =*2, the contrast of the first-order carryover effects by *γ* ¼ *γ<sup>A</sup>* � *γ<sup>B</sup>* ð Þ*=*2, the contrast of the self carryover effects by *γ<sup>s</sup>* ¼ *γ<sup>s</sup>*,*<sup>A</sup>* � *γ<sup>s</sup>*,*<sup>B</sup>* � �*=*2 and the contrast of the mixed carryover effects by *γ<sup>m</sup>* ¼ *γ<sup>m</sup>*,*<sup>A</sup>* � *γ<sup>m</sup>*,*<sup>B</sup>* � �*=*2.

To construct a model-based optimal design, we commonly use design criteria such as *A*-, *D*-, and *E*-optimality. The *A*-, *D*-, and *E*-optimal design maximizes the trace, the determinant, or the eigenvalue of the information matrix among a class of all competing designs. The information matrix measures the amount of information about the unknown model parameters. Formally, given the model and the design, the elements in the information matrix are found by first taking the expectation of the second derivatives of the complete log-likelihood function with respect to the parameters and multiplying them by �1. In practice, not all model parameters are of interest. In this case, we would first partition the information matrix and work with the submatrix corresponding to the parameters of interest.

Specifically, we first partition the design matrix **X** ¼ ½ � **X**1, **X**<sup>2</sup> , where **X**<sup>1</sup> contains the columns of the design matrix pertaining to nuisance parameters and **X**<sup>2</sup> contains columns corresponding to the parameters of interest. The vector of model parameters *θ* is likewise partitioned as *θ* = (*τ*,*γ*)<sup>0</sup> or (*τ*, *γs*, *γm*)<sup>0</sup> , representing the direct treatment effects and carryover effects. Then, with Σ denoting the covariance matrix, the information matrix can be written as

$$I\_d(\theta) = X\_2' \Sigma^{-1} X\_2 - X\_2' \Sigma^{-1} X\_1 \left(X\_1' \Sigma^{-1} X\_1\right)^{-1} X\_1' \Sigma^{-1} X\_2. \tag{3}$$

Following [13], a design is universally optimal if (i) its information matrix is completely symmetric, and (ii) it maximizes the trace of the information matrix. To study the universal optimality of treatment effects in the *t* treatments N-of-1 designs, we obtain the information matrix for the parameters of interest under the traditional model. Then the universally optimal designs could be constructed as long as the conditions given by [13] are satisfied.

## **2.1 Cycles and sequences**

We first discuss how to find N-of-1 designs for comparing two treatments by minimizing the variance of the estimated direct treatment effect contrast, *τ*. To this end, it is helpful to define sequence feature parameters and show the association between them and the sequences in N-of-1 designs is useful for finding optimal N-of-1 trials for model (1) and (2) .

For N-of-1 trials involving two treatments, the design sequences consist of crossover pairs, *AB* and *BA*. Within each crossover pair, the two treatments are distinct. For two consecutive crossover pairs, the treatments assigned to the second period in the previous pair and the first period in the latter pair can be different or the same.

Further, if an *AB* pair is followed by a *BA* pair, as in *ABBA* (or *BAAB*), we define the design as having alternating pairs in the sequence. The performance of an N-of-1 trial sequence is related to how the pairs *AB* and *BA* alternate. The following feature parameters define how *AB* and *BA* alternate in a sequence.


When we define *s* and *m*, the subsequences can be constructed by either the treatments from a crossover pair, or be the treatments assigned to the second period in *Optimal N-of-1 Clinical Trials for Individualized Patient Care and Aggregated N-of-1… DOI: http://dx.doi.org/10.5772/intechopen.106352*


**Table 1.**

*Feature parameters of a sequence in a 2-treatment N-of-1 design*.


#### **Table 2.**

*Sequences for p* ¼ 8 *with corresponding design parameter values.*

the previous pair and the first period in the latter pair. Therefore, in a *p*-period sequence, there are *p* � 1 such subsequences with a length of 2. By the definition of feature parameters, we have *s* þ *m* ¼ *p* � 1. Determined by how a sequence is constructed, the value of *h* is negative and takes on possible values in �1, �3, ⋯, �ð Þ *p* � 1 . **Table 1** displays the relationship among *h*, *s* and *m*.

For a particular *h*, we calculate *s* and *m* by setting *s* ¼ ð Þ *p* � 1 þ *h =*2 and *m* ¼ ð Þ *p* � 1 � *h =*2. Further, for any given *p*, the N-of-1 designs can be classified by *h*. As an example, for *p* ¼ 8, **Table 2** shows the relationship between the design sequences and the feature parameters.

In the next section, we show that the information matrix of the parameters of interest are only dependent on the feature parameters. That is, sequences with the same *h* values have the same information matrix. For instance, when *h* ¼ �3, the three sequences *ABABBAAB*, *ABBAABAB*, *ABBABAAB* and their dual sequences share the same information matrix. If this *h* is the optimum value, the 8 period N-of-1 trials can use any of these three sequences and their duals.
