A Benchmark for Multiple Endpoints

In the setting with several endpoints, the correlation between them is important. Below, we describe the algorithm generating correlated outcomes in the bench-

mark framework. The approach described below has been known for a long time (Tate, 1955; Molenberghs et al., 2001). We apply it to an arbitrary distribution of outcomes to generate the complete information vector. We start with the case of binary toxicity and continuous efficacy responses that has attracted a lot of attention in the literature recently (see e.g. Hirakawa, 2012; Yeung et al., 2015, 2017).

Consider a Phase I/II clinical trial with toxicity outcomeY_ij(1) and efficacy outcome

Y_ij(2) with CDFsF_j(1) and F_j(2), respectively, at dose level dj for patient i. We will use the setting studied by Bekele and Shen (2005) to illustrate the construction of the benchmark for multiple endpoints throughout this section.

Example 2. Bekele and Shen (2005) considered a setting with m= 4 dose levels, an efficacy outcome at dosedj having Gamma distribution Γ(λjω, ω) where λjω is

the shape parameter, ω = 0.1 is the rate parameter (i.e., the mean equals to λj),

and a DLT outcome having probability pj. In one of the simulation scenarios the

following parameters are assumed λ1 = 25, λ2 = 60, λ3 = 115, λ4 = 127

and p1 = 0.01, p2 = 0.10, p3 = 0.25, p4 = 0.60. Then, Fj(1) is the CDF of

a Bernoulli random variable with parameter pj and G(·, λjω, ω) is the CDF of a

Gamma random variable with parameter λj, j = 1,2,3,4.

The toxicity/efficacy profile of patientiis given by two characteristics: u(1)_i ∈(0,1) corresponding to toxicity and u(2)_i ∈ (0,1) corresponding to efficacy. Firstly, we generate a bivariate standard normal vector (x(1)_i , x(2)_i ) with the mean µ = (0,0) and the covariance matrix

Σ =    1 ρ ρ 1    (2.3.3)

whereρ is the correlation coefficient. In simulation studies, the correlation coeffi- cient,ρ, is specified by a statistician as part of the scenario. By applying the CDF

of the standard normal random variable

(u(1)_i , u(2)_i ) = (Φ(x(1)_i ),Φ(x(2)_i )),

one can obtain two correlated random variables having uniform distributions. Then, the corresponding quantile transformations are applied to u(1)_i and u(2)_i

marginally as described in Section 2.3.1 and responses for patient i given the dose leveldj are obtained asy

(1) ij =F −1(1) j (u (1) i ), y (2) ij =F −1(2) j (u (2) i ). This results in the complete information vector of toxicity and efficacy outcomes at all dose level for the patient i. The procedure is repeated for N patients and pairs of vectors

y(1)_j = (y₁(1)_j , . . . , y(1)_{N j}) and y(2)_j = (y₁(2)_j , . . . , y(2)_{N j}) are obtained for each dose level

dj, j = 1, . . . , m.

Example 2 (Continued). The correlation coefficient considered by Bekele and Shen (2005) is ρ = 0.25. The bivariate normal vector with mean µ = (0,0)

and covariance matrix Σ is initially generated: (x1, x2) = (−0.892,0.292). Then,

the first patient has a toxicity profile u(1)₁ = Φ(−0.892) = 0.186 and an effi- cacy profile u(2)₁ = Φ(0.292) = 0.615 which corresponds to the toxicity re- sponse F₁−1(1)(u(1)₁ = 0.186, p1 = 0.01) = 0 and to the efficacy response

G−1_(u(2)

1 = 0.615, λ1ω = 2.5, ω = 0.1) = 26.3. Subsequently, the vector of the

complete toxicity information is (0,0,1,1) and the vector of the complete efficacy information is (26.3,74.6,121.8,134.3). The complete information for 5 patients with randomly generated profiles u(1)₁ , u(2)₁ , . . . , u(1)₅ , u(2)₅ is given in Table 2.7.

Similar to the single endpoint case, the TD selection is based on a pre-specified decision criterion, T(y(1)_j ,y(2)_j ), which takes the minimum (maximum) value for the most desirable dose level. This would, however, involve the information for all endpoints of interest and can have more complicated structure. In the context of the Phase I/II clinical trial the decision criterion is also known as a trade-off function (see e.g. Thall and Cook, 2004).

Table 2.7: Complete information for 5 patients with randomly generated toxicity and efficacy profiles.

Patient’s profile ui Patient’s response d1 d2 d3 d4 u(1)₁ = 0.186 0 0 1 1 u(2)₁ = 0.615 26.3 74.6 121.8 134.3 u(1)₂ = 0.390 0 0 0 1 u(2)₂ = 0.214 12.2 48.4 87.3 97.3 u(1)₃ = 0.618 0 0 0 0 u(2)₃ = 0.898 45.7 104.7 159.3 173.5 u(1)₄ = 0.456 0 0 0 1 u(2)₄ = 0.545 23.6 70.0 112.9 128.1 u(1)₅ = 0.683 0 0 0 0 u(2)₅ = 0.869 42.5 99.9 153.5 167.4 Number of toxicities 0 0 1 3 Mean (efficacy) 30.1 79.5 127.5 140.2 Standard Deviation (efficacy) 13.8 23.0 29.5 30.8

Example 2(Continued). Bekele and Shen (2005) defined the TD as the dose with the highest expected efficacy while being safe (pj <0.35) and efficacious (λj >5).

This translates into the criterion

T(y(1)_j ,y(2)_j ) = Pn i=1y (2) ij n I Z 5 0 g(2)_j (v|y(2)_j )dv < θ(2), Z 0.35 0 g_j(1)(v|y(1)_j )dv > θ(1) (2.3.4)

whereg_j(1)(·|y(1)_j )andg(2)_j (·|y(2)_j )are probability density functions of a toxicity prob- ability and of an efficacy response given the datay(1)_j ,y(2)_j , respectively, andθ(1)_,θ(2)

are threshold probabilities. This decision criterion is used to construct the bench- mark in this setting. Applied to the benchmark, the integrals in Equation (2.3.4) are computed using density functions of Beta and Normal random variables for toxicity and efficacy outcomes, respectively. Using summary statistics given in Ta- ble 2.7 and threshold probabilities θ(1) =θ(2) = 0.50, the values of the criterion are

T(y(1)₁ ,y(2)₁ ) = 0.30; T(y(1)₂ ,y(2)₂ ) = 0.79; T(y(1)₃ ,y(2)₃ ) = 1.28; T(y(1)₄ ,y(2)₄ ) = 0.

The criterion is maximised for dose level d3 which is selected as the TD in this

single trial. The procedure is repeated for s = 1, . . . , S simulated trials to obtain the proportion of correct selections. The evaluation of the method by Bekele and

Shen (2005) using the proposed benchmark is provided in Section 2.3.4.

Similarly, the benchmark can be applied to an arbitrary number of endpoints. For instance, consider a Phase I/II trial in which toxicity and efficacy are evaluated in four cycles. Then, the profile of patient i is given by u(1)_i , . . . , u(8)_i each drawn fromU(0,1) and the rest of the construction remains unchanged. The procedure to generate the benchmark for K endpoints is given in Algorithm 2.

Algorithm 2Computing a benchmark for multiple outcomes

1. Specify K×K covariance matrix Σ and define objective function T(·). 2. Generate xi = (x(1)i , . . . , x (K) i ), i = 1, . . . , N from N(µ,Σ) where µ = [0, . . . ,0]1×K. 3. Computeui = (u (1) i , . . . , u (K)

i ), i= 1, . . . , N applying CDF Φ to each compo- nent of xi.

4. Apply the quantile transformation y_ij(k) = F_j−1(k)(u(_ik)) for k = 1, . . . , K at each dose leveldj,j = 1, . . . , mand fori= 1, . . . , N as described in Algorithm 1 and store y(_jk) = (y₁(_jk), . . . , y_{N j}(k))

5. Compute T(y(1)_j , . . . ,y_j(K)), j = 1, . . . , m, find dose level J for which T(·) is maximised (minimised) and set Zs=J.

6. Repeat steps 2-5 for s= 1, . . . , S simulated trials 7. Use ¯Z(j) = PS

s=1I(Zs =j)/S as the selection proportion of dose dj,

j = 1, . . . , m

In the following section, we illustrate the implementation of Algorithm 1 (Section 2.3.3) and Algorithm 2 (Section 2.3.4) in different clinical contexts.