Operating Characteristics

We conducted extensive simulations to evaluate the operating characteristics of the proposed phase I/II design. Step II3 in our design encourages exploration of untried dose combinations when sample size is small. This is an important feature of the proposed dose-finding algorithm. To evaluate the impact of this feature, we compared the proposed design to a “greedy” design that is otherwise identical except that it always assigns patients to the dose with the highest estimate of efficacy.

Technically, this means that the greedy design replaces the condition (2.4) with ˆ

q_j∗k^∗ > 0 so that the dose with the highest efficacy among admissible dose setAg^∗ is always selected.

We also compared our design with the phase I/II combination trial design proposed by Mandrekar, Cui, and Sargent(2007) [31]. For convenience, we refer to the latter design as the MCS design. The MCS design converts toxicity and efficacy into a mutually exclusive trinary outcome (namely, “no efficacy and no toxicity,”

“efficacy without toxicity” and “toxicity”) and then uses a continuation ratio model to describe the relationship between this trinary outcome and the dose. To conduct a trial, the MCS design continuously updates the posterior estimates of the model parameters based on the observed data and assigns patients to the dose combination with the highest estimate of the probability of efficacy without toxicity (i.e., the MCS design adopts a greedy dose-finding algorithm).

We considered trials combining two biological agents, A and B, with a max-imum sample size of 45 patients and a cohort size of 3. We investigated 8 different dose-toxicity and dose-efficacy scenarios (see Table 2.1). The first four scenarios consider the 4× 4 combination trials with 4 dose levels for both agents A and B,

which were (0.075, 0.15, 0.225, 0.3) and (0.08, 0.16, 0.24, 0.32), respectively. We set the toxicity upper limit ϕ = 0.3. The last four scenarios were taken from the work of Mandrekar, Cui, and Sargent (2007) [31], which involves the analysis of 5×3 combination trials with 5 doses of agent A, (0.60, 0.75, 0.90, 1.05, 1.35), and 3 doses of agent B, (0.60, 0.90, 1.20). The toxicity upper limit was ϕ = 0.33.

In the proposed design, we set the safety cutoff δ = 0.4 and the tuning pa-rameter α = 2, and used 2,000 posterior samples of unknown papa-rameters ω, β, and γ to make inference after 1,000 burn-in iterations based on the adaptive rejection

Metropolis sampling algorithm [14]. Under each scenario, we carried out 2,000 simu-lated trials for each of the designs. We used C++ to implement the proposed design;

the simulation code is available upon request.

The simulation results under scenarios 1-4 are summarized in Table 2.2, in-cluding the selection percentage for each dose combination as the BODC and the percentage of patients allocated to each dose combination (shown as subscripts). In scenario 1, the dose-toxicity surface initially increases with the dose levels of agents A and B and then plateaus in the right upper corner of the dose combination matrix with a toxicity probability of 0.25; the dose-efficacy relationship is non-monotonic, characterized by efficacy monotonically increasing with agent A but not with agent B.

The true BODC is (a4, b2). Among the three designs, the proposed design performs the best with the highest selection probability (31.0%) and allocates the highest per-centage of patients (15.9%) to the target dose combination. The greedy design is often trapped in the doses on the diagonal since it does not encourage exploration of untried dose combinations. As a result, it incorrectly selects the dose combination (a4, b4) as the BODC with the highest percentage. Moreover, the greedy design only allocates 10.0% patients to the true BODC, which is more than 1/3 lower than the

Table 2.1: Eight dose-toxicity and dose-efficacy scenarios for the simulation studies.

proposed design. The MCS design does not perform well, selecting the true BODC only 9.2% of the times. Scenarios 2 and 3 share the same dose-toxicity surface as scenario 1, i.e., toxicity initially increases and then plateaus, but possesses different shapes of the dose-efficacy surface. In scenario 2, combination (a3, b4) has the highest efficacy and is the true BODC. Our proposed design identifies (a3, b4) with the high-est selection percentage 33.1% and assigns 18.5% patients to that dose combination.

The greedy and MCS designs identify the true BODC 17.9% and 14.5% of the times and assign only 9.3% and 9.4% of the patients to the target, respectively. In scenario 3, a monotonic dose-efficacy relationship is assumed for agent B but not for agent A and the highest dose combination (a1, b4) is the true BODC. The proposed design again outperforms the other two designs. Scenario 4 is constructed to examine the case in which only toxicity monotonically increases with dose, but not efficacy. The proposed design yields a selection percentage of 46.3%, which is higher than those of the greedy design (39.1%) and the MCS design (26.5%).

The simulation results for trials with 5× 3 combinations are shown in Table 2.3, indexed as scenarios 5-8. In scenario 5, toxicity is negligible for all dose combi-nations and efficacy monotonically increases with dose. The greedy design exhibits the best performance. This is mainly due to the coincidence that the greedy design would first escalate from dose combination (a1, b1) to (a3, b3) along the diagonal, then escalate up to the dose combination (a5, b3) during the run-in period in stage I.

Therefore, after the initial dose escalation the greedy design would quickly identify (a5, b3) as the most desirable dose without exploring off-diagonal untried doses. Nev-ertheless, the proposed design exhibits a better performance than the MCS design.

In scenario 6, toxicity monotonically increases with doses of both agents A and B, whereas efficacy only increases with agent B and is not affected by agent A. The

selection percentage of the proposed design is lower than that of the MSC design by 6.2%, but higher than that of the greedy design. In scenario 7, the selection per-centage of the proposed design is higher than that of the MCS design (37.2% versus 30.2%), and in scenario 8, the selection percentage of the MCS design is 9.6% higher than the proposed design. In addition to the eight scenarios shown in Tables 2.2 and 2.3, we also considered additional scenarios (see Table 2.4) with different shapes of dose-toxicity and dose-efficacy relationships. The simulation results are summarized in Table 2.5 and demonstrate that the proposed design performs consistently well.

Figure 2.1: Surface of the toxicity probabilities for combinational agents using the proposed change-point model. Toxicity initially increases with dose level and plateaus after reaching the change line.

dose le

vels of agent A

−2

−1

0

1

2

dose le vels of agent B

−2

−1 0

1 2 toxicity probabilities

0.1 0.2 0.3 0.4

Table2.2:Theselectionpercentageandthepercentageofpatientstreatedateachdosecombination(shownasthesubscripts) forscenarios1-4undertheproposed,greedyandMCSdesigns.ThetargetBODCsarebolded. AgentB AgentProposedDesignGreedyDesignMCSDesign ScenarioA123412341234 1423.814.131.015.910.89.48.98.518.29.521.510.07.85.321.826.511.15.09.25.011.16.129.99.8 33.53.95.56.01.26.91.14.64.53.04.33.01.19.52.23.25.14.71.11.81.43.28.37.0 20.92.32.78.10.83.70.52.31.21.64.211.40.91.60.61.90.69.20.20.60.83.78.48.1 10.77.62.12.81.02.10.91.80.58.42.21.91.42.12.11.20.89.31.09.43.89.06.68.0 241.62.13.23.24.16.417.013.72.51.63.12.33.93.730.132.03.83.34.93.44.23.629.89.8 32.52.12.84.37.19.233.118.52.42.33.12.39.013.917.99.32.63.71.11.91.42.314.59.4 20.71.61.57.83.45.39.68.50.80.91.19.03.02.68.25.10.49.30.60.71.22.814.910.6 10.37.30.81.62.52.76.05.70.17.70.60.92.22.37.13.90.59.41.49.66.09.812.010.3 341.32.10.92.5184.64.26.94.61.20.61.91.93.011.820.71.21.61.91.72.22.322.68.5 30.82.10.73.51.26.54.36.20.71.40.71.41.29.13.54.21.02.80.41.91.42.611.18.8 20.82.80.97.91.33.96.36.01.51.60.89.81.21.65.84.20.29.31.51.31.33.318.811.9 13.89.53.84.69.46.554.024.58.217.02.22.37.84.742.916.00.59.51.79.99.911.323.813.1 4446.318.96.85.53.45.21.36.139.113.87.15.23.33.60.99.826.510.025.611.111.38.95.24.6 37.85.52.75.03.18.62.24.57.33.92.62.93.513.22.93.94.96.81.22.62.24.77.05.6 25.35.01.98.21.54.53.13.43.92.73.012.01.82.53.93.60.48.90.00.20.63.59.25.9 15.510.22.33.61.72.72.93.08.616.12.52.02.51.84.92.90.28.50.28.60.95.64.54.6

Table2.3:Theselectionpercentageandthepercentageofpatientstreatedateachdosecombination(shownasthesubscripts) forscenarios5-8undertheproposed,greedyandMCSdesigns.ThetargetBODCsarebolded. AgentB AgentProposeddesignGreedydesignMCSdesign ScenarioA123123123 556.17.88.210.762.934.02.81.83.12.770.952.87.02.610.24.947.615.5 40.41.30.64.911.111.30.30.20.40.214.116.50.81.10.82.312.29.9 30.10.20.42.04.79.10.20.20.10.25.110.00.01.60.42.39.69.7 20.10.20.06.71.72.40.00.00.16.91.21.10.09.20.22.68.210.1 10.16.70.30.43.42.20.16.70.20.01.20.70.09.30.410.02.48.9 651.72.41.03.83.66.23.21.91.32.93.19.10.60.70.20.40.00.2 42.13.61.95.57.29.82.13.11.33.18.614.43.23.21.21.31.21.1 32.33.31.85.716.213.12.52.92.33.417.217.05.46.81.22.76.04.1 21.23.61.48.114.58.80.82.41.79.612.77.76.814.41.64.224.611.2 11.79.03.53.830.013.33.110.63.12.227.29.83.413.67.217.736.218.4 750.40.40.22.01.77.60.50.20.40.781.89.62.83.20.81.80.81.7 41.21.12.77.623.818.30.80.41.11.126.627.88.86.34.45.47.05.8 33.01.86.87.837.223.02.00.93.21.641.428.54.85.33.64.730.211.4 20.91.11.77.814.310.30.80.70.87.814.59.80.29.60.23.725.012.3 10.46.90.50.74.63.60.57.10.40.44.93.30.29.31.210.58.09.1 850.20.20.10.50.12.30.20.20.00.20.02.10.00.20.00.00.00.1 41.01.20.32.80.44.90.80.90.51.60.46.00.01.60.00.30.00.1 31.94.01.95.43.08.42.12.61.23.53.411.13.47.30.61.80.00.9 24.57.24.010.38.36.53.03.95.114.28.46.516.420.01.24.05.84.5 120.521.013.09.134.216.324.126.010.75.834.515.36.616.419.823.943.819.0

Table 2.4: Additional dose-toxicity and dose-efficacy scenarios for the simulation

Table 2.4 continued.

Table2.5:Theselectionpercentageandthepercentageofpatientstreatedateachdosecombination(shownasthesubscripts) foradditionalscenariosundertheproposed,greedyandMCSdesigns.ThetargetBODCsarebolded. AgentB AgentProposedDesignGreedyDesignMCSDesign ScenarioA123412341234 941.52.04.54.912.811.48.713.70.90.92.92.412.68.013.120.71.22.55.33.99.46.253.916.9 32.01.69.18.130.317.017.618.50.80.75.53.139.133.014.39.10.82.70.41.21.12.514.911.3 20.10.40.97.45.56.94.58.50.20.21.78.53.22.23.92.90.08.90.00.10.31.59.510.1 10.16.70.20.40.91.31.05.70.16.90.10.20.40.41.10.70.08.90.28.90.27.12.37.2 10425.714.937.217.611.810.78.625.720.59.824.010.26.65.328.231.83.02.17.63.911.07.355.117.1 31.42.44.55.62.17.61.410.01.51.33.82.71.410.42.03.60.52.10.41.21.03.211.210.0 20.51.31.47.30.83.60.94.40.90.94.310.20.71.30.91.80.09.00.00.00.42.37.09.3 10.66.91.31.60.61.11.03.20.47.62.11.10.51.11.90.90.08.90.08.90.27.72.47.1 11443.118.75.65.34.76.96.48.732.610.95.33.84.94.212.522.311.16.213.28.114.910.545.415.3 37.44.71.94.23.17.64.36.56.02.92.12.03.312.34.34.00.84.80.51.81.04.17.16.1 25.04.21.87.61.34.22.73.84.72.52.211.41.51.62.72.70.19.00.00.10.22.84.04.9 15.29.62.42.91.62.03.23.18.114.51.61.82.31.35.81.80.18.80.08.90.15.21.23.4 1241.72.23.54.77.29.63.68.71.81.33.82.96.56.74.014.75.13.46.23.78.95.224.710.3 33.22.511.18.934.218.116.311.82.21.38.14.943.533.615.19.64.73.71.82.12.73.222.211.8 20.41.11.87.86.87.65.85.80.10.41.28.54.73.35.33.60.48.60.30.41.12.514.611.5 10.26.70.60.91.91.61.32.00.06.90.40.21.10.91.61.10.08.50.18.62.18.15.18.6 1341.21.80.92.20.84.42.67.12.11.31.02.01.73.34.815.43.03.34.03.63.43.720.89.5 31.42.00.73.81.97.811.810.02.42.01.12.52.411.88.07.11.44.71.03.62.23.723.912.0 22.42.42.17.85.36.237.917.63.12.53.510.95.73.832.312.70.28.70.40.41.02.221.811.4 10.77.10.31.84.33.325.514.71.18.70.91.14.22.525.512.30.08.60.58.83.57.312.68.6

Table2.5continued. AgentB AgentProposedDesignGreedyDesignMCSDesign ScenarioA123412341234 1441.21.80.42.21.24.94.08.22.11.80.86.02.63.315.222.90.72.73.33.76.05.948.015.6 31.41.60.83.31.77.312.610.51.21.60.75.62.611.87.06.20.54.40.42.61.23.518.411.1 22.02.01.67.64.85.638.118.03.010.73.97.14.63.127.711.60.08.90.00.10.61.714.19.4 11.17.10.51.63.03.325.414.90.71.10.60.73.22.023.811.60.08.90.19.01.26.05.36.4 15425.114.837.217.211.610.79.09.618.99.822.97.17.05.829.332.23.82.59.04.511.67.552.616.2 31.22.54.75.81.47.41.75.42.12.73.85.81.110.41.93.30.82.60.61.60.63.210.49.3 20.31.32.17.60.43.40.82.50.910.23.96.90.81.11.31.80.18.90.00.10.52.86.98.6 10.76.91.91.80.41.30.91.80.31.32.10.71.41.31.90.90.08.80.08.90.77.72.16.8 1641.52.03.55.113.411.69.511.11.32.22.46.212.07.811.920.41.52.44.53.910.36.650.815.7 32.21.88.27.829.416.717.913.11.43.35.54.540.633.313.98.60.82.60.41.21.52.814.110.6 20.30.71.17.66.36.93.65.10.38.01.47.13.42.74.12.90.08.80.00.20.72.39.810.0 10.16.70.10.71.11.31.12.00.10.20.10.40.90.70.60.90.08.80.08.91.27.64.27.6 17427.615.239.017.46.88.12.26.830.514.132.811.05.06.11.513.210.04.218.46.512.27.19.45.6 32.42.96.46.61.98.41.65.03.04.56.07.01.812.32.64.12.43.41.62.03.94.411.19.0 20.41.82.17.90.94.11.42.70.810.94.67.41.32.31.63.20.48.70.20.32.23.916.410.5 10.57.22.22.00.61.81.52.00.41.62.51.61.51.61.91.60.38.50.68.61.88.38.99.1 1840.91.62.12.91.75.00.96.11.53.13.69.22.43.40.47.63.83.28.04.56.65.110.15.8 30.92.02.75.79.811.12.75.41.53.83.56.313.619.63.13.40.84.01.43.04.04.717.810.8 21.92.97.010.029.514.912.77.92.716.911.17.422.010.811.76.30.28.50.20.42.02.925.112.1 11.17.52.93.415.28.16.05.40.61.82.41.311.85.26.24.90.28.41.08.74.07.914.510.0

Table2.5continued. AgentB AgentProposedDesignGreedyDesignMCSDesign ScenarioA123412341234 1941.21.81.12.51.34.31.96.52.22.01.82.71.23.20.910.64.24.27.95.16.85.68.75.5 31.52.00.54.12.28.16.58.12.01.81.22.32.512.75.15.71.24.80.83.02.74.318.910.6 22.22.52.57.96.26.838.716.93.62.34.011.36.65.233.014.00.28.50.00.31.52.626.211.5 10.97.40.82.04.03.826.515.10.48.60.91.45.72.927.213.30.48.60.48.84.37.215.39.2 2048.15.31.92.41.24.91.97.19.34.42.82.21.72.21.110.00.64.71.05.62.08.895.127.7 336.213.77.27.14.58.23.55.524.47.68.94.96.715.83.93.60.05.20.01.00.13.21.03.6 214.99.35.19.11.95.31.02.98.14.28.316.93.52.92.12.40.09.50.00.00.02.60.02.4 18.511.32.14.20.52.01.21.812.618.03.32.41.41.31.71.10.09.60.09.60.04.40.01.9 2144.93.61.11.80.94.21.36.75.12.70.91.30.91.60.78.90.95.11.16.03.49.293.527.0 311.06.51.74.00.97.10.54.211.13.81.61.31.312.50.41.60.07.20.02.70.04.30.53.5 237.517.43.58.51.54.21.82.217.16.25.715.42.21.63.12.40.09.50.00.00.01.80.21.1 127.120.52.84.91.42.21.92.043.636.32.41.81.21.12.61.60.09.50.09.50.22.70.20.9 2247.75.82.12.61.63.70.95.68.25.11.92.11.42.10.97.416.69.715.29.414.59.425.69.0 335.415.37.27.03.97.72.54.429.911.18.25.15.213.13.53.210.08.70.82.41.74.24.23.7 216.210.43.59.31.85.11.02.67.55.17.016.12.53.01.21.82.09.50.40.61.03.74.03.3 17.612.12.44.40.52.61.11.612.419.82.82.50.91.61.80.90.89.10.28.91.15.61.22.9 2341.51.80.10.90.22.80.45.01.10.90.10.50.10.70.25.68.36.08.05.68.86.123.07.4 30.92.80.43.40.16.00.12.81.11.20.20.50.29.80.10.74.56.81.63.41.65.36.75.4 25.58.00.87.60.23.40.11.11.71.41.312.80.30.50.40.71.28.80.40.71.44.28.96.4 173.439.06.38.52.13.73.63.282.360.52.41.91.61.41.71.21.88.71.68.86.18.615.37.9

CHAPTER 3

Bayesian Adaptive Phase II Screening Design for Combination Trials

In this chapter, we propose a Bayesian adaptive screening design for com-bination trials. There is an increasing trend to use the comcom-bination therapies for cancer treatment. Combination therapies can lead to treatment synergies that result in improved patient outcomes. Therefore, the number of treatment combinations that must be tested is often quite large. Conducting separate phase II trials on every possible combination of treatments is often not practical. Novel designs that can test the efficacy all combinations in a single trial are imperative.

Toward this goal, we describe a Bayesian adaptive trial design that facilitates the pooling of information obtained across treatment combinations. We model the main and synergistic effects of the treatment agents using a linear model, which facilitates borrowing information across the combinations. We cast the screening problem into a Bayesian hypothesis testing problem. We construct a series of hy-potheses, each of which appoints one of the combinations as the most efficacious treatment. We utilize an encompassing prior with non-local constraints to accom-modate the complex parameter constraints imposed by the hypotheses. During the trial conduct, based on the observed data, we continuously update the posterior probabilities of the hypotheses and use them to adaptively allocate patients to

effec-tive combinations and select the best treatment. We conduct extensive simulation studies to evaluate the performance of the proposed design. The comparison to the standard (multi-arm) balanced factorial design show that our proposed design selects the best treatment with a significantly higher probability and allocates more patients to efficacious treatments.

In following sections, we describe our model, prior specification and trial design. We examine the operating characteristics of our design using simulation studies.

3.1 Methods

3.1.1 Probability Model

We consider trials to evaluate the treatment effects of all possible combi-nations of k treatment agents, A₁, A₂,· · · , and Ak. We assume that each drug combination is assigned to one treatment arm, although it is straightforward to ex-tend our design to trials where some combinations are excluded. Given k agents, there are (_k

)= 2^kcombinations, including placebo group, to be evaluated. The goal

of the trial is to identify the most efficacious treatment combination.

The outcome variable in the trial that motivates our research represents the mean change in the patient-reported symptom score. We assume that the outcome for the ith patient, y_i, is continuous and follows a linear model of the form

(3.1)

y_i = β₀+β₁I_i(A₁)+β₂I_i(A₂)+· · ·+β1,2I_i(A₁, A₂)+· · ·+β1,2,··· ,kI_i(A₁, A₂,· · · , Ak)+ϵ_i

where β₀ is the intercept of the linear model and I_i(·) is an indicator of whether patient i receives the given agents. For example, if patient i receives a combination

of A₁ and A₂, then I_i(A₁) = I_i(A₂) = I_i(A₁, A₂) = 1; whereas all the other indicator functions are then 0. Model (3.1) is flexible and accounts for the main and interaction effects of combining agents. Specifically, βk represents the main treatment effect of Ak, βk,k^′ represents the two-way interaction or synergistic effect between Ak and Ak^′

when k ̸= k^′, and so on. We assume that the residual ϵi follows a normal distribution with mean 0 and variance σ². Binary and time-to-event outcomes can be modeled using a similar linear structure within a generalized linear model framework.

To cast the problem into a hypothesis testing framework, we define the null hypothesis H0 to assert that no treatment is better than the placebo, and a series of alternative hypotheses H1, . . . , Hp−1, where Hj asserts that the jth treatment combination is superior to all others. In our trial, for example, treatment j is superior to treatment k if it leads to a greater reduction in symptom burden. Let θ0(β) denote the effect of the placebo and let θj(β), j = 1, . . . , p− 1, denote the net treatment effect of the jth combination (or treatment arm). Under the linear model (3.1), the treatment effect, θj(β), is a linear combination of the regression parameters, β’s. For example, the treatment effect of the combination of A1 and A2 is given by θj(β) = β0+ β1+ β2+ β1,2; and the treatment effect of the three-agent combination of A1, A2 and A3 is θj(β) = β0+ β1+ β2+ β3+ β1,2+ β1,3+ β2,3+ β1,2,3. To be consistent with the lung cancer trial described in Section 1.2, we assume that a smaller value of θj(β) (i.e., less symptom burden) represents a better response. Then the hypotheses can be formally expressed as

H_j : θ_j(β) = min(θ₀(β),· · · , θp−1(β)), j = 0,· · · , p − 1.

We let π_j(β, σ²) denote the prior distribution assigned to the unknown parameters β and σ² under H_j. Further discussion of the prior specification is provided in Section 2.3; for the moment we note that the domain of each prior is restricted to values of

β that are consistent with the hypothesis under which they are defined [21]. Given these prior densities, the marginal density of the observed data y under Hj is

(3.2) m_j(y) =

If p(H_j) denotes the prior probability of H_j, then the posterior probability of H_j given the data y is

If we assume that all hypotheses are equally likely a priori, then the posterior prob-ability of H_j simplifies to

The value of p(H_j|y) has a very intuitive probability interpretation—the probability that the jth combination is the best treatment conditional on the observed data.

Meanwhile, the value of p(H₀|y) is the probability that the placebo is the best treatment. Therefore, it provides a natural evidence-based mechanism to adaptively assign patients to efficacious combinations and select the most promising combina-tion.

3.1.2 Trial Design

We propose the following adaptive randomization scheme for the conduct of the trial. We assume that a total of N patients are available for testing, and that the first m× p patients are equally randomized into the p treatment arms using m

replications of a complete factorial design, i.e., m patients are randomized to each of p arms. The advantage of using a factorial design is that it allows us to rapidly obtain preliminary estimates of the main treatment effects. Following the lead-in factorial phase of the design, subsequent patients are assigned to a treatment according to the posterior probability that each treatment is best. The resulting design can be described as follows.

1. Assign m×p patients to the p treatment arms using m replications of a factorial design.

2. For i = m× p + 1, . . . , N, randomize the ith patient to the jth treatment arm with probability p(H_j|y), j = 0, . . . , p − 1, where y = (y1,· · · , yi−1)^′ are the observed outcomes data from the first i− 1 patients.

3. At the end of the trial, we select the combination j^∗ that has the highest pos-terior model probability, i.e., j^∗ = argmax_jp(H_j|y), j = 1, · · · , p − 1.

During the trial, we impose the following futility stopping rule: the trial is terminated for futility if

max{p(θ0− θj > δ|y)} < α, j = 1, · · · , p − 1

where δ and α are the prespecified minimal effect size and threshold, respectively.

That is, at any time during the trial, conditional on the observed data, if the proba-bility of achieving an effect size of δ for the best treatment arm is below the threshold α, we terminate the trial. In practice, the value of δ can be elicited from investiga-tors, and the values of design parameters m and α can be chosen by examining the operating characteristics of the trial in simulation studies.

3.1.3 Delayed Outcomes

In general, outcome-dependent adaptive randomization, such as the one we have proposed, assumes that the outcome is quickly ascertainable so that when an incoming patient is ready for randomization, the previous patients have been as-sessed and their outcomes are completely observed. This assumption may not hold in practice. In many cases, the patient outcomes require a long follow-up time to be assessed (or the accrual is fast), so their outcomes are not available when a new patient is randomized. To address this delayed outcome issue, one approach is to suspend accrual and wait until the outcomes of patients treated in the trial are fully observed. However, this approach is often not practical because it causes lengthy delays in a trial, wastes patient resources, and causes administrative problems. Al-ternatively, we propose to base our adaptive randomization scheme only on those patient outcomes that are available at the time that each new patient is random-ized. Our simulation studies in Section 3.2.1 show that, with finite samples, this observed-data approach is competitive to the approach of suspending accrual.

3.1.4 Prior Specification and Derivation of Bayes Factor

We adopt the encompassing prior approach proposed by Klugkist et al. [25]

and Klugkist and Hoijtink [24] to set the prior distributions on β and σ² under each hypothesis. In this approach, we first specify a prior distribution for the un-constrained model, and then based on that prior define prior densities under each hypothesis. More specifically, we begin by assigning a noninformative prior to σ² of the form π(σ²)∝ 1/σ². Given σ² and a hyperparameter g, we assume that β has a normal prior density of the form π(β|σ²) ∼ N(0, gσ²I_p), where I_p denotes a p× p identity matrix.

To modify the unconstrained priors for application to hypothesis H_j, j = 0, . . . , p− 1, we restrict the domain of β under each hypothesis so that it is con-sistent with the assumptions of the given hypothesis [21]. That is, under hypoth-esis Hj, the domain of β is restricted to the value space satisfying the condition θj(β) = min(θ0(β),· · · , θp−1(β)). This leads to the encompassing prior for Hj

The prior densities used to define each hypothesis are thus non-local with respect to one another, which enables us to more rapidly exclude hypotheses that are inconsis-tent with the data [21].

Under model (3.1) and the encompassing prior (3.6), the marginal density of data y under hypothesis H_j is given by

m_j =

Letting β^′ = β/σ, it follows that the normalizing constant c_j =

∫

N (β^′|0, gIp)I_min(θ_j(σβ^′))dβ^′

Recall that θ_j(β), j = 0,· · · , p − 1, is a linear function of β, thus Imin(θ_j(σβ^′)) = I_min(σθ_j(β^′)). Since σ > 0, the order of {σθj(β^′)} is the same as that of {θj(β^′)}.

Thus

We can see that c_j is independent of σ² , which greatly simplifies the evaluation of the marginal density of y. Then it follows that

m_j =

Here tν(·) denotes a multivariate student distribution with degree of freedom ν = n, scale matrix Σ = V (y^Ty − µ^TV⁻¹µ)/n and median µ = V X^Ty, where V = (¹_gI_p + X^TX)⁻¹, X is the design matrix in model (3.1) and n is the number of patients who have completed the assessment during the course of the trial.

3.2 Numerical Studies

3.2.1 Operating Characteristics

We evaluated the operating characteristics of the proposed Bayesian adaptive screening (BAS) trial design through extensive simulation studies. In the context of the lung cancer trial, we considered a total of 16 combinational treatments, including the placebo control, that result from 4 agents (Table 3.1). Two hundred patients were available for enrollment (i.e., N=200), and we performed m = 2 replications of the factorial design to obtain preliminary estimates of the treatment effects. The accrual rate was 12 patients per month, and it took 10 days to obtain the symptoms outcome.

Because the accrual was fast, we faced the delayed-outcome problem, that is, when a new patient is accrued and ready for randomization, some patients treated in the trial may have not finished their 10-day assessment and their outcomes are not available for calculating the randomization probabilities for the new patient. To deal with this issue, we adopted the observed-data approach described previously and calculated the randomization probabilities based on observed data when the outcomes of some patients are not available. Because the observed-data approach supports continuous accrual, it took approximately 17 months to complete the trial. For this trial, the approach of suspending accrual apparently is not feasible because it would lead to an infeasibly long trial lasting at least 4.8 years. Although the accrual-suspension approach is not useful in practice, it provides a theoretical benchmark for compar-ison because it represents the optimal case that the complete data are available to determine treatment assignment. For convenience, we denote the BAS design based on the accrual-suspension approach as BAS_susp. We configured the simulation pa-rameters, β, to generate 12 different efficacy scenarios. The simulation results of the selection percentage of each treatment under these 12 different efficacy scenarios are

displayed in Table 3.2. The total selection percentage of target treatments and the total percentage of patients assigned to targets are summarized in Table 3.3. Under each scenario, the most efficacious combination was defined as the combination with the smallest value of symptom burden, θ(β). We set the residual variance σ² = 130 based on previous symptom report data. The two parameters involved in the futility stopping rule, α and δ, were set to 0.35 and 10, respectively. We also compared the proposed BAS design to a design based on 12 replications of the complete factorial design on the 16 treatments, randomly allocating the last 8 available patients to treatments. For the factorial design (FD), the treatment with the lowest value of the least square estimate of θ(β) was selected as the best treatment at the end of the trial. We carried out 2,000 simulations for each scenario.

Table 3.1: The 16 combinations of four agents (A₁, A₂, A₃ and A₄) investigated in the lung cancer trial.

Treatment T0 T1 T2 T3 T4 T5 T6 T7

A1 0 1 0 0 0 1 1 1

A2 0 0 1 0 0 1 0 0

A3 0 0 0 1 0 0 1 0

A₄ 0 0 0 0 1 0 0 1

Treatment T8 T9 T10 T11 T12 T13 T14 T15

A₁ 0 0 0 1 1 1 0 1

A2 1 1 0 1 1 0 1 1

A₃ 1 0 1 1 0 1 1 1

A4 0 1 1 0 1 1 1 1

Scenarios 1 to 4 simulated the cases in which there was a single best treatment.

In scenario 1, the best (or most efficacious) treatment was T1 (i.e., single agent A1), and the BAS design substantially outperformed FD. The selection probability of the target treatment T1 under the BAS design was 86.1%, while under FD it was 64.9%. In addition, compared to FD, the BAS design allocated a significantly higher percentage of patients to the best treatment (6.3% versus 39.3%, respectively). The performance of the BAS design was rather similar to that of the optimal BASsusp

In scenario 1, the best (or most efficacious) treatment was T1 (i.e., single agent A1), and the BAS design substantially outperformed FD. The selection probability of the target treatment T1 under the BAS design was 86.1%, while under FD it was 64.9%. In addition, compared to FD, the BAS design allocated a significantly higher percentage of patients to the best treatment (6.3% versus 39.3%, respectively). The performance of the BAS design was rather similar to that of the optimal BASsusp