Comparing three regularization methods to avoid extreme allocation probability in response-adaptive randomization

Abstract

We examine three variations of the regularization methods for response-adaptive randomization (RAR) and compare their operating characteristics. A power transformation (PT) is applied to refine the randomization probability. The clip method is used to bound the randomization probability within specified limits. A burn-in period of equal randomization (ER) can be added before adaptive randomization (AR). For each method, more patients are assigned to the superior arm and overall response rate increase as the scheme approximates simple AR, while statistical power increases as it approximates ER. We evaluate the performance of the three methods by varying the tuning parameter to control the extent of AR to achieve the same statistical power. When there is no early stopping rule, PT method generally performed the best in yielding higher proportion to the superior arm and higher overall response rate, but with larger variability. The burn-in method showed smallest variability compared with the clip method and the PT method. With the efficacy early stopping rule, all three methods performed more similarly. The PT and clip methods are better than the burn-in method in achieving higher proportion randomized to the superior arm and higher overall response rate but burn-in method required fewer patients in the trial. By carefully choosing the method and the tuning parameter, RAR methods can be tailored to strike a balance between achieving the desired statistical power and enhancing the overall response rate.

Keywords: Bayesian methods; clinical trial design; early stopping; operating characteristics.

Supplemental Figure 1:

Operating characteristics without early stopping for efficacy. Proportion of patients assigned to arm 2 (the superior treatment), overall response rate, and statistical power are shown for the 3 methods under 4 scenarios, where θ₁ = 0.2 and θ₂ ∈ {0.2,0.3,0.4,0.5} as t varies from 0 to 1 in increments of 0.1. The red solid lines represent the PT method; blue dashed lines represent the clip method; green dashed lines represent the burn-in method. Each line was derived from 100,000 simulations.

Supplemental Figure 2:

Operating characteristics with early stopping for efficacy. Proportion of patients assigned to arm 2 (the superior treatment), overall response rate, statistical power, and total patients are shown for the 3 methods under 4 scenarios, where θ₁ = 0.2 and θ₂ ∈{0.2,0.3,0.4,0.5} as t varies from 0 to 1 in increments of 0.1. The red solid lines represent the PT method; blue dashed lines represent the clip method; green dashed lines represent the burn-in method. Each line was derived from 100,000 simulations.

Supplemental Figure 3:

Performance without early stopping as assessed by the 10% and 90% quantile estimates of the proportion to arm 2 (the superior treatment) and overall response rate for the 3 methods under 3 scenarios, where θ₁ = 0.2 and θ₂ ∈ {0.2,0.4,0.5}. The red solid lines represent the PT method; blue dashed lines represent the clip method; green dashed lines represent the burn-in method. Each one was derived from 100,000 simulations.

Supplemental Figure 4:

Performance with early stopping as assessed by the 10% and 90% quantile estimates of the proportion to arm 2 (the superior treatment), overall response rate, and total patients in the trial for the 3 methods under 3 scenarios, where θ₁ = 0.2 and θ₂ ∈{0.2,0.4,0.5}. The red solid lines represent the PT method; blue dashed lines represent the clip method; green dashed lines represent the burn-in method. Each one was derived from 100,000 simulations.

Supplemental Figure 5:

Distributions of total patients in the trial for the 3 methods with early stopping under the scenario θ₁ = 0.2 and θ₂ = 0.5 setting power to 72%, 75%, and 80%, respectively. Each plot was derived from 100,000 simulations.

Figure 1:

Performances of the allocation probability during the trial for the three regularization methods for adaptive randomization yielding 80% power. A sample of 20 realizations of the trial are shown. The x-axis represents the total number of patients; the y-axis represents the probability of allocation to arm 2, the superior treatment. The true response rates are θ₁ = 0.2 and θ₂ = 0.5.

Figure 2:

Box plots for the distributions of the overall response rate and proportion of patients assigned to the better arm (arm 2) with no early stopping rule implemented. The left, middle, and right panels show the results at power equals to 75%, 80%, and 85%, respectively. The red plots represent the PT method; blue plots represent the clip method; green plots represent the burn-in method. All the plots are from the scenario θ₁ = 0.2 and θ₂ = 0.5. Each plot was derived from 100,000 simulations.

Figure 3:

Box plots for the distributions of the overall response rate, proportion of patients assigned to the better arm (arm 2), and total patients in the trial with early stopping rule implemented. The left, middle, and right panels show the result at power equals to 72%, 75%, and 80%, respectively. The red plots represent the PT method; blue plots represent the clip method; green plots represent the burn-in method. All the plots are from the scenario θ₁ = 0.2 and θ₂ = 0.5. Each plot was derived from 100,000 simulations.