On the inclusion of non-concurrent controls in platform trials with an interim analysis

Abstract

The analysis of platform trials can be enhanced by utilizing non-concurrent controls. Since including this data might also introduce bias in the treatment effect estimators if time trends are present, methods for incorporating non-concurrent controls adjusting for time have been proposed. However, so far their behavior has not been systematically investigated in platform trials that include interim analyses. To evaluate the impact of an interim analysis in trials utilizing non-concurrent controls, we consider a platform trial featuring two experimental arms and a shared control, with the second experimental arm entering later. We focus on a frequentist regression model that uses non-concurrent controls to estimate the treatment effect of the second arm and adjusts for time using a step function to account for temporal changes. We show that performing an interim analysis in Arm 1 may introduce bias in the point estimation of the effect in Arm 2, if the regression model is used without adjustment, and investigate how the marginal bias and bias conditional on the first arm continuing after the interim depend on different trial design parameters. Moreover, we propose a new estimator of the treatment effect in Arm 2, aiming to eliminate the bias introduced by both the interim analysis in Arm 1 and the time trends, and evaluate its performance in a simulation study. The newly proposed estimator is shown to substantially reduce the bias and type I error rate inflation while leading to power gains compared to an analysis using only concurrent controls.

Figures (8)

• Figure 1: The considered platform trial design with an interim analysis for Arm 1 at the time point where Arm 2 joins the platform.
• Figure 2: Marginal and conditional bias of the unadjusted model-based treatment effect estimator for Arm 2 \ref{['eq_trt2_estimate']} when varying different design parameters. In all cases, no treatment effect for Arm 1 is assumed ($\theta_1=0$), and a unit variance $\sigma^2=1$ in each arm is used. In the first row, only stopping for futility is applied ($\alpha_E = 0$ in all cases), while in the second row, only efficacy stopping is used ($\alpha_F = 1$ in all cases). A) Varying the futility bound $\alpha_F$ and $\alpha_E$. Sample sizes are set to 150 per arm and period. B) Varying the sample sizes ratio between period 1 and period 2 $r = n_{01}/n_{02} = n_{11}/n_{12}$. Sample sizes per arm in period 2 are fixed ($n_{02} = n_{12} = 150$). Stopping bounds $\alpha_F = 0.5$ (first row) and $\alpha_E = 0.5$ (second row) are used. C) Varying the allocation ratio in Arm 1 vs control $a = n_{11}/n_{01} = n_{12}/n_{02}$. The sample size in the control arm is fixed to 150 in each period ($n_{01} = n_{02} = 150$). Stopping bounds $\alpha_F = 0.5$ (first row) and $\alpha_E = 0.5$ (second row) are used.
• Figure 3: Conditional (left figure) and marginal (right figure) bias of the unadjusted model-based treatment effect estimator for Arm 2 \ref{['eq_trt2_estimate']} when varying the treatment effect in Arm 1 ($\theta_1$). In all cases, sample sizes of 150 in each arm and period are assumed, and unit variance $\sigma^2=1$ in each arm is used. Different lines show different stopping rules: futility stopping only ($\alpha_F=0.5$, $\alpha_E=0$, red line), efficacy stopping only ($\alpha_F=1$, $\alpha_E=0.00258$, blue line), and stopping for both futility and efficacy ($\alpha_F=0.5$, $\alpha_E=0.00264$, grey line). The chosen efficacy boundary $\alpha_E$ corresponds to the O'Brien-Fleming boundary using binding futility stopping at $\alpha_F$, assuming a significance level of 0.025 in the final analysis. Note that for better readability, the y-axis scale is different in each figure.
• Figure 4: Marginal bias in the treatment effect estimator for Arm 2 for varying the treatment effect in Arm 1 ($\theta_1$) using the unadjusted estimator and the mean adjusted estimator with $\theta_1$ estimated from both periods, only period 1, only period 2, or using the CUMVUE. This figure corresponds to a platform trial with sample sizes per arm and period set to 150. The futility boundary was set to $\alpha_F = 0.5$, while for the efficacy boundary, $\alpha_E = 0.00264$ was chosen, corresponding to the O'Brien-Fleming boundary for the given sample sizes and $\alpha_F$, assuming a significance level of 0.025 in the final analysis. The strength of the time trend is $\lambda=0.15$ for all considered patterns.
• Figure 5: Conditional bias in the treatment effect estimator for Arm 2 using the unadjusted estimator $\tilde{\theta}_2$ and the mean adjusted estimator $\tilde{\theta}^A_2$ with $\theta_1$ estimated from both periods, only period 1, only period 2, or using the CUMVUE. A) Varying treatment effect in Arm 1 ($\theta_1$). This figure corresponds to a platform trial without time trends, with sample sizes per arm and period set to 150. The futility boundary was set to $\alpha_F = 0.5$, while for the efficacy boundary, $\alpha_E = 0.00264$ was chosen, corresponding to the O'Brien-Fleming boundary for the given sample sizes and $\alpha_F$, assuming a significance level of 0.025 in the final analysis. Note that we have restricted the range of the values for $\theta_1$, such that Arm 1 has a reasonable probability to continue after the interim analysis ($>4\%$). B) Varying ratio between the sample sizes in period 1 and period 2 ($r$), where the period 2 sample sizes per arm are fixed. Sample sizes per arm in period 1 are determined by the chosen ratio $r$, i.e., increase or decrease with respect to the period 2 sample sizes by a factor of $r$. This figure corresponds to a platform trial without time trends, with sample sizes per arm in period 2 set to 150. The futility boundary was set to $\alpha_F = 0.5$, while the efficacy boundary $\alpha_E$ was set in each case to the corresponding O'Brien-Fleming boundary for the given sample sizes and $\alpha_F$, assuming a significance level of 0.025 in the final analysis. The treatment effect in Arm 1 was in each case chosen such that a fixed design analysis of Arm 1 would yield approx. 80% power. C) Varying ratio between the sample sizes in Arm 1 and control ($a$), where the control arm sample sizes per period are fixed. Sample sizes in Arm 1 per period are determined by the chosen ratio $a$, i.e., increase or decrease with respect to the control arm sample sizes by a factor of $a$. This figure corresponds to a platform trial without time trends, with sample sizes in the control arm per period set to 150. The futility boundary was set to $\alpha_F = 0.5$, while for the efficacy boundary, $\alpha_E = 0.00264$ was chosen, corresponding to the O'Brien-Fleming boundary for the given sample sizes and $\alpha_F$, assuming a significance level of 0.025 in the final analysis. The treatment effect in Arm 1 was in each case chosen such that a fixed design analysis of Arm 1 would yield approx. 80% power. D) Varying strength of the time trend ($\lambda$). This figure corresponds to a platform trial with a linear time trend of strength $\lambda$, with sample sizes per arm and period set to 150. The futility boundary was set to $\alpha_F = 0.5$, while for the efficacy boundary, $\alpha_E = 0.00264$ was chosen, corresponding to the O'Brien-Fleming boundary for the given sample sizes and $\alpha_F$, assuming a significance level of 0.025 in the final analysis. The treatment effect in Arm 1 was in each case chosen such that a fixed design analysis of Arm 1 would yield approx. 80% power.