. Author manuscript; available in PMC: 2022 Oct 1.

Published in final edited form as: Biom J. 2021 Jul 17;63(7):1406–1433. doi: 10.1002/bimj.202000043

Table 5.

Performance of Proposed Method when a two-arm trial is designed using a Piecewise Exponential Model with various hazard shapes and constant HR of 0.75

Scenario	Piecewise Exponential Model				Proposed Method (Parameter estimates based on plotting H(t) vs log(t) for Control arm)							Proposed Method (10000 Simulations)
Scenario	Interval	Hazard in each interval	Cumulative hazard at interval midpoint	$\hat{H R}$ and Empirical power when using 150 events in each arm (10000 simulations)	Shape ${\hat{β}}_{0} = {\hat{β}}_{1}$ (due to PH)	Control Arm scale ${\hat{θ}}_{0}$	Control arm median ${\hat{t}}_{m e d, C}$	Treatment Arm scale ${\hat{θ}}_{1}$	Treatment arm median ${\hat{t}}_{m e d, E}$	$\hat{R T} (p) = P T = \frac{{\hat{t}}_{m e d, E}}{{\hat{t}}_{m e d, C}}$	Number of Events^*	$\hat{R T} (p)$ and Empirical power (when using 150 events in each arm – data from PE model)
Decreasing hazard	0 – 2	0.70	0.70	0.7549; 77.23%	0.2982	1.9328	0.5655	5.0720	1.4839	2.6242	150	2.1416; 82.56%
	2 – 4	0.10	1.50
	4 – 24	0.001	1.61
Decreasing hazard	0 – 2	0.90	0.90	0.7535; 81.38%	0.4913	0.9839	0.4666	1.7670	0.8380	1.7959	150	1.7049; 87.17%
	2 – 4	0.30	2.10
	4 – 24	0.10	3.40
Decreasing hazard	0 – 2	0.30	0.30	0.7536; 79.89%	0.7486	4.6197	2.8313	6.7845	4.1580	1.4686	150	1.4952; 81.84%
	2 – 4	0.20	0.80
	4 – 24	0.12	2.20
Constant hazard	0 – 2	0.20	0.20	0.7535; 80.10%	1.000	5.000	3.4656	6.6665	4.6209	1.3333	150	1.3441; 80.55%
	2 – 4	0.20	0.60
	4 – 24	0.20	2.80
Increasing hazard	0 – 2	0.50	0.50	0.7548; 79.68%	1.2487	1.8528	1.3816	2.3329	1.7395	1.2591	150	1.2625; 74.58%
	2 – 4	0.60	1.60
	4 – 24	1.10	13.2
Increasing hazard	0 – 2	0.20	0.20	0.7548; 79.68%	1.5133	2.8068	2.2031	3.3945	2.6644	1.2094	150	1.1882; 77.98%
	2 – 4	0.80	1.20
	4 – 24	0.90	11.0
Increasing hazard	0 – 2	0.10	0.10	0.7548; 79.68%	1.7500	3.9817	3.2293	4.6929	3.8062	1.1787	150	1.1454; 75.78%
	2 – 4	0.30	0.50
	4 – 24	0.90	9.80
Hazard decreases constant	0 – 2	0.80	0.80	0.7538; 80.05%	0.5407	1.3285	0.6745	2.2618	1.1483	1.7024	150	1.6546; 85.10%
	2 – 4	0.15	1.75
	4 – 24	0.15	3.40
Bathtub shaped hazard	0 – 2	0.80	0.80	0.7528; 80.15%	0.7536	1.3943	0.8573	2.0425	1.2559	1.4648	150	1.4827; 82.86%
	2 – 4	0.10	1.70
	4 – 24	0.40	5.80
Arc shaped hazard	0 – 2	0.10	0.10	0.7540; 79.62%	1.2730	5.4465	4.0839	6.8276	5.1195	1.2536	150	1.3135; 83.54%
	2 – 4	0.40	0.60
	4 – 24	0.20	3.00
Arc shaped hazard	0 – 2	0.10	0.10	0.7528; 80.22%	1.4379	4.1210	3.1938	5.0335	3.9009	1.2215	150	1.2871; 86.51%
	2 – 4	0.80	1.00
	4 – 24	0.30	4.80

Sample size for proposed method is exactly same as that obtained by Schoenfeld formula with HR = 0.75 as in each scenario the Weibull property of ${\hat{γ}}_{P H} = - {\hat{γ}}_{A F T}$ . β is satisfied where ${\hat{γ}}_{P H}$ is the log-hazard ratio, ${\hat{γ}}_{A F T}$ is the time ratio and β is the shape parameter corresponding to a Weibull model. Target power for all scenarios in the table is 80%. Type I error is 5% for one-sided test.