Table 3.

Empirical type I, Empirical Power, Coverage for different settings of Extended Cox model compared to Nominal type I error of 5%, Nominal Power of 80% for one-sided test, a =12, f = 12, r = 1, using 10,000 simulations

Relative Time Method (Design parameters and Quantities of Interest)					Extended Cox Model: ${\mathit{h}}_1\left(\mathit{t}\right)={\mathit{h}}_0\left(\mathit{t}\right).\mathit{e}\mathit{x}\mathit{p}\left\{{\mathit{\gamma }}_0+{\mathit{\gamma }}_1\mathbf{l}\mathbf{o}\mathbf{g}\left(\mathit{t}\right)\right\}$
Control arm Shape parameter β0	Effect Size definitions for survival curves	Treatment arm Shape β1 calculated via defined effect size	Event Size/ Sample Size (Each arm)	True HR calculated at ${\mathit{t}}_{\mathit{a}\mathit{v}\mathit{g}}=\frac{{\mathit{t}}_{\mathit{m}\mathit{e}\mathit{d},\mathit{C}}+{\mathit{t}}_{\mathit{m}\mathit{e}\mathit{d},\mathit{E}}}{2}$ $\mathbf{H}\mathbf{R}=\frac{{\mathit{\beta }}_1{{\mathit{\theta }}_0}^{{\mathit{\beta }}_0}}{{\mathit{\beta }}_0{{\mathit{\theta }}_1}^{{\mathit{\beta }}_1}}{{\mathit{t}}_{\mathit{a}\mathit{v}\mathit{g}}}^{{\mathit{\beta }}_1-{\mathit{\beta }}_0}$	Coverage % for HR at ${\mathit{t}}_{\mathit{a}\mathit{v}\mathit{g}}=\frac{{\mathit{t}}_{\mathit{m}\mathit{e}\mathit{d},\mathit{C}}+{\mathit{t}}_{\mathit{m}\mathit{e}\mathit{d},\mathit{E}}}{2}$ using 90% CI: exp $\left\{\left[{\widehat{\mathit{\gamma }}}_0+{\widehat{\mathit{\gamma }}}_1\mathbf{log}\left(\mathit{t}\right)\right]\pm 1.645\cdot \mathit{S}\mathit{E}\left[{\widehat{\mathit{\gamma }}}_0+{\widehat{\mathit{\gamma }}}_1\mathbf{l}\mathbf{o}\mathbf{g}\left(\mathit{t}\right)\right]\right\}$	Coverage % for: Line 1: log(HR) at t=1 using ${\widehat{\mathit{\gamma }}}_0\pm 1.645.\mathit{S}\mathit{E}\left({\widehat{\mathit{\gamma }}}_0\right)$ Line 2: ${\mathit{\beta }}_1-{\mathit{\beta }}_0$ using ${\widehat{\mathit{\gamma }}}_1\pm 1.645.\mathit{S}\mathit{E}\left({\widehat{\mathit{\gamma }}}_1\right)$	Approximation for $\widehat{\mathbf{R}\mathbf{T}}\left(\mathit{p}\right)$ calculated at ${\mathit{t}}_{\mathit{m}\mathit{e}\mathit{d},\mathit{C}}$ using ${\widehat{\mathbf{R}\mathbf{T}}}_{\mathit{a}\mathit{p}\mathit{p}\mathit{r}\mathit{o}\mathit{x}}\left(\mathit{p}\right)={\left\{\frac{\left({\mathit{\beta }}_0+{\widehat{\mathit{\gamma }}}_1\right)\mathbf{e}\mathbf{x}\mathbf{p}\left(-{\widehat{\mathit{\gamma }}}_0\right)}{{\mathit{\beta }}_0{\left({\widehat{\mathit{t}}}_{\mathit{m}\mathit{e}\mathit{d},\mathit{C}}\right)}^{{\widehat{\mathit{\gamma }}}_0}}\right\}}^{1/\left({\mathit{\beta }}_0+{\widehat{\mathit{\gamma }}}_1\right)}$
							Empirical Type I error %	Empirical Power %
β0=0.25	RT(p1)=1.52 RT(p2)=1.98	β1=0.2448	601/991	0.8479	90.28%	90.49% 90.27%	4.56%	79.58%
	RT(p1)=2.00 RT(p2)=1.50	β1=0.2560	722/1182	0.8984	90.01%	89.86% 89.68%	4.65%	80.16%
β0=0.50	RT(p1)=1.52 RT(p2)=1.98	β1=0.4795	154/216	0.7211	90.05%	89.80% 90.14%	5.18%	78.45%
	RT(p1)=2.00 RT(p2)=1.50	β1=0.5247	177/244	0.8054	89.91%	89.83% 90.19%	5.56%	79.88%
β0=0.75	RT(p1)=1.52 RT(p2)=1.98	β1=0.7047	70/87	0.6150	90.27%	90.32% 90.87%	5.49%	76.87%
	RT(p1)=2.00 RT(p2)=1.50	β1=0.8064	77/93	0.7202	89.75%	90.00% 90.52%	6.25%	77.41%
β0=1.00	RT(p1)=1.52 RT(p2)=1.98	β1=0.9211	41/46	0.5258	89.58%	90.50% 90.62%	7.00%	77.00%
	RT(p1)=2.00 RT(p2)=1.50	β1=1.1029	43/47	0.6423	90.04%	90.57% 90.11%	7.35%	77.75%
β0=1.25	RT(p1)=1.52 RT(p2)=1.98	β1=1.1290	27/29	0.4507	90.00%	90.95% 90.82%	7.31%	75.32%
	RT(p1)=2.00 RT(p2)=1.50	β1=1.4150	27/27	0.5712	89.55%	90.79% 90.23%	7.26%	76.11%
β0=1.50	RT(p1)=1.52 RT(p2)=1.98	β1=1.3291	19/20	0.3872	89.52%	92.47% 91.48%	7.89%	74.41%
	RT(p1)=2.00 RT(p2)=1.50	β1=1.7440	18/19	0.5064	89.64%	92.58% 90.95%	7.60%	77.09%
β0=2.00	RT(p1)=1.52 RT(p2)=1.98	β1=1.7073	11/12	0.2877	91.41%	94.35% 93.44%	6.65%	70.72%
	RT(p1)=2.00 RT(p2)=1.50	β1=2.4586	10/10	0.3938	91.14%	93.37% 92.44%	8.49%	75.58%