. Author manuscript; available in PMC: 2014 Dec 11.

Published in final edited form as: Stat Med. 2011 Dec 5;30(30):3532–3545. doi: 10.1002/sim.4401

Table 4.

Summary of approximate 95% two-sided confidence bounds of BTP, BTII and transformed GI for P₂ under gamma distributions (based on 5,000 simulations).

	Three Independent Gamma Distributions
	(α₁, β₁) = (1, 6)′, (α₂, β₂) = (4, 6)′, (α₃, β₃) = (6.2, 6)′, P₂ = 0.5
	Coverage Probability			Lower Tail			Upper Tail			Length of CI
Sample Sizes	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI
(10, 10, 10)	0.9674	0.9498	0.9628	0.0306	0.0358	0.0038	0.0020	0.0144	0.0334	0.8131	0.6409	0.7133
(30, 30, 30)	0.9762	0.9632	0.9628	0.0176	0.0216	0.0130	0.0062	0.0152	0.0242	0.5550	0.5021	0.4271
(20, 10, 10)	0.9656	0.9518	0.9644	0.0318	0.0350	0.0074	0.0026	0.0132	0.0282	0.7986	0.6246	0.6627
(50, 30, 30)	0.9696	0.9620	0.9586	0.0234	0.0254	0.0172	0.0070	0.0126	0.0242	0.5347	0.4876	0.3988
(50, 50, 50)	0.9734	0.9654	0.9548	0.0176	0.0206	0.0174	0.0090	0.0140	0.0278	0.4370	0.4138	0.3231
	(α₁, β₁) = (1, 6)′, (α₂, β₂) = (4, 6)′, (α₃, β₃) = (7.7, 6)′, P₂ = 0.7
	Coverage Probability			Lower Tail			Upper Tail			Length of CI
Sample Sizes	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI
(10, 10, 10)	0.9634	0.9600	0.9578	0.0342	0.0228	0.0026	0.0024	0.0172	0.0396	0.7463	0.5823	0.7318
(30, 30, 30)	0.9726	0.9696	0.9638	0.0204	0.0136	0.0088	0.0070	0.0168	0.0274	0.4905	0.4476	0.3874
(20, 10, 10)	0.9640	0.9652	0.9536	0.0340	0.0184	0.0048	0.0020	0.0164	0.0416	0.7179	0.5565	0.6386
(50, 30, 30)	0.9712	0.9660	0.9594	0.0224	0.0186	0.0108	0.0064	0.0154	0.0298	0.4667	0.4216	0.3571
(50, 50, 50)	0.9714	0.9736	0.9594	0.0208	0.0148	0.0148	0.0078	0.0116	0.0258	0.3819	0.3622	0.2905
	(α₁, β₁) = (1, 6)′, (α₂, β₂) = (4, 6)′, (α₃, β₃) = (9.0, 6)′, P₂ = 0.8
	Coverage Probability			Lower Tail			Upper Tail			Length of CI
Sample Sizes	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI
(10, 10, 10)	0.9680	0.9572	0.9570	0.0228	0.0190	0.0022	0.0092	0.0238	0.0408	0.6473	0.5053	0.6967
(30, 30, 30)	0.9776	0.9760	0.9582	0.0136	0.0104	0.0084	0.0088	0.0136	0.0334	0.4166	0.3777	0.3378
(20, 10, 10)	0.9742	0.9626	0.9490	0.0162	0.0136	0.0036	0.0096	0.0238	0.0474	0.6096	0.4795	0.5771
(50, 30, 30)	0.9740	0.9738	0.9508	0.0136	0.0116	0.0102	0.0124	0.0146	0.0390	0.3864	0.3529	0.3053
(50, 50, 50)	0.9746	0.9744	0.9554	0.0104	0.0122	0.0118	0.0150	0.0134	0.0328	0.3213	0.3055	0.2518
	(α₁, β₁) = (1, 6)′, (α₂, β₂) = (4, 6)′, (α₃, β₃) = (12.1, 6)′, P₂ = 0.9
	Coverage Probability			Lower Tail			Upper Tail			Length of CI
Sample Sizes	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI	BTP	BTII	GI
(10, 10, 10)	0.9292	0.8888	0.9526	0.0626	0.0000	0.0056	0.0082	0.1112	0.0418	0.4958	0.3872	0.6012
(30, 30, 30)	0.9754	0.9704	0.9568	0.0146	0.0100	0.0130	0.0100	0.0196	0.0302	0.3072	0.2840	0.2696
(20, 10, 10)	0.9264	0.8924	0.9502	0.0670	0.0000	0.0072	0.0066	0.1076	0.0426	0.4332	0.3544	0.4592
(50, 30, 30)	0.9726	0.9678	0.9598	0.0138	0.0122	0.0142	0.0136	0.0200	0.0260	0.2740	0.2561	0.2305
(50, 50, 50)	0.9678	0.9706	0.9520	0.0206	0.0120	0.0184	0.0116	0.0174	0.0296	0.2401	0.2295	0.1973

BTP: The confidence interval based on bootstrap percentiles.

BTII: The confidence interval presented in equ (15).

GI: The generalized confidence interval for Box-Cox transferred data.

Low tail (upper tail): One-sided coverage errors, i.e. the proportion of runs in which the lower (or upper) limit of the confidence interval excluded the true P₂ at nominal level 0.025.

Length of CI: the average length of two-sided confidence intervals for P₂.