Incorporating the sample correlation into the testing of two endpoints in clinical trials

Figures & data

Table 1. One-sided critical values ($\alpha$= 0.025; β = 0.05 for n < 1,000, β = 0.01 for n ≥ 1,000)

r	Sample Size, n (n1:n2)
	10 (5:5)	20 (10:10)	30 (15:15)	50 (25:25)	80 (40:40)	150 (75:75)	500 (250:250)	2,000 (1,000:1,000)
−1	0.01250	0.01250	0.01250	0.01250	0.01250	0.01250	0.01250	0.01250
−0.3	0.01250	0.01250	0.01250	0.01250	0.01250	0.0125	0.01250	0.01250
−0.25	0.01250	0.01250	0.01250	0.01250	0.01250	0.0125	0.01250	0.01250
−0.2	0.01250	0.01250	0.01250	0.01250	0.01250	0.0125	0.01251	0.01251
−0.15	0.01250	0.01250	0.01250	0.01250	0.01250	0.01251	0.01251	0.01251
−0.10	0.01250	0.01250	0.01250	0.01250	0.01251	0.01251	0.01252	0.01252
−0.05	0.01250	0.01250	0.01250	0.01251	0.01251	0.01252	0.01253	0.01253
0.0	0.01250	0.01250	0.01251	0.01251	0.01252	0.01253	0.01254	0.01254
0.05	0.01250	0.01251	0.01251	0.01252	0.01253	0.01254	0.01256	0.01256
0.10	0.01250	0.01251	0.01252	0.01253	0.01254	0.01256	0.01259	0.01259
0.15	0.01250	0.01251	0.01252	0.01254	0.01256	0.01258	0.01262	0.01262
0.20	0.01250	0.01252	0.01253	0.01256	0.01258	0.01261	0.01266	0.01267
0.25	0.01251	0.01253	0.01255	0.01258	0.01261	0.01266	0.01272	0.01272
0.30	0.01251	0.01254	0.01257	0.01261	0.01265	0.01271	0.01278	0.01278
0.35	0.01251	0.01256	0.01260	0.01265	0.01271	0.01277	0.01286	0.01286
0.40	0.01252	0.01258	0.01263	0.01271	0.01277	0.01285	0.01296	0.01296
0.45	0.01253	0.01261	0.01268	0.01278	0.01286	0.01295	0.01308	0.01308
0.50	0.01254	0.01266	0.01275	0.01287	0.01297	0.01308	0.01323	0.01323
0.55	0.01256	0.01272	0.01284	0.01298	0.01310	0.01323	0.01341	0.01341
0.60	0.01259	0.01280	0.01295	0.01313	0.01327	0.01343	0.01362	0.01362
0.65	0.01263	0.01292	0.01311	0.01332	0.01349	0.01367	0.01388	0.01389
0.70	0.01269	0.01308	0.01332	0.01358	0.01377	0.01397	0.01421	0.01421
0.75	0.01278	0.01331	0.01360	0.01391	0.01413	0.01435	0.01462	0.01462
0.80	0.01295	0.01365	0.01400	0.01436	0.01461	0.01485	0.01514	0.01514
0.85	0.01323	0.01416	0.01458	0.01499	0.01526	0.01553	0.01583	0.01583
0.90	0.01378	0.01500	0.01549	0.01593	0.01622	0.0165	0.01681	0.01681
0.95	0.01507	0.01657	0.01709	0.01753	0.01781	0.01807	0.01836	0.01836

Table 2. Two-sided critical values ($\alpha$= 0.05; β = 0.05 for n < 1,000, β = 0.01 for n ≥ 1,000)

|r|	Sample Size, n (n1:n2)
	10 (5:5)	20 (10:10)	30 (15:15)	50 (25:25)	80 (40:40)	150 (75:75)	500 (250:250)	2,000 (1,000:1,000)
0.0	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530
0.05	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530	0.02531
0.10	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530	0.02531	0.02533
0.15	0.02530	0.02530	0.02530	0.02530	0.02530	0.02530	0.02533	0.02537
0.20	0.02530	0.02530	0.02530	0.02530	0.02530	0.02533	0.02539	0.02545
0.25	0.02530	0.02530	0.02530	0.02530	0.02532	0.02538	0.02547	0.02555
0.30	0.02530	0.02530	0.02530	0.02532	0.02538	0.02546	0.02559	0.02569
0.35	0.02530	0.02530	0.02530	0.02537	0.02546	0.02557	0.02574	0.02586
0.40	0.02530	0.02530	0.02534	0.02546	0.02557	0.02572	0.02593	0.02608
0.45	0.02530	0.02530	0.02542	0.02558	0.02573	0.02592	0.02617	0.02634
0.50	0.02530	0.02537	0.02553	0.02575	0.02594	0.02616	0.02646	0.02665
0.55	0.02530	0.02548	0.02569	0.02597	0.02621	0.02647	0.02681	0.02703
0.60	0.02530	0.02563	0.02592	0.02627	0.02655	0.02685	0.02724	0.02749
0.65	0.02530	0.02585	0.02622	0.02665	0.02698	0.02733	0.02777	0.02804
0.70	0.02539	0.02617	0.02664	0.02715	0.02754	0.02794	0.02842	0.02872
0.75	0.02559	0.02663	0.02721	0.02782	0.02826	0.02870	0.02924	0.02956
0.80	0.02590	0.02730	0.02801	0.02872	0.02921	0.02971	0.03029	0.03063
0.85	0.02647	0.02833	0.02917	0.02998	0.03053	0.03106	0.03167	0.03202
0.90	0.02756	0.03000	0.03097	0.03186	0.03244	0.03299	0.03362	0.03397
0.95	0.03013	0.03314	0.03418	0.03507	0.03563	0.03615	0.03673	0.03704

Table 3. Power (%) comparison. One-sided $\alpha$= 0.025

Sample Size, n (n1:n2)	Effect sizes	$\rho$	Adaptive. Bonferroni	Bonferroni	Simes	Šidák
30 (15:15)	1.3,0	0	87.8	87.8	87.8	87.8
		0.5	87.9	87.6	87.6	87.7
		0.9	89.7	87.6	87.6	87.7
	1.2,0.6	0	86.2	86.2	86.9	86.2
		0.5	83.0	82.7	83.2	82.8
		0.9	84.1	81.4	81.5	81.5
	1.1,1.1	0	93.2	93.1	94.0	93.2
		0.5	87.7	87.4	88.6	87.5
		0.9	82.6	79.8	82.2	79.9
500 (250:250)	0.3,0	0	86.7	86.7	86.7	86.7
		0.5	87.1	86.6	86.8	86.6
		0.9	88.9	86.5	86.5	86.6
	0.28,0.14	0	85.8	85.7	86.4	85.8
		0.5	82.8	82.2	82.7	82.3
		0.9	84.1	81.1	81.2	81.2
	0.25,0.25	0	91.5	91.4	92.3	91.5
		0.5	85.5	84.9	86.0	85.0
		0.9	80.5	77.0	79.2	77.1