Tabular Results for Arbitrary Upstrapping (AU)

This section (Table S1:Table S4) includes tabular results summaries for the arbitrary upstrapping (AU) approach to interim monitoring. For each sample size (\(N\)) and power (\(\beta\)) based scenario three evaluation metrics are reported: expected total sample size (Expected \(N\)), proportion of trials that stopped early at any interim stopping point (Proportion Stopped), and proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Proportion Rejected). All tables also include fixed sample rejection rate results for comparison. Table S1 reports results for trials designed with a single interim stopping point at 25% of the total sample size. Table S2 reports results for trials designed with a single interim stopping point at 50% of the total sample size. Table S3 reports results for trials designed with a single interim stopping point at 75% of the total sample size. Table S4 reports results for trials designed with sequential interim stopping points at 25%, 50%, and 75% of the total sample size.

Table S1: Simulation Results Using Arbitrary Upstrapping Thresholds for 0.25 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 33 (12.94) 0.247 0.031 35 (11.51) 0.179 0.196 28 (14.84) 0.426 0.247 0.179 0.193 0.034
40 0.50 36 (10.92) 0.157 0.357 30 (14.31) 0.349 0.532 25 (15.01) 0.506 0.157 0.349 0.507 0.382
40 0.80 38 (8.80) 0.095 0.640 25 (15.01) 0.502 0.771 23 (14.72) 0.597 0.095 0.502 0.727 0.684
40 0.95 39 (5.95) 0.041 0.884 20 (14.01) 0.679 0.943 19 (13.48) 0.720 0.041 0.679 0.912 0.915
160 0.05 121 (56.58) 0.333 0.034 142 (43.22) 0.153 0.172 102 (60.01) 0.486 0.333 0.153 0.168 0.038
160 0.50 138 (47.19) 0.191 0.371 120 (56.63) 0.334 0.526 97 (59.95) 0.525 0.191 0.334 0.488 0.409
160 0.80 145 (40.38) 0.130 0.660 102 (60.02) 0.491 0.783 86 (58.25) 0.621 0.130 0.491 0.722 0.721
160 0.95 152 (31.03) 0.072 0.863 84 (57.66) 0.639 0.938 75 (54.42) 0.711 0.072 0.639 0.885 0.916
600 0.05 501 (187.11) 0.222 0.025 528 (165.89) 0.162 0.178 428 (218.97) 0.384 0.222 0.162 0.175 0.028
600 0.50 545 (147.87) 0.123 0.447 438 (216.37) 0.362 0.613 382 (225.01) 0.485 0.123 0.362 0.571 0.489
600 0.80 559 (130.76) 0.093 0.726 380 (225.08) 0.491 0.820 338 (221.91) 0.584 0.093 0.491 0.773 0.773
600 0.95 582 (88.23) 0.040 0.913 306 (214.02) 0.655 0.959 288 (207.29) 0.695 0.040 0.655 0.923 0.949
2000 0.05 1687 (610.20) 0.209 0.039 1739 (568.95) 0.174 0.197 1426 (729.54) 0.383 0.209 0.174 0.196 0.040
2000 0.50 1799 (511.23) 0.134 0.457 1459 (720.80) 0.361 0.596 1258 (750.34) 0.495 0.134 0.361 0.561 0.492
2000 0.80 1886 (397.70) 0.076 0.766 1208 (749.20) 0.528 0.859 1094 (733.96) 0.604 0.076 0.528 0.817 0.808
2000 0.95 1942 (290.54) 0.039 0.932 968 (695.31) 0.688 0.969 910 (668.59) 0.727 0.039 0.688 0.941 0.960
Table S2: Simulation Results Using Arbitrary Upstrapping Thresholds for 0.50 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 29 (9.90) 0.572 0.034 39 (4.44) 0.052 0.074 28 (9.69) 0.624 0.572 0.052 0.074 0.034
40 0.50 35 (8.70) 0.253 0.367 34 (9.18) 0.301 0.457 29 (9.95) 0.554 0.253 0.301 0.442 0.382
40 0.80 39 (5.81) 0.093 0.672 30 (10.00) 0.503 0.734 29 (9.82) 0.596 0.093 0.503 0.722 0.684
40 0.95 40 (3.06) 0.024 0.908 26 (8.84) 0.734 0.936 25 (8.57) 0.758 0.024 0.734 0.929 0.915
160 0.05 121 (40.02) 0.495 0.037 156 (18.71) 0.058 0.081 116 (39.79) 0.553 0.495 0.058 0.080 0.038
160 0.50 143 (33.32) 0.223 0.393 136 (36.71) 0.301 0.478 119 (39.97) 0.524 0.223 0.301 0.462 0.409
160 0.80 152 (24.22) 0.102 0.703 118 (39.96) 0.527 0.768 110 (38.67) 0.629 0.102 0.527 0.750 0.721
160 0.95 158 (14.91) 0.036 0.905 101 (35.15) 0.739 0.930 98 (33.42) 0.775 0.036 0.739 0.919 0.916
600 0.05 456 (149.98) 0.482 0.024 586 (64.79) 0.049 0.069 441 (149.79) 0.531 0.482 0.049 0.065 0.028
600 0.50 550 (112.48) 0.169 0.464 501 (141.24) 0.331 0.549 450 (150.08) 0.500 0.169 0.331 0.524 0.489
600 0.80 571 (89.24) 0.098 0.748 435 (149.29) 0.551 0.797 406 (143.26) 0.649 0.098 0.551 0.772 0.773
600 0.95 595 (40.98) 0.019 0.937 373 (128.37) 0.759 0.959 367 (124.74) 0.778 0.019 0.759 0.947 0.949
2000 0.05 1555 (497.21) 0.445 0.036 1940 (237.61) 0.060 0.084 1495 (500.23) 0.505 0.445 0.060 0.080 0.040
2000 0.50 1822 (382.70) 0.178 0.480 1660 (473.95) 0.340 0.555 1482 (499.93) 0.518 0.178 0.340 0.543 0.492
2000 0.80 1929 (256.95) 0.071 0.779 1444 (497.10) 0.556 0.842 1373 (483.84) 0.627 0.071 0.556 0.813 0.808
2000 0.95 1983 (129.34) 0.017 0.950 1225 (417.79) 0.775 0.966 1208 (406.08) 0.792 0.017 0.775 0.956 0.960
Table S3: Simulation Results Using Arbitrary Upstrapping Thresholds for 0.75 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 33 (4.10) 0.786 0.034 40 (1.62) 0.027 0.046 32 (3.90) 0.813 0.786 0.027 0.046 0.034
40 0.50 37 (4.63) 0.311 0.382 38 (4.43) 0.268 0.410 35 (4.94) 0.579 0.311 0.268 0.410 0.382
40 0.80 39 (3.04) 0.103 0.679 35 (4.99) 0.538 0.710 34 (4.80) 0.641 0.103 0.538 0.705 0.684
40 0.95 40 (1.59) 0.026 0.911 32 (3.87) 0.817 0.927 32 (3.64) 0.843 0.026 0.817 0.923 0.915
160 0.05 131 (17.40) 0.747 0.036 159 (7.35) 0.035 0.053 129 (16.52) 0.782 0.747 0.035 0.051 0.038
160 0.50 148 (18.62) 0.317 0.401 149 (17.91) 0.277 0.441 137 (19.65) 0.594 0.317 0.277 0.433 0.409
160 0.80 156 (12.96) 0.119 0.717 138 (19.88) 0.557 0.742 133 (18.73) 0.676 0.119 0.557 0.738 0.721
160 0.95 159 (6.37) 0.026 0.913 129 (16.39) 0.787 0.925 128 (15.60) 0.813 0.026 0.787 0.922 0.916
600 0.05 495 (68.64) 0.702 0.027 597 (23.43) 0.025 0.044 491 (66.86) 0.727 0.702 0.025 0.043 0.028
600 0.50 564 (64.28) 0.242 0.478 546 (72.08) 0.361 0.522 510 (73.43) 0.603 0.242 0.361 0.511 0.489
600 0.80 589 (40.48) 0.079 0.768 506 (72.50) 0.629 0.788 494 (68.24) 0.708 0.079 0.629 0.783 0.773
600 0.95 598 (17.63) 0.014 0.944 474 (54.17) 0.846 0.956 471 (52.07) 0.860 0.014 0.846 0.951 0.949
2000 0.05 1645 (226.76) 0.711 0.039 1984 (88.04) 0.032 0.058 1629 (218.60) 0.743 0.711 0.032 0.057 0.040
2000 0.50 1890 (207.23) 0.220 0.486 1819 (240.41) 0.362 0.521 1709 (246.74) 0.582 0.220 0.362 0.515 0.492
2000 0.80 1967 (125.07) 0.067 0.801 1671 (237.14) 0.659 0.830 1637 (223.12) 0.726 0.067 0.659 0.823 0.808
2000 0.95 1995 (49.77) 0.010 0.959 1569 (172.01) 0.863 0.963 1564 (166.57) 0.873 0.010 0.863 0.962 0.960
Table S4: Simulation Results Using Arbitrary Upstrapping Thresholds for Sequential Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 23 (10.08) 0.839 0.031 35 (11.65) 0.207 0.217 19 (9.06) 0.933 0.731 0.202 0.211 0.034
40 0.50 31 (11.66) 0.424 0.350 28 (13.86) 0.494 0.570 19 (10.98) 0.848 0.361 0.487 0.540 0.382
40 0.80 36 (9.76) 0.202 0.632 22 (13.01) 0.717 0.800 18 (10.58) 0.876 0.183 0.693 0.746 0.684
40 0.95 39 (6.56) 0.072 0.876 16 (9.93) 0.903 0.957 15 (8.64) 0.941 0.061 0.880 0.920 0.915
160 0.05 90 (44.15) 0.815 0.032 139 (44.37) 0.195 0.202 74 (39.46) 0.911 0.720 0.191 0.194 0.038
160 0.50 122 (48.29) 0.440 0.355 111 (55.12) 0.475 0.562 76 (44.19) 0.855 0.399 0.456 0.507 0.409
160 0.80 138 (42.86) 0.233 0.649 88 (52.50) 0.703 0.809 68 (41.38) 0.885 0.219 0.666 0.737 0.721
160 0.95 150 (32.99) 0.102 0.856 68 (42.27) 0.875 0.945 60 (34.88) 0.934 0.096 0.838 0.882 0.916
600 0.05 368 (159.90) 0.778 0.023 521 (168.13) 0.188 0.199 306 (154.54) 0.873 0.690 0.183 0.190 0.028
600 0.50 492 (163.13) 0.353 0.428 393 (205.42) 0.543 0.645 297 (169.15) 0.837 0.314 0.523 0.583 0.489
600 0.80 536 (144.81) 0.180 0.709 319 (190.46) 0.740 0.840 264 (158.61) 0.876 0.170 0.706 0.775 0.773
600 0.95 577 (95.51) 0.062 0.901 245 (150.94) 0.894 0.964 226 (131.86) 0.936 0.060 0.876 0.917 0.949
2000 0.05 1244 (527.49) 0.778 0.035 1715 (577.12) 0.202 0.220 1015 (503.02) 0.892 0.690 0.202 0.215 0.040
2000 0.50 1630 (558.83) 0.347 0.447 1310 (684.42) 0.543 0.635 979 (566.66) 0.831 0.309 0.522 0.587 0.492
2000 0.80 1824 (440.22) 0.156 0.742 1020 (624.09) 0.769 0.875 866 (520.51) 0.892 0.146 0.746 0.809 0.808
2000 0.95 1931 (305.80) 0.051 0.925 781 (479.86) 0.913 0.974 729 (427.46) 0.943 0.050 0.893 0.940 0.960

Tabular Results for Calibrated Upstrapping (CU)

This section (Table S5:Table S8) includes tabular results summaries for the calibrated upstrapping (CU) approach to interim monitoring. For each sample size (\(N\)) and power (\(\beta\)) based scenario three evaluation metrics are reported: expected total sample size (Expected \(N\)), proportion of trials that stopped early at any interim stopping point (Proportion Stopped), and proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Proportion Rejected). All tables also include fixed sample rejection rate results for comparison. Table S5 reports results for trials designed with a single interim stopping point at 25% of the total sample size. Table S6 reports results for trials designed with a single interim stopping point at 50% of the total sample size. Table S7 reports results for trials designed with a single interim stopping point at 75% of the total sample size. Table S8 reports results for trials designed with sequential interim stopping points at 25%, 50%, and 75% of the total sample size.

Table S5: Simulation Results Using Calibrated Upstrapping Thresholds for 0.25 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 27 (14.90) 0.441 0.026 40 (0.00) 0.000 0.034 27 (14.90) 0.441 0.441 0.000 0.026 0.034
40 0.50 31 (14.00) 0.320 0.312 40 (0.00) 0.000 0.382 31 (14.00) 0.320 0.320 0.000 0.312 0.382
40 0.80 35 (11.89) 0.195 0.592 40 (0.00) 0.000 0.684 35 (11.89) 0.195 0.195 0.000 0.592 0.684
40 0.95 38 (7.81) 0.073 0.861 40 (0.00) 0.000 0.915 38 (7.81) 0.073 0.073 0.000 0.861 0.915
160 0.05 106 (59.82) 0.458 0.034 155 (25.66) 0.048 0.078 100 (60.03) 0.506 0.458 0.048 0.074 0.038
160 0.50 127 (53.73) 0.277 0.348 142 (42.87) 0.150 0.448 109 (59.39) 0.427 0.277 0.150 0.387 0.409
160 0.80 138 (47.00) 0.189 0.625 129 (52.66) 0.260 0.738 107 (59.72) 0.449 0.189 0.260 0.642 0.721
160 0.95 146 (39.15) 0.121 0.820 113 (58.72) 0.396 0.921 98 (60.00) 0.517 0.121 0.396 0.825 0.916
600 0.05 394 (224.32) 0.458 0.020 579 (96.24) 0.048 0.072 373 (225.10) 0.506 0.458 0.048 0.064 0.028
600 0.50 479 (200.11) 0.271 0.404 529 (164.64) 0.159 0.527 407 (222.90) 0.430 0.271 0.159 0.442 0.489
600 0.80 511 (179.41) 0.198 0.672 484 (197.24) 0.259 0.786 395 (224.28) 0.457 0.198 0.259 0.685 0.773
600 0.95 555 (135.67) 0.101 0.864 426 (219.29) 0.387 0.953 381 (225.05) 0.488 0.101 0.387 0.868 0.949
2000 0.05 1294 (749.11) 0.471 0.033 1930 (317.62) 0.047 0.078 1223 (749.89) 0.518 0.471 0.047 0.071 0.040
2000 0.50 1543 (690.96) 0.305 0.408 1786 (525.37) 0.143 0.518 1328 (746.31) 0.448 0.305 0.143 0.434 0.492
2000 0.80 1702 (599.17) 0.199 0.687 1610 (658.28) 0.260 0.823 1312 (747.85) 0.459 0.199 0.260 0.702 0.808
2000 0.95 1840 (463.90) 0.107 0.869 1361 (742.11) 0.426 0.962 1201 (748.74) 0.533 0.107 0.426 0.871 0.960
Table S6: Simulation Results Using Calibrated Upstrapping Thresholds for 0.50 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 26 (8.74) 0.743 0.031 40 (4.10) 0.044 0.070 25 (8.19) 0.787 0.743 0.044 0.067 0.034
40 0.50 33 (9.62) 0.363 0.353 35 (9.14) 0.297 0.452 27 (9.48) 0.660 0.363 0.297 0.423 0.382
40 0.80 37 (7.90) 0.193 0.646 30 (10.00) 0.506 0.733 27 (9.18) 0.699 0.193 0.506 0.695 0.684
40 0.95 39 (5.04) 0.068 0.886 26 (8.92) 0.726 0.932 25 (8.09) 0.794 0.068 0.726 0.903 0.915
160 0.05 102 (35.62) 0.728 0.033 156 (17.44) 0.050 0.075 98 (33.26) 0.778 0.728 0.050 0.070 0.038
160 0.50 130 (38.91) 0.383 0.359 138 (35.94) 0.280 0.466 107 (37.83) 0.663 0.383 0.280 0.416 0.409
160 0.80 146 (30.41) 0.175 0.672 120 (40.02) 0.504 0.763 106 (37.37) 0.679 0.175 0.504 0.714 0.721
160 0.95 154 (21.21) 0.076 0.876 103 (36.09) 0.716 0.928 97 (32.49) 0.792 0.076 0.716 0.888 0.916
600 0.05 378 (131.49) 0.741 0.022 586 (64.79) 0.049 0.069 363 (122.25) 0.790 0.741 0.049 0.063 0.028
600 0.50 490 (144.75) 0.368 0.418 501 (141.24) 0.331 0.549 391 (137.68) 0.699 0.368 0.331 0.478 0.489
600 0.80 541 (119.83) 0.199 0.710 435 (149.29) 0.551 0.797 375 (129.97) 0.750 0.199 0.551 0.734 0.773
600 0.95 577 (80.02) 0.077 0.889 373 (128.37) 0.759 0.959 350 (111.14) 0.836 0.077 0.759 0.899 0.949
2000 0.05 1246 (430.89) 0.754 0.031 1950 (218.05) 0.050 0.076 1196 (397.17) 0.804 0.754 0.050 0.067 0.040
2000 0.50 1611 (487.77) 0.389 0.425 1679 (467.09) 0.321 0.545 1290 (453.99) 0.710 0.389 0.321 0.478 0.492
2000 0.80 1801 (399.45) 0.199 0.716 1453 (498.04) 0.547 0.840 1254 (435.52) 0.746 0.199 0.547 0.748 0.808
2000 0.95 1924 (265.13) 0.076 0.904 1234 (423.58) 0.766 0.966 1158 (364.92) 0.842 0.076 0.766 0.910 0.960
Table S7: Simulation Results Using Calibrated Upstrapping Thresholds for 0.75 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 32 (3.25) 0.880 0.031 40 (2.18) 0.050 0.062 31 (2.55) 0.930 0.880 0.050 0.059 0.034
40 0.50 36 (4.96) 0.436 0.367 37 (4.90) 0.399 0.467 32 (3.71) 0.835 0.436 0.399 0.452 0.382
40 0.80 39 (3.99) 0.198 0.662 34 (4.75) 0.656 0.745 32 (3.53) 0.854 0.198 0.656 0.723 0.684
40 0.95 40 (2.24) 0.053 0.902 32 (3.38) 0.869 0.933 31 (2.68) 0.922 0.053 0.869 0.920 0.915
160 0.05 125 (13.14) 0.877 0.035 159 (8.64) 0.049 0.063 123 (10.48) 0.926 0.877 0.049 0.060 0.038
160 0.50 142 (19.97) 0.470 0.378 146 (19.19) 0.358 0.468 127 (15.10) 0.828 0.470 0.358 0.437 0.409
160 0.80 153 (15.98) 0.199 0.692 135 (19.26) 0.636 0.759 127 (14.85) 0.835 0.199 0.636 0.730 0.721
160 0.95 158 (9.65) 0.062 0.898 126 (14.17) 0.853 0.932 124 (11.16) 0.915 0.062 0.853 0.914 0.916
600 0.05 467 (46.96) 0.890 0.022 593 (32.71) 0.050 0.065 459 (35.64) 0.940 0.890 0.050 0.059 0.028
600 0.50 538 (74.05) 0.419 0.449 532 (74.77) 0.458 0.554 469 (49.29) 0.877 0.419 0.458 0.514 0.489
600 0.80 572 (59.23) 0.193 0.737 494 (68.03) 0.711 0.798 465 (44.21) 0.904 0.193 0.711 0.762 0.773
600 0.95 593 (33.32) 0.052 0.920 466 (45.62) 0.897 0.965 458 (33.02) 0.949 0.052 0.897 0.936 0.949
2000 0.05 1541 (136.49) 0.919 0.029 1976 (106.94) 0.048 0.068 1517 (89.36) 0.967 0.919 0.048 0.057 0.040
2000 0.50 1775 (248.92) 0.451 0.437 1769 (249.40) 0.462 0.554 1544 (140.99) 0.913 0.451 0.462 0.499 0.492
2000 0.80 1901 (199.35) 0.198 0.746 1627 (217.76) 0.746 0.848 1528 (115.02) 0.944 0.198 0.746 0.786 0.808
2000 0.95 1972 (115.02) 0.056 0.928 1546 (143.88) 0.909 0.967 1518 (91.94) 0.965 0.056 0.909 0.935 0.960
Table S8: Simulation Results Using Calibrated Upstrapping Thresholds for Sequential Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 19 (9.32) 0.916 0.023 39 (4.41) 0.077 0.089 18 (7.96) 0.968 0.905 0.063 0.069 0.034
40 0.50 27 (13.10) 0.582 0.297 33 (8.87) 0.444 0.504 20 (8.54) 0.941 0.566 0.375 0.396 0.382
40 0.80 32 (12.26) 0.355 0.562 28 (8.79) 0.694 0.766 21 (7.61) 0.944 0.340 0.604 0.629 0.684
40 0.95 37 (8.63) 0.136 0.837 24 (6.68) 0.895 0.942 22 (5.64) 0.970 0.133 0.837 0.859 0.915
160 0.05 74 (38.38) 0.910 0.029 151 (29.41) 0.108 0.121 68 (32.42) 0.965 0.871 0.094 0.100 0.038
160 0.50 107 (50.58) 0.591 0.308 125 (46.18) 0.425 0.514 75 (35.95) 0.943 0.564 0.379 0.410 0.409
160 0.80 128 (48.49) 0.339 0.586 101 (47.24) 0.688 0.788 74 (36.32) 0.937 0.326 0.611 0.646 0.721
160 0.95 142 (41.02) 0.181 0.795 80 (40.65) 0.877 0.937 67 (31.87) 0.967 0.180 0.787 0.808 0.916
600 0.05 272 (135.05) 0.938 0.015 564 (110.59) 0.114 0.124 248 (114.44) 0.979 0.889 0.090 0.094 0.028
600 0.50 409 (192.21) 0.551 0.355 444 (172.43) 0.520 0.596 273 (128.42) 0.957 0.522 0.435 0.452 0.489
600 0.80 475 (185.34) 0.340 0.626 363 (168.56) 0.750 0.820 261 (120.04) 0.971 0.325 0.646 0.663 0.773
600 0.95 541 (144.71) 0.155 0.824 290 (140.39) 0.912 0.967 248 (110.40) 0.982 0.150 0.832 0.843 0.949
2000 0.05 891 (447.73) 0.937 0.025 1888 (359.98) 0.099 0.117 811 (370.70) 0.983 0.894 0.089 0.095 0.040
2000 0.50 1319 (651.66) 0.572 0.354 1496 (563.07) 0.514 0.590 887 (410.53) 0.972 0.538 0.434 0.447 0.492
2000 0.80 1588 (615.61) 0.339 0.625 1197 (549.99) 0.775 0.865 862 (388.08) 0.982 0.326 0.656 0.669 0.808
2000 0.95 1797 (488.44) 0.158 0.833 934 (462.45) 0.924 0.972 794 (355.29) 0.989 0.153 0.836 0.844 0.960

Tabular Results for Group Sequential Calibrated Upstrapping (GU)

This section (Table S9:Table S12) includes tabular results summaries for the group sequential calibrated upstrapping (GU) approach to interim monitoring. For each sample size (\(N\)) and power (\(\beta\)) based scenario three evaluation metrics are reported: expected total sample size (Expected \(N\)), proportion of trials that stopped early at any interim stopping point (Proportion Stopped), and proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Proportion Rejected). All tables also include fixed sample rejection rate results for comparison. Table S9 reports results for trials designed with a single interim stopping point at 25% of the total sample size. Table S10 reports results for trials designed with a single interim stopping point at 50% of the total sample size. Table S11 reports results for trials designed with a single interim stopping point at 75% of the total sample size. Table S12 reports results for trials designed with sequential interim stopping points at 25%, 50%, and 75% of the total sample size.

Table S9: Simulation Results Using Group Sequential O’Brien Fleming P Values and Calibrated Upstrapping Proportion Thresholds for 0.25 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 40 (0.00) 0 0.044 40 (5.04) 0.029 0.057 40 (5.04) 0.029 0 0.029 0.063 0.034
40 0.50 40 (0.00) 0 0.472 38 (8.92) 0.098 0.412 38 (8.92) 0.098 0 0.098 0.438 0.382
40 0.80 40 (0.00) 0 0.758 35 (11.22) 0.168 0.702 35 (11.22) 0.168 0 0.168 0.728 0.684
40 0.95 40 (0.00) 0 0.925 33 (13.26) 0.266 0.914 33 (13.26) 0.266 0 0.266 0.924 0.915
160 0.05 160 (0.00) 0 0.050 155 (25.92) 0.049 0.078 155 (25.92) 0.049 0 0.049 0.084 0.038
160 0.50 160 (0.00) 0 0.469 142 (42.87) 0.150 0.447 142 (42.87) 0.150 0 0.150 0.449 0.409
160 0.80 160 (0.00) 0 0.763 129 (52.92) 0.264 0.739 129 (52.92) 0.264 0 0.264 0.742 0.721
160 0.95 160 (0.00) 0 0.933 113 (58.74) 0.397 0.921 113 (58.74) 0.397 0 0.397 0.922 0.916
600 0.05 600 (0.00) 0 0.038 578 (98.12) 0.050 0.065 578 (98.12) 0.050 0 0.050 0.074 0.028
600 0.50 600 (0.00) 0 0.512 531 (162.51) 0.154 0.499 531 (162.51) 0.154 0 0.154 0.526 0.489
600 0.80 600 (0.00) 0 0.788 484 (196.99) 0.258 0.759 484 (196.99) 0.258 0 0.258 0.787 0.773
600 0.95 600 (0.00) 0 0.956 427 (219.18) 0.386 0.947 427 (219.18) 0.386 0 0.386 0.957 0.949
2000 0.05 2000 (0.00) 0 0.044 1928 (320.81) 0.048 0.076 1928 (320.81) 0.048 0 0.048 0.080 0.040
2000 0.50 2000 (0.00) 0 0.531 1786 (525.37) 0.143 0.499 1786 (525.37) 0.143 0 0.143 0.536 0.492
2000 0.80 2000 (0.00) 0 0.832 1613 (656.63) 0.258 0.806 1613 (656.63) 0.258 0 0.258 0.829 0.808
2000 0.95 2000 (0.00) 0 0.968 1361 (742.11) 0.426 0.958 1361 (742.11) 0.426 0 0.426 0.964 0.960
Table S10: Simulation Results Using Group Sequential O’Brien Fleming P Values and Calibrated Upstrapping Proportion Thresholds for 0.50 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 26 (8.79) 0.739 0.035 40 (4.28) 0.048 0.070 25 (8.19) 0.787 0.739 0.048 0.069 0.034
40 0.50 33 (9.62) 0.362 0.421 34 (9.19) 0.302 0.452 27 (9.45) 0.664 0.362 0.302 0.441 0.382
40 0.80 37 (7.80) 0.187 0.700 30 (10.00) 0.508 0.732 27 (9.21) 0.695 0.187 0.508 0.710 0.684
40 0.95 39 (4.83) 0.062 0.899 26 (8.83) 0.735 0.931 25 (8.05) 0.797 0.062 0.735 0.912 0.915
160 0.05 104 (36.64) 0.701 0.044 157 (15.87) 0.041 0.066 101 (35.02) 0.742 0.701 0.041 0.066 0.038
160 0.50 132 (38.18) 0.350 0.414 142 (33.48) 0.226 0.446 114 (39.56) 0.576 0.350 0.226 0.409 0.409
160 0.80 148 (28.97) 0.155 0.717 125 (39.77) 0.444 0.750 113 (39.23) 0.599 0.155 0.444 0.713 0.721
160 0.95 156 (19.01) 0.060 0.905 108 (37.89) 0.661 0.925 103 (35.90) 0.721 0.060 0.661 0.900 0.916
600 0.05 383 (134.02) 0.725 0.030 589 (58.11) 0.039 0.052 371 (127.45) 0.764 0.725 0.039 0.054 0.028
600 0.50 494 (143.72) 0.356 0.437 509 (138.32) 0.306 0.514 402 (141.98) 0.662 0.356 0.306 0.473 0.489
600 0.80 543 (117.99) 0.191 0.717 441 (149.80) 0.530 0.773 384 (134.62) 0.721 0.191 0.530 0.737 0.773
600 0.95 578 (78.57) 0.074 0.897 379 (131.98) 0.738 0.950 357 (117.27) 0.812 0.074 0.738 0.902 0.949
2000 0.05 1291 (454.45) 0.709 0.036 1954 (209.59) 0.046 0.069 1235 (424.21) 0.765 0.719 0.046 0.065 0.040
2000 0.50 1645 (478.75) 0.355 0.464 1689 (463.13) 0.311 0.524 1329 (470.09) 0.671 0.360 0.311 0.490 0.492
2000 0.80 1836 (370.46) 0.164 0.753 1469 (499.29) 0.531 0.822 1303 (459.79) 0.697 0.166 0.531 0.768 0.808
2000 0.95 1938 (241.28) 0.062 0.922 1260 (438.85) 0.740 0.962 1197 (397.93) 0.803 0.063 0.740 0.921 0.960
Table S11: Simulation Results Using Group Sequential O’Brien Fleming P Values and Calibrated Upstrapping Proportion Thresholds for 0.75 Stopping Point Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 32 (3.44) 0.863 0.037 40 (2.18) 0.050 0.063 31 (2.60) 0.927 0.877 0.050 0.061 0.034
40 0.50 36 (4.93) 0.413 0.438 37 (4.90) 0.397 0.465 32 (3.75) 0.831 0.434 0.397 0.467 0.382
40 0.80 39 (3.70) 0.164 0.728 34 (4.79) 0.645 0.744 32 (3.69) 0.838 0.193 0.645 0.737 0.684
40 0.95 40 (2.03) 0.043 0.914 32 (3.58) 0.849 0.929 31 (2.99) 0.901 0.052 0.849 0.921 0.915
160 0.05 126 (13.63) 0.866 0.042 159 (8.12) 0.043 0.057 124 (11.16) 0.915 0.872 0.043 0.054 0.038
160 0.50 142 (19.97) 0.469 0.413 146 (19.16) 0.356 0.467 127 (15.17) 0.826 0.470 0.356 0.438 0.409
160 0.80 153 (15.98) 0.199 0.717 135 (19.28) 0.634 0.756 127 (14.89) 0.834 0.200 0.634 0.727 0.721
160 0.95 158 (9.65) 0.062 0.910 127 (14.33) 0.849 0.932 124 (11.40) 0.911 0.062 0.849 0.915 0.916
600 0.05 467 (46.77) 0.891 0.027 593 (31.76) 0.047 0.054 463 (40.95) 0.919 0.872 0.047 0.059 0.028
600 0.50 536 (74.36) 0.433 0.457 533 (74.65) 0.449 0.533 470 (49.97) 0.873 0.424 0.449 0.507 0.489
600 0.80 571 (59.80) 0.198 0.743 494 (68.30) 0.707 0.780 465 (44.00) 0.905 0.198 0.707 0.762 0.773
600 0.95 592 (34.51) 0.056 0.921 466 (45.81) 0.896 0.958 458 (31.76) 0.953 0.057 0.896 0.932 0.949
2000 0.05 1544 (141.72) 0.912 0.034 1979 (101.48) 0.043 0.061 1521 (100.34) 0.958 0.915 0.043 0.054 0.040
2000 0.50 1784 (247.80) 0.432 0.467 1782 (248.07) 0.436 0.531 1557 (158.38) 0.887 0.450 0.437 0.496 0.492
2000 0.80 1909 (193.02) 0.182 0.768 1637 (223.12) 0.726 0.831 1539 (134.15) 0.922 0.194 0.728 0.780 0.808
2000 0.95 1975 (109.03) 0.050 0.938 1551 (150.74) 0.899 0.963 1524 (105.87) 0.953 0.054 0.899 0.937 0.960
Table S12: Simulation Results Using Group Sequential O’Brien Fleming P Values and Calibrated Upstrapping Proportion Thresholds for Sequential Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 24 (6.67) 0.894 0.032 39 (6.31) 0.093 0.104 23 (5.38) 0.966 0.876 0.090 0.097 0.034
40 0.50 32 (9.26) 0.480 0.405 32 (10.70) 0.460 0.517 24 (7.72) 0.907 0.465 0.442 0.483 0.382
40 0.80 36 (7.87) 0.243 0.685 27 (10.86) 0.698 0.774 22 (8.03) 0.924 0.244 0.680 0.725 0.684
40 0.95 39 (4.96) 0.080 0.891 22 (9.26) 0.885 0.940 20 (7.70) 0.956 0.080 0.876 0.910 0.915
160 0.05 97 (27.32) 0.888 0.040 152 (28.88) 0.097 0.110 90 (24.70) 0.949 0.857 0.092 0.101 0.038
160 0.50 126 (36.12) 0.518 0.386 127 (45.36) 0.411 0.506 94 (34.10) 0.898 0.497 0.401 0.455 0.409
160 0.80 145 (29.67) 0.244 0.689 104 (47.79) 0.673 0.777 89 (37.89) 0.896 0.233 0.663 0.720 0.721
160 0.95 155 (19.80) 0.086 0.892 82 (42.03) 0.867 0.934 77 (35.92) 0.944 0.086 0.858 0.899 0.916
600 0.05 354 (93.60) 0.920 0.024 566 (108.96) 0.105 0.108 330 (86.84) 0.962 0.867 0.095 0.101 0.028
600 0.50 473 (137.56) 0.493 0.416 450 (171.11) 0.507 0.571 333 (117.34) 0.933 0.455 0.478 0.505 0.489
600 0.80 533 (119.10) 0.262 0.697 367 (169.30) 0.745 0.804 306 (119.08) 0.961 0.246 0.715 0.739 0.773
600 0.95 575 (80.69) 0.096 0.883 293 (142.58) 0.909 0.962 272 (115.46) 0.977 0.093 0.884 0.898 0.949
2000 0.05 1185 (312.85) 0.921 0.032 1891 (359.01) 0.095 0.111 1096 (275.15) 0.974 0.881 0.093 0.100 0.040
2000 0.50 1577 (458.38) 0.492 0.423 1511 (566.30) 0.491 0.567 1110 (380.01) 0.940 0.473 0.467 0.497 0.492
2000 0.80 1802 (376.82) 0.233 0.725 1216 (560.02) 0.754 0.847 1031 (405.57) 0.958 0.231 0.727 0.757 0.808
2000 0.95 1927 (249.47) 0.084 0.907 952 (481.28) 0.910 0.967 887 (394.83) 0.976 0.086 0.890 0.911 0.960

Tabular Results for Group Sequential Approaches (OBF, PO)

This section (Table S13:Table S14) includes tabular results summaries for the traditional group sequential design based approaches to interim monitoring including both O’Brien-Fleming (OBF) and Pocock (PO). For each sample size (\(N\)) and power (\(\beta\)) based scenario three evaluation metrics are reported: expected total sample size (Expected \(N\)), proportion of trials that stopped early at any interim stopping point (Proportion Stopped), and proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Proportion Rejected). Both tables also include fixed sample rejection rate results for comparison. Table S13 reports results for trials designed with O’Brien-Fleming p-value boundaries for sequential interim stopping points at 25%, 50%, and 75% of the total sample size. Table S14 reports results for trials designed with Pocock p-value boundaries for sequential interim stopping points at 25%, 50%, and 75% of the total sample size.

Table S13: Simulation Results Using Group Sequential O’Brien Fleming P Value Thresholds for Sequential Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 25 (6.68) 0.898 0.032 40 (0.94) 0.009 0.036 24 (6.49) 0.908 0.899 0.009 0.032 0.034
40 0.50 32 (9.06) 0.493 0.402 39 (4.18) 0.156 0.385 30 (8.42) 0.650 0.494 0.156 0.373 0.382
40 0.80 36 (7.43) 0.251 0.683 36 (5.98) 0.372 0.688 32 (7.57) 0.632 0.260 0.372 0.658 0.684
40 0.95 39 (4.32) 0.074 0.897 33 (6.55) 0.645 0.913 31 (6.66) 0.732 0.087 0.645 0.888 0.915
160 0.05 103 (28.78) 0.864 0.042 160 (6.64) 0.019 0.043 102 (27.33) 0.891 0.872 0.019 0.043 0.038
160 0.50 131 (33.52) 0.484 0.407 151 (19.99) 0.215 0.428 121 (30.50) 0.703 0.488 0.215 0.391 0.409
160 0.80 147 (26.47) 0.223 0.704 138 (26.68) 0.463 0.732 125 (28.50) 0.686 0.225 0.461 0.690 0.721
160 0.95 156 (16.61) 0.070 0.904 123 (28.83) 0.712 0.918 118 (27.79) 0.784 0.072 0.712 0.895 0.916
600 0.05 398 (107.90) 0.855 0.032 599 (17.69) 0.008 0.022 397 (106.53) 0.863 0.855 0.008 0.027 0.028
600 0.50 514 (116.48) 0.396 0.463 559 (78.44) 0.243 0.471 472 (111.69) 0.642 0.399 0.243 0.457 0.489
600 0.80 558 (95.88) 0.184 0.738 505 (106.49) 0.500 0.753 462 (110.74) 0.688 0.188 0.500 0.737 0.773
600 0.95 591 (47.17) 0.046 0.928 444 (111.51) 0.745 0.944 433 (106.94) 0.797 0.053 0.744 0.925 0.949
2000 0.05 1348 (357.24) 0.851 0.035 1992 (68.43) 0.016 0.043 1333 (340.31) 0.880 0.864 0.016 0.038 0.040
2000 0.50 1724 (393.64) 0.367 0.489 1855 (271.20) 0.246 0.483 1571 (381.58) 0.628 0.381 0.247 0.479 0.492
2000 0.80 1888 (284.67) 0.149 0.782 1655 (364.71) 0.533 0.804 1541 (366.68) 0.687 0.154 0.533 0.776 0.808
2000 0.95 1975 (142.81) 0.033 0.951 1439 (369.34) 0.787 0.958 1417 (352.71) 0.821 0.034 0.787 0.946 0.960
Table S14: Simulation Results Using Group Sequential Pocock P Value Thresholds for Sequential Monitoring
Futility Only Monitoring
Efficacy Only Monitoring
Futility and Efficacy Monitoring
Fixed Sample
N \(\beta\) Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Rejected Expected N mean(sd) Proportion Stopped Proportion Stopped (Futility) Proportion Stopped (Efficacy) Proportion Rejected Proportion Rejected
40 0.05 17 (9.44) 0.925 0.024 40 (1.92) 0.012 0.020 16 (8.45) 0.966 0.955 0.011 0.021 0.034
40 0.50 25 (13.61) 0.618 0.312 38 (6.58) 0.173 0.296 22 (11.44) 0.835 0.678 0.157 0.261 0.382
40 0.80 30 (13.46) 0.396 0.558 34 (8.68) 0.394 0.572 24 (11.40) 0.792 0.440 0.352 0.493 0.684
40 0.95 35 (11.41) 0.209 0.777 30 (9.22) 0.664 0.828 24 (9.94) 0.845 0.230 0.615 0.735 0.915
160 0.05 78 (39.34) 0.896 0.038 158 (14.40) 0.030 0.036 69 (35.07) 0.956 0.928 0.028 0.035 0.038
160 0.50 112 (49.20) 0.559 0.355 144 (33.28) 0.240 0.355 86 (43.74) 0.849 0.630 0.219 0.306 0.409
160 0.80 132 (45.98) 0.313 0.628 126 (41.00) 0.496 0.652 90 (43.03) 0.844 0.385 0.459 0.557 0.721
160 0.95 145 (37.53) 0.153 0.828 105 (42.56) 0.730 0.866 86 (40.11) 0.881 0.203 0.678 0.770 0.916
600 0.05 316 (148.35) 0.888 0.028 590 (64.32) 0.031 0.035 282 (125.55) 0.964 0.936 0.028 0.033 0.028
600 0.50 450 (178.65) 0.474 0.427 526 (134.40) 0.284 0.398 350 (161.64) 0.820 0.553 0.267 0.360 0.489
600 0.80 506 (167.34) 0.268 0.676 457 (160.61) 0.531 0.680 344 (159.44) 0.842 0.341 0.501 0.619 0.773
600 0.95 561 (118.40) 0.107 0.874 369 (162.05) 0.771 0.902 320 (148.03) 0.891 0.153 0.738 0.822 0.949
2000 0.05 1085 (497.35) 0.877 0.039 1966 (203.43) 0.032 0.039 996 (432.34) 0.947 0.915 0.032 0.042 0.040
2000 0.50 1501 (602.38) 0.465 0.454 1717 (486.98) 0.295 0.408 1176 (547.10) 0.793 0.506 0.287 0.381 0.492
2000 0.80 1742 (517.88) 0.223 0.727 1458 (554.75) 0.568 0.726 1168 (520.14) 0.833 0.261 0.572 0.679 0.808
2000 0.95 1884 (383.18) 0.090 0.902 1186 (527.84) 0.804 0.921 1064 (484.20) 0.894 0.112 0.782 0.867 0.960

AU Single Stopping Point Data Visualizations

This section includes graphical results summaries comparing between the various versions of arbitrary upstrapping (AU) presented in Table S5:Table S8. For each sample size (\(N\), shown on the y axis) and power (\(\beta\), shown on the x axis) based scenario three evaluation metrics are reported: expected total sample size (Expected Sample Size), proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Rejection Rate), and proportion of trials that stopped early at any interim stopping point (Interim Stopping Rate). The AU method refers to trials designed with sequential stopping points at 25%, 50%, and 75% of the total sample size. The AU 0.25 method refers to trials designed with a single interim stopping point at 25% of the total sample size. The AU 0.50 method refers to trials designed with a single interim stopping point at 50% of the total sample size. The AU 0.75 method refers to trials designed with a single interim stopping point at 75% of the total sample size.

**Figure S1:** AU single stopping point results**Figure S1:** AU single stopping point results**Figure S1:** AU single stopping point results

Figure S1: AU single stopping point results

CU Single Stopping Point Data Visualizations

This section includes graphical results summaries comparing between the various versions of calibrated upstrapping (CU) presented in Table S1:Table S4. For each sample size (\(N\), shown on the y axis) and power (\(\beta\), shown on the x axis) based scenario three evaluation metrics are reported: expected total sample size (Expected Sample Size), proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Rejection Rate), and proportion of trials that stopped early at any interim stopping point (Interim Stopping Rate). The CU method refers to trials designed with sequential stopping points at 25%, 50%, and 75% of the total sample size. The CU 0.25 method refers to trials designed with a single interim stopping point at 25% of the total sample size. The CU 0.50 method refers to trials designed with a single interim stopping point at 50% of the total sample size. The CU 0.75 method refers to trials designed with a single interim stopping point at 75% of the total sample size.

**Figure S2:** CU single stopping point results**Figure S2:** CU single stopping point results**Figure S2:** CU single stopping point results

Figure S2: CU single stopping point results

GU Single Stopping Point Data Visualizations

This section includes graphical results summaries comparing between the various versions of group sequential calibrated upstrapping (GU) presented in Table S9:Table S12. For each sample size (\(N\), shown on the y axis) and power (\(\beta\), shown on the x axis) based scenario three evaluation metrics are reported: expected total sample size (Expected Sample Size), proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Rejection Rate), and proportion of trials that stopped early at any interim stopping point (Interim Stopping Rate). The GU method refers to trials designed with sequential stopping points at 25%, 50%, and 75% of the total sample size. The GU 0.25 method refers to trials designed with a single interim stopping point at 25% of the total sample size. The GU 0.50 method refers to trials designed with a single interim stopping point at 50% of the total sample size. The GU 0.75 method refers to trials designed with a single interim stopping point at 75% of the total sample size.

**Figure S3:** GU single stopping point results**Figure S3:** GU single stopping point results**Figure S3:** GU single stopping point results

Figure S3: GU single stopping point results

Combined Data Visualizations

This section includes graphical results summaries comparing simultaneously between all the various versions of upstrapping (CU, AU, or GU) presented in Table S1:Table S12. For each sample size (\(N\), shown on the y axis) and power (\(\beta\), shown on the x axis) based scenario three evaluation metrics are reported: expected total sample size (Expected Sample Size), proportion of trials that either stopped early for efficacy or rejected the null hypothesis with the full sample data (Rejection Rate), and proportion of trials that stopped early at any interim stopping point (Interim Stopping Rate). The CU, AU, GU methods refer to trials designed with sequential stopping points at 25%, 50%, and 75% of the total sample size. The CU 0.25, AU 0.25, GU 0.25 methods refer to trials designed with a single interim stopping point at 25% of the total sample size. The CU 0.50, AU 0.50, GU 0.50 methods refer to trials designed with a single interim stopping point at 50% of the total sample size. The CU 0.75, AU 0.75, GU 0.75 methods refer to trials designed with a single interim stopping point at 75% of the total sample size.

**Figure S4:** Combined results for all methods**Figure S4:** Combined results for all methods**Figure S4:** Combined results for all methods

Figure S4: Combined results for all methods

Sensitivity Analysis: Upstrapping Results Excluding the 25% Monitoring Point

Due to concerns that method performance might be inhibited by lack of sufficient data at particularly early monitoring points, simulations dropping the earliest interim analysis (at 25% of the full sample) were implemented. Results from this sensitivity analysis are consistent with the main results findings overall, with small to moderate gains in operating characteristics across methods. Without the possibility of stopping for futility at 25% of the full sample ESS increased for all methods proportionally.

While CP 1% still performs best out of all four CP based methods, the decrease in power under the alternative is lessened compared to the main results findings, with power reductions of up to 1.6% and 7.2% for CP 1% and CP 5% respectively. Type I error rate was consistent with previous results. For ESS the difference between the four CP methods was narrower, ranging from 52-65% of FS under the null and 83-100% under the alternative.

AU and CU both showed significant gains in power after excluding the first interim look, with power reductions between 1.2-3.3% compared to FS for AU and 5.2-11.6% for CU. Type I error rate showed slight improvement, remaining within 0.5% of FS for AU and within 1.3% for CU. Under the null scenario ESS was around 69% of FS for AU and around 59% for CU. For the alternative ESS was approximately 95% of FS for AU and 89% for CU. Results remain unchanged for GU since it’s specific calibration approach already excludes the first interim monitoring point.

Excluding the first interim look does have a positive effect on performance of the upstrapping based designs. Both AU and CU achieved improvements in power and type I error rate and higher ESS under the alternative scenario. While ESS under the null scenario showed more modest results compared to the main findings, it was still significantly reduced compared to FS. After excluding the first interim look the upstrapping designs were more comparable to both OBF, CP 5%, and CP 1%.

The full results of this sensitivity analysis are shown below in Table S15 and Figure S5 below.

Table S15 describes sensitivity analysis results from alternative and null scenario simulations for the performance of the interim analysis calibration method, excluding the first interim monitoring point, relative to the fixed sample design. CU is the calibrated upstrap, AU is the arbitrary upstrap, GU is the group-sequential upstrap, OBF is the O’Brien-Fleming alpha-spending function, PO is the Pocok alpha-spending function, CP is the conditional power method. ESS refers here to expected sample size, while TIE refers to the type I error rate. Any positive increases in TIE and power for GU, OBF, and PO relative to the fixed sample occur because group sequential designs involve an adjusted p-value boundary for the final analysis.

Table S15: Sensitivity Analysis Results with Two Stopping Points (50% and 75%)
Null
Alternative
Method N Expected N Ratio Type I Error Difference Expected N Ratio Power Difference
0.50-0.75 AU 40 0.68 0.000 0.95 -0.016
0.50-0.75 AU 160 0.68 -0.002 0.94 -0.020
0.50-0.75 AU 600 0.69 -0.004 0.94 -0.026
0.50-0.75 AU 2000 0.70 -0.005 0.95 -0.033
0.50-0.75 CU 40 0.60 -0.006 0.90 -0.052
0.50-0.75 CU 160 0.59 -0.005 0.89 -0.065
0.50-0.75 CU 600 0.58 -0.009 0.88 -0.080
0.50-0.75 CU 2000 0.58 -0.013 0.88 -0.116
0.50-0.75 GU 40 0.60 -0.002 0.90 0.001
0.50-0.75 GU 160 0.61 0.002 0.91 -0.032
0.50-0.75 GU 600 0.59 -0.004 0.89 -0.076
0.50-0.75 GU 2000 0.59 -0.008 0.90 -0.083
0.50-0.75 CP 1% 40 0.65 -0.013 1.00 -0.002
0.50-0.75 CP 1% 160 0.63 -0.020 0.96 -0.008
0.50-0.75 CP 1% 600 0.63 -0.012 0.96 -0.008
0.50-0.75 CP 1% 2000 0.62 -0.019 0.97 -0.016
0.50-0.75 CP 5% 40 0.58 -0.017 0.98 -0.010
0.50-0.75 CP 5% 160 0.58 -0.021 0.92 -0.032
0.50-0.75 CP 5% 600 0.56 -0.015 0.91 -0.059
0.50-0.75 CP 5% 2000 0.56 -0.024 0.92 -0.072
0.50-0.75 CP 10% 40 0.55 -0.020 0.92 -0.082
0.50-0.75 CP 10% 160 0.54 -0.026 0.90 -0.061
0.50-0.75 CP 10% 600 0.54 -0.020 0.88 -0.100
0.50-0.75 CP 10% 2000 0.54 -0.029 0.88 -0.123
0.50-0.75 CP 20% 40 0.55 -0.024 0.88 -0.119
0.50-0.75 CP 20% 160 0.52 -0.031 0.84 -0.140
0.50-0.75 CP 20% 600 0.52 -0.024 0.83 -0.181
0.50-0.75 CP 20% 2000 0.52 -0.033 0.83 -0.220

Figure S5 describes the results for a sensitivity analysis excluding the first interim monitoring point (i.e. interim analyses are performed at 50% and 75% of the total planned sample). The panels from left to right show mean expected sample size, rejection rate, and interim stopping rate.

**Figure S5:** Sensitivity Analysis Results**Figure S5:** Sensitivity Analysis Results**Figure S5:** Sensitivity Analysis Results

Figure S5: Sensitivity Analysis Results