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Table 5 Sample scenarios assuming beta priors p(π 1) and p(π 2) where \(\tau _{1}^{2}=0.001\). Hypothesized values of π 1 and π 2 set equal to m 1 and m 2, respectively, under the traditional design. Two-sided α=0.05 and 1−β=0.80 assumed

From: Sample size determination for a binary response in a superiority clinical trial using a hybrid classical and Bayesian procedure

  (\(\tau _{1}^{2}=0.001\)) Traditional design CEP design  
(m 1,m 2) \(\tau _{2}^{2}\) \(\hat {N}\) Performance CEP N Performance E(π 2π 1|π 2>π 1) P(π 2>π 1) Marginal benefit
(0.1,0.9) 0.01 10 0.344 0.706 14 0.640 0.728 1 0.0740
  0.02 10 0.280 0.639 16 0.615 0.682 1 0.0559
  0.03 10 0.239 0.586 20 0.655 0.640 0.999 0.0416
  0.04 10 0.208 0.539 24 0.659 0.601 0.997 0.0322
  0.05 10 0.184 0.497 30 0.671 0.564 0.991 0.0244
  0.06 10 0.163 0.459 38 0.687 0.528 0.979 0.0187
  0.07 10 0.144 0.424 50 0.693 0.494 0.955 0.0137
  0.08 10 0.127 0.389 70 0.703 0.459 0.913 0.0096
(0.1,0.8) 0.01 14 0.439 0.748 16 0.586 0.656 1 0.0735
  0.02 14 0.389 0.697 20 0.623 0.624 1 0.0390
  0.03 14 0.353 0.652 24 0.651 0.592 0.999 0.0298
  0.04 14 0.324 0.611 28 0.667 0.561 0.997 0.0245
  0.05 14 0.299 0.574 34 0.674 0.533 0.990 0.0187
  0.06 14 0.278 0.540 44 0.687 0.505 0.975 0.0136
  0.07 14 0.259 0.509 56 0.695 0.480 0.950 0.0104
  0.08 14 0.241 0.480 72 0.703 0.455 0.910 0.0080
(0.1,0.7) 0.01 20 0.507 0.773 22 0.603 0.572 1 0.0481
  0.02 20 0.466 0.732 26 0.623 0.552 1 0.0262
  0.03 20 0.438 0.695 30 0.643 0.532 0.999 0.0205
  0.04 20 0.416 0.662 36 0.667 0.513 0.995 0.0157
  0.05 20 0.397 0.633 42 0.680 0.494 0.986 0.0129
  0.06 20 0.382 0.607 52 0.691 0.478 0.969 0.0097
  0.07 20 0.369 0.585 62 0.696 0.464 0.943 0.0078
  0.08 20 0.358 0.566 76 0.705 0.450 0.906 0.0062
(0.2,0.8) 0.01 20 0.430 0.734 24 0.601 0.561 1 0.0428
  0.02 20 0.382 0.676 30 0.641 0.528 0.999 0.0259
  0.03 20 0.349 0.628 38 0.666 0.499 0.995 0.0176
  0.04 20 0.324 0.589 48 0.681 0.473 0.982 0.0127
  0.05 20 0.305 0.557 62 0.693 0.452 0.960 0.0092
  0.06 20 0.289 0.530 76 0.699 0.434 0.927 0.0073
  0.07 20 0.276 0.508 96 0.706 0.418 0.881 0.0057
  0.08 20 0.264 0.488 118 0.710 0.403 0.821 0.0046
(0.1,0.6) 0.01 28 0.511 0.770 32 0.604 0.483 1 0.0231
  0.02 28 0.490 0.737 36 0.631 0.474 1 0.0175
  0.03 28 0.477 0.708 40 0.651 0.465 0.998 0.0145
  0.04 28 0.467 0.685 46 0.667 0.457 0.991 0.0111
  0.05 28 0.461 0.667 54 0.685 0.452 0.978 0.0086
  0.06 28 0.458 0.653 62 0.691 0.448 0.959 0.0069
  0.07 28 0.457 0.643 70 0.698 0.446 0.934 0.0057
  0.08 28 0.458 0.636 78 0.704 0.446 0.903 0.0049
(0.2,0.7) 0.01 30 0.479 0.750 36 0.608 0.477 1 0.0214
  0.02 30 0.447 0.702 44 0.649 0.458 0.999 0.0144
  0.03 30 0.426 0.664 54 0.671 0.441 0.991 0.0102
  0.04 30 0.411 0.636 66 0.686 0.427 0.974 0.0076
  0.05 30 0.401 0.615 78 0.695 0.417 0.947 0.0061
  0.06 30 0.395 0.599 92 0.702 0.410 0.912 0.0050
  0.07 30 0.391 0.587 106 0.706 0.404 0.868 0.0042
  0.08 30 0.388 0.578 122 0.711 0.400 0.816 0.0035
(0.1,0.5) 0.01 40 0.490 0.751 48 0.614 0.392 1 0.0155
  0.02 40 0.493 0.725 54 0.643 0.392 0.999 0.0107
  0.03 40 0.498 0.708 60 0.666 0.395 0.993 0.0084
  0.04 40 0.504 0.697 66 0.678 0.400 0.981 0.0067
  0.05 40 0.514 0.692 72 0.689 0.407 0.965 0.0055
  0.06 40 0.525 0.692 76 0.695 0.417 0.945 0.0047
  0.07 40 0.538 0.694 80 0.701 0.429 0.923 0.0041
  0.08 40 0.552 0.699 80 0.705 0.442 0.899 0.0038
(0.2,0.6) 0.01 46 0.487 0.742 56 0.624 0.388 1 0.0137
  0.02 46 0.475 0.704 68 0.659 0.380 0.996 0.0083
  0.03 46 0.471 0.679 82 0.678 0.376 0.981 0.0057
  0.04 46 0.472 0.665 94 0.691 0.376 0.956 0.0046
  0.05 46 0.477 0.658 106 0.698 0.379 0.925 0.0037
  0.06 46 0.484 0.655 114 0.704 0.384 0.890 0.0032
  0.07 46 0.493 0.655 120 0.707 0.390 0.851 0.0029
  0.08 46 0.502 0.658 126 0.711 0.396 0.810 0.0026
(0.3,0.7) 0.01 48 0.465 0.724 62 0.630 0.379 1 0.0118
  0.02 48 0.440 0.673 82 0.670 0.363 0.990 0.0068
  0.03 48 0.429 0.643 104 0.689 0.354 0.963 0.0046
  0.04 48 0.426 0.626 124 0.698 0.349 0.924 0.0036
  0.05 48 0.425 0.616 144 0.705 0.347 0.879 0.0029
  0.06 48 0.427 0.610 158 0.708 0.347 0.829 0.0026
  0.07 48 0.431 0.607 172 0.711 0.348 0.776 0.0023
  0.08 48 0.435 0.605 182 0.713 0.349 0.718 0.0021
(0.1,0.4) 0.01 64 0.497 0.735 82 0.635 0.300 0.999 0.0077
  0.02 64 0.519 0.720 92 0.667 0.312 0.992 0.0053
  0.03 64 0.540 0.717 98 0.680 0.328 0.977 0.0041
  0.04 64 0.563 0.722 102 0.691 0.346 0.960 0.0034
  0.05 64 0.585 0.730 100 0.696 0.366 0.942 0.0031
  0.06 64 0.608 0.741 96 0.700 0.388 0.925 0.0029
  0.07 64 0.632 0.754 90 0.703 0.412 0.909 0.0027
  0.08 64 0.656 0.768 84 0.706 0.437 0.895 0.0025
(0.2,0.5) 0.01 78 0.496 0.726 104 0.644 0.297 0.998 0.0057
  0.02 78 0.505 0.701 126 0.677 0.302 0.982 0.0036
  0.03 78 0.521 0.695 140 0.691 0.313 0.954 0.0027
  0.04 78 0.540 0.698 148 0.699 0.327 0.922 0.0023
  0.05 78 0.559 0.705 148 0.704 0.342 0.890 0.0021
  0.06 78 0.579 0.715 144 0.707 0.358 0.860 0.0019
  0.07 78 0.599 0.726 138 0.709 0.375 0.831 0.0018
  0.08 78 0.620 0.738 130 0.711 0.393 0.804 0.0018
(0.3,0.6) 0.01 84 0.474 0.708 120 0.652 0.291 0.997 0.0049
  0.02 84 0.477 0.677 156 0.684 0.290 0.970 0.0029
  0.03 84 0.489 0.668 180 0.697 0.297 0.928 0.0022
  0.04 84 0.504 0.669 194 0.704 0.305 0.882 0.0018
  0.05 84 0.520 0.674 198 0.707 0.315 0.838 0.0016
  0.06 84 0.536 0.681 198 0.710 0.325 0.795 0.0015
  0.07 84 0.551 0.689 194 0.712 0.336 0.753 0.0015
  0.08 84 0.566 0.698 188 0.713 0.346 0.712 0.0014
(0.1,0.3) 0.01 124 0.525 0.718 182 0.671 0.214 0.987 0.0025
  0.02 124 0.574 0.730 186 0.688 0.242 0.960 0.0018
  0.03 124 0.614 0.749 174 0.695 0.271 0.937 0.0016
  0.04 124 0.650 0.768 156 0.699 0.301 0.920 0.0016
  0.05 124 0.681 0.787 138 0.702 0.332 0.908 0.0015
  0.06 124 0.710 0.805 120 0.704 0.363 0.899 0.0015
  0.07 124 0.738 0.822 102 0.704 0.397 0.894 0.0015
  0.08 124 0.765 0.840 86 0.705 0.433 0.891 0.0016
(0.2,0.4) 0.01 164 0.524 0.711 254 0.678 0.210 0.979 0.0017
  0.02 164 0.565 0.718 272 0.695 0.233 0.934 0.0012
  0.03 164 0.602 0.735 260 0.702 0.258 0.895 0.0010
  0.04 164 0.635 0.753 236 0.706 0.283 0.864 0.0010
  0.05 164 0.665 0.771 208 0.708 0.309 0.841 0.0010
  0.06 164 0.692 0.789 182 0.710 0.335 0.822 0.0010
  0.07 164 0.718 0.805 156 0.710 0.362 0.809 0.0010
  0.08 164 0.742 0.822 134 0.711 0.389 0.798 0.0010
(0.3,0.5) 0.01 186 0.509 0.697 314 0.683 0.205 0.971 0.0014
  0.02 186 0.545 0.701 348 0.699 0.224 0.909 0.0010
  0.03 186 0.580 0.717 336 0.705 0.244 0.857 0.0008
  0.04 186 0.610 0.734 310 0.708 0.265 0.815 0.0008
  0.05 186 0.638 0.751 280 0.711 0.285 0.780 0.0008
  0.06 186 0.663 0.767 250 0.711 0.304 0.751 0.0008
  0.07 186 0.686 0.782 222 0.713 0.324 0.726 0.0008
  0.08 186 0.707 0.796 194 0.714 0.343 0.704 0.0008
(0.4,0.6) 0.01 194 0.495 0.684 354 0.686 0.200 0.962 0.0012
  0.02 194 0.526 0.685 406 0.702 0.214 0.885 0.0008
  0.03 194 0.556 0.698 402 0.707 0.229 0.820 0.0007
  0.04 194 0.582 0.712 382 0.710 0.244 0.767 0.0007
  0.05 194 0.605 0.726 354 0.712 0.258 0.722 0.0007
  0.06 194 0.626 0.739 326 0.713 0.272 0.682 0.0007
  0.07 194 0.645 0.751 300 0.714 0.284 0.646 0.0007
  0.08 194 0.662 0.763 274 0.715 0.296 0.610 0.0007
(0.1,0.2) 0.01 398 0.626 0.752 562 0.701 0.145 0.897 0.0005
  0.02 398 0.703 0.799 402 0.705 0.190 0.870 0.0005
  0.03 398 0.751 0.830 302 0.706 0.230 0.861 0.0005
  0.04 398 0.786 0.853 234 0.706 0.268 0.861 0.0005
  0.05 398 0.814 0.872 184 0.707 0.305 0.864 0.0005
  0.06 398 0.837 0.888 146 0.707 0.343 0.870 0.0005
  0.07 398 0.859 0.903 116 0.707 0.384 0.878 0.0005
  0.08 398 0.879 0.917 88 0.706 0.429 0.887 0.0006
(0.2,0.3) 0.01 588 0.623 0.747 874 0.704 0.140 0.864 0.0003
  0.02 588 0.697 0.792 630 0.708 0.180 0.813 0.0003
  0.03 588 0.743 0.823 470 0.710 0.216 0.791 0.0003
  0.04 588 0.778 0.846 362 0.711 0.250 0.782 0.0003
  0.05 588 0.806 0.865 284 0.711 0.283 0.779 0.0003
  0.06 588 0.829 0.881 224 0.711 0.316 0.780 0.0003
  0.07 588 0.849 0.895 176 0.711 0.350 0.785 0.0003
  0.08 588 0.868 0.909 138 0.711 0.386 0.793 0.0004
(0.3,0.4) 0.01 712 0.614 0.739 1126 0.706 0.136 0.841 0.0002
  0.02 712 0.684 0.783 830 0.710 0.171 0.774 0.0002
  0.03 712 0.730 0.813 628 0.711 0.202 0.740 0.0002
  0.04 712 0.764 0.836 488 0.712 0.231 0.720 0.0002
  0.05 712 0.791 0.854 388 0.713 0.259 0.708 0.0002
  0.06 712 0.814 0.870 310 0.713 0.286 0.701 0.0003
  0.07 712 0.834 0.884 250 0.713 0.313 0.697 0.0003
  0.08 712 0.852 0.896 200 0.714 0.341 0.697 0.0003
(0.4,0.5) 0.01 776 0.605 0.732 1298 0.707 0.132 0.823 0.0002
  0.02 776 0.672 0.773 984 0.711 0.163 0.742 0.0002
  0.03 776 0.715 0.802 762 0.712 0.189 0.698 0.0002
  0.04 776 0.747 0.824 608 0.713 0.213 0.668 0.0002
  0.05 776 0.773 0.841 494 0.714 0.234 0.646 0.0002
  0.06 776 0.794 0.856 406 0.714 0.255 0.629 0.0002
  0.07 776 0.813 0.869 338 0.715 0.275 0.614 0.0002
  0.08 776 0.829 0.880 282 0.715 0.294 0.602 0.0002