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Explanations and References Studies to detect rare events are usually carried out after the introduction of a new drug or treatment into general use, to exclude the occurrence of unusual adverse events. These are sometimes called post marketing studies, or Phase IV Trials. The idea is to require all users to report side effects or complications, so that rare adverse outcomes are recorded and dealt with. The sample size represents a count of the number of negative cases that accompany a nominated number of positive cases, assuming the data has a negative binomial distribution, which is very similar to the Poisson distribution. It is calculated with the following parameters. Power (1-β), where β is the probability of Type II Error, so that power is the likelihood that the rare event will be detected if it exists. The table in the next panel provides sample size for powers of 0.8, 0.9, 0.95, and 0.99. Although power=0.08 is commonly used in clinical research, the importance of detecting unusual and unexpected adverse outcome is such that the power used is commonly set at 0.95, or even 0.99 for those adverse outcomes that are life threatening. The Mean Event Rate λ, (number of events / number of observations). This is the critical level of probability, above which is considered unacceptable and requiring remedy. The table in the next panel provides sample size for λ=0.0001 (1 in 10,000) to λ=0.01 (1%) with 0.0001 intervals. The Critical Event Limit r is the pre-determined number of events that, if occurred within the sample size observed, will led to the conclusion that λ is exceeded. Although r is usually set to 1, the table in the next panel provides for up to r=3. The algorithm calculates sample size required using iterative approximation until all the parameters are satisfied. Clinically, the sample size is usually rounded upwards to the next thousand. An example : Using the very first row of the exmple data : If we wish to detect an event as rare as 1 in 10,000 (λ=0.0001) and if the power of detection is set at 0.95 (95% sure we will detect it if it exists as frequently as 1 in 10,000) then we can conclude that the event is more frequent than 1 in 10,000 if we observe 1 incident in 30,000 cases (rounded up from 29958), or two incidents in 48,000 cases (rounded up from 47439), or 3 incidences in 63,000 cases (rounded up from 62960). The sample size table is constructed using the algorithm described in the reference Reference Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 144-145 Sample Size Table Power=0.8 Power=0.9 Power=0.95 Power=0.99 Power=0.8 Power=0.9 Power=0.95 Power=0.99 λ r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 λ r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 0.0001 16095 29943 42791 23026 38897 53225 29958 47439 62960 46052 66389 84062 0.0002 8048 14972 21396 11513 19450 26612 14979 23721 31479 23026 33187 42032 0.0003 5365 9982 14264 7676 12966 17741 9986 15813 20987 15351 22126 28022 0.0004 4024 7486 10698 5757 9725 13306 7490 11860 15740 11513 16595 21017 0.0005 3219 5989 8559 4606 7780 10645 5992 9489 12592 9211 13276 16814 0.0006 2683 4991 7132 3838 6483 8871 4993 7907 10493 7676 11064 14012 0.0007 2300 4278 6113 3290 5557 7604 4280 6778 8995 6579 9484 12007 0.0008 2012 3743 5349 2879 4863 6653 3745 5930 7871 5757 8299 10507 0.0009 1789 3328 4755 2559 4322 5914 3329 5271 6996 5117 7377 9340 0.0010 1610 2995 4280 2303 3890 5323 2996 4744 6296 4606 6639 8406 0.0011 1464 2723 3890 2094 3537 4839 2724 4313 5724 4187 6036 7642 0.0012 1342 2496 3566 1919 3242 4436 2497 3954 5247 3838 5533 7005 0.0013 1239 2304 3292 1772 2993 4094 2305 3650 4843 3543 5106 6467 0.0014 1150 2139 3057 1645 2779 3802 2140 3389 4498 3290 4742 6005 0.0015 1073 1997 2853 1536 2594 3549 1998 3163 4198 3071 4426 5605 0.0016 1006 1872 2675 1440 2432 3327 1873 2966 3935 2879 4149 5255 0.0017 947 1762 2518 1355 2289 3131 1763 2791 3704 2709 3906 4946 0.0018 895 1664 2378 1280 2161 2957 1665 2636 3498 2559 3689 4671 0.0019 848 1576 2253 1212 2048 2802 1577 2497 3314 2424 3495 4425 0.0020 805 1498 2140 1152 1945 2662 1498 2372 3148 2303 3319 4203 0.0021 767 1426 2038 1097 1853 2535 1427 2259 2999 2193 3161 4003 0.0022 732 1362 1946 1047 1769 2420 1362 2157 2862 2094 3018 3821 0.0023 700 1302 1861 1002 1692 2314 1303 2063 2738 2003 2887 3655 0.0024 671 1248 1783 960 1621 2218 1249 1977 2624 1919 2766 3503 0.0025 644 1198 1712 922 1556 2129 1199 1898 2519 1843 2656 3363 0.0026 620 1152 1646 886 1497 2048 1153 1825 2422 1772 2554 3234 0.0027 597 1110 1585 853 1441 1972 1110 1757 2332 1706 2459 3114 0.0028 575 1070 1529 823 1390 1901 1070 1695 2249 1645 2372 3003 0.0029 555 1033 1476 794 1342 1836 1034 1636 2172 1588 2290 2899 0.0030 537 999 1427 768 1297 1775 999 1582 2099 1536 2214 2803 0.0031 520 966 1381 743 1255 1717 967 1531 2031 1486 2142 2712 0.0032 503 936 1338 720 1216 1664 937 1483 1968 1440 2075 2628 0.0033 488 908 1297 698 1179 1613 908 1438 1908 1396 2012 2548 0.0034 474 881 1259 678 1145 1566 882 1396 1852 1355 1953 2473 0.0035 460 856 1223 658 1112 1521 856 1356 1799 1316 1897 2402 0.0036 448 832 1189 640 1081 1479 833 1318 1749 1280 1845 2335 0.0037 435 810 1157 623 1052 1439 810 1283 1702 1245 1794 2272 0.0038 424 788 1127 606 1024 1401 789 1249 1657 1212 1747 2213 0.0039 413 768 1098 591 998 1365 769 1217 1615 1181 1702 2156 0.0040 403 749 1070 576 973 1331 749 1186 1574 1152 1660 2102 Power=0.8 Power=0.9 Power=0.95 Power=0.99 Power=0.8 Power=0.9 Power=0.95 Power=0.99 λ r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 λ r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 0.0041 393 731 1044 562 949 1299 731 1158 1536 1124 1620 2051 0.0042 384 713 1019 549 927 1268 714 1130 1500 1097 1581 2002 0.0043 375 697 996 536 905 1238 697 1104 1465 1071 1544 1956 0.0044 366 681 973 524 885 1210 681 1079 1431 1047 1509 1911 0.0045 358 666 951 512 865 1183 666 1055 1400 1024 1476 1869 0.0046 350 651 931 501 846 1158 652 1032 1369 1002 1444 1828 0.0047 343 638 911 490 828 1133 638 1010 1340 980 1413 1789 0.0048 336 624 892 480 811 1109 625 989 1312 960 1384 1752 0.0049 329 612 874 470 794 1087 612 969 1285 940 1355 1716 0.0050 322 599 856 461 778 1065 600 949 1260 922 1328 1682 0.0051 316 588 840 452 763 1044 588 931 1235 903 1302 1649 0.0052 310 576 823 443 749 1024 577 913 1211 886 1277 1617 0.0053 304 565 808 435 734 1005 566 896 1188 869 1253 1587 0.0054 299 555 793 427 721 986 555 879 1166 853 1230 1557 0.0055 293 545 779 419 708 968 545 863 1145 838 1207 1529 0.0056 288 535 765 412 695 951 535 848 1125 823 1186 1502 0.0057 283 526 751 404 683 934 526 833 1105 808 1165 1475 0.0058 278 517 738 397 671 918 517 818 1086 794 1145 1450 0.0059 273 508 726 391 660 903 508 805 1068 781 1126 1425 0.0060 269 500 714 384 649 888 500 791 1050 768 1107 1401 0.0061 264 491 702 378 638 873 492 778 1033 755 1089 1378 0.0062 260 483 691 372 628 859 484 766 1016 743 1071 1356 0.0063 256 476 680 366 618 845 476 754 1000 731 1054 1335 0.0064 252 468 669 360 608 832 469 742 984 720 1038 1314 0.0065 248 461 659 355 599 819 461 730 969 709 1022 1294 0.0066 244 454 649 349 590 807 454 719 954 698 1006 1274 0.0067 241 447 639 344 581 795 448 709 940 688 991 1255 0.0068 237 441 630 339 573 783 441 698 926 678 977 1237 0.0069 234 434 621 334 564 772 435 688 913 668 963 1219 0.0070 230 428 612 329 556 761 428 678 900 658 949 1201 0.0071 227 422 603 325 548 750 422 669 887 649 936 1185 0.0072 224 416 595 320 541 740 417 659 875 640 923 1168 0.0073 221 411 587 316 533 730 411 650 863 631 910 1152 0.0074 218 405 579 312 526 720 405 642 851 623 897 1136 0.0075 215 400 571 308 519 710 400 633 840 615 886 1121 0.0076 212 394 564 303 512 701 395 625 829 606 874 1106 0.0077 210 389 556 300 506 692 390 617 818 599 863 1092 0.0078 207 384 549 296 499 683 385 609 808 591 852 1078 0.0079 204 380 542 292 493 674 380 601 797 583 841 1065 0.0080 202 375 535 288 487 666 375 593 787 576 830 1051 Power=0.8 Power=0.9 Power=0.95 Power=0.99 Power=0.8 Power=0.9 Power=0.95 Power=0.99 λ r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 λ r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3 0.0081 199 370 529 285 481 658 370 586 778 569 820 1038 0.0082 197 366 522 281 475 650 366 579 768 562 810 1026 0.0083 194 361 516 278 469 642 361 572 759 555 800 1013 0.0084 192 357 510 275 464 634 357 565 750 549 791 1001 0.0085 190 353 504 271 458 627 353 559 741 542 781 990 0.0086 188 349 498 268 453 619 349 552 733 536 772 978 0.0087 185 345 492 265 448 612 345 546 724 530 764 967 0.0088 183 341 487 262 443 605 341 540 716 524 755 956 0.0089 181 337 481 259 438 599 337 533 708 518 746 945 0.0090 179 333 476 256 433 592 333 528 700 512 738 934 0.0091 177 330 471 254 428 585 330 522 692 507 730 924 0.0092 175 326 466 251 423 579 326 516 685 501 722 914 0.0093 174 322 461 248 419 573 323 511 677 496 714 904 0.0094 172 319 456 245 414 567 319 505 670 490 707 895 0.0095 170 316 451 243 410 561 316 500 663 485 699 885 0.0096 168 312 446 240 406 555 313 495 656 480 692 876 0.0097 166 309 442 238 402 549 309 490 650 475 685 867 0.0098 165 306 437 235 397 544 306 485 643 470 678 858 0.0099 163 303 433 233 393 538 303 480 636 466 671 850 0.0100 161 300 428 231 389 533 300 475 630 461 664 841 Javascript Program Data Entry   - The data is in 3 columns separated by white spaces   - Each row is a sepaqrate test     - Column 1 is Power (1-β)     - Column 2 is the Expected Event Rate, the probability or proportion to detect     - Column 3 is the Critical tolerance limit, the number of positives within the sample size        for a decision that the Expected Event Rate is Exceeded R Codes R codess for sample size to detect rare events fuctions for calculation FindB <- function(lambda,n,a) # Poisson coefficient { b = 0 for(y in 1:a) { x = y - 1 b = b + (n * lambda)^x * exp(-n * lambda) / factorial(x) } return (b) } # interative approximation for the correct sample size nm SSizAInc <- function(lambda,beta,r) { if(r==1) { return (-log(beta) / lambda) } nL = 2 nR = r * -log(beta) / lambda; nM = (nL + nR) / 2 bL = FindB(lambda, nL, r) - beta bR = FindB(lambda, nR, r) - beta bM = FindB(lambda, nM, r) - beta if(bL*bR>0) { return (0) } if(bL*bM>0) { nL = nM } else { nR = nM } j=0; while(j<100 & abs(bM)>0.00001) { nM = (nL + nR) / 2 bL = FindB(lambda,nL,r) - beta bM = FindB(lambda,nM,r) - beta if(bL*bM>0) { nL = nM } else { nR = nM } j = j + 1 } return (nM) } Main program # Data entry as a table myDat = (" pow lambda r # power, expected proportion, critical tolerance limit 0.90 0.001 1 0.90 0.001 2 0.99 0.002 1 0.99 0.002 3 ") df <- read.table(textConnection(myDat),header=TRUE) # conversion to data frame df # display input data # Calculations ssiz <- vector() # vector (array) to contain sample sizes for(i in 1:nrow(df)) { beta = 1 - df$pow[i] lambda = df$lambda[i] r = df$r[i] ssiz <- append(ssiz,ceiling(SSizAInc(lambda,beta,r))) } df$SSiz <- ssiz # add sample size column to data frame df # display input data + sample size calculated The results are > df # display input data pow lambda r 1 0.90 0.001 1 2 0.90 0.001 2 3 0.99 0.002 1 4 0.99 0.002 3 > df # display input data + sample size calculated pow lambda r SSiz 1 0.90 0.001 1 2303 2 0.90 0.001 2 3890 3 0.99 0.002 1 2303 4 0.99 0.002 3 4204