Stat 149_Exercise 2
Leonessa T. Adobas
2023-11-10
DATA: German Breast Cancer Study
GBCD = data.frame(read.csv("gbcs_short.csv"))
head(GBCD)## id meno horm size grade nodes rectime censrec prog estr
## 1 1 1 1 18 3 5 1337 1 1 1
## 2 2 1 1 20 1 1 1420 1 1 0
## 3 3 1 1 30 2 1 1279 1 1 1
## 4 4 1 1 24 1 3 148 0 1 0
## 5 5 2 2 19 2 1 1863 0 0 0
## 6 6 2 2 56 1 3 1933 0 1 1str(GBCD)## 'data.frame': 686 obs. of 10 variables:
## $ id : int 1 2 3 4 5 6 7 8 9 10 ...
## $ meno : int 1 1 1 1 2 2 2 2 1 2 ...
## $ horm : int 1 1 1 1 2 2 1 2 1 2 ...
## $ size : int 18 20 30 24 19 56 52 22 30 20 ...
## $ grade : int 3 1 2 1 2 1 2 2 2 2 ...
## $ nodes : int 5 1 1 3 1 3 9 2 1 1 ...
## $ rectime: int 1337 1420 1279 148 1863 1933 358 2372 2563 2372 ...
## $ censrec: int 1 1 1 0 0 0 1 1 0 0 ...
## $ prog : int 1 1 1 1 0 1 0 0 1 1 ...
## $ estr : int 1 0 1 0 0 1 1 1 0 1 ...1.Using the Kaplan-Meier method, solve for the survival probabilities of patients regardless of the treatment assignment.
library(survival)
library(survminer)
#For Conversion:
with(GBCD, Surv(rectime, censrec))## [1] 1337 1420 1279 148+ 1863+ 1933+ 358 2372 2563+ 2372+ 1989 579
## [13] 1043 2234+ 2297+ 2014+ 518 1763 889 357 547 1722+ 2372+ 251
## [25] 1959+ 1897+ 348 275 1329 1193 698 436 552 2551+ 754 819
## [37] 1280 663+ 722 322+ 1150 446 1855+ 238 1838+ 1826+ 2353+ 2471+
## [49] 893 2093 740+ 632 1866+ 491 1918 72 2556+ 1753 417 956
## [61] 1846+ 2449+ 2286 456 536 612 2034 1990 1976+ 2539+ 2467+ 876
## [73] 2132+ 426 537 2217+ 296+ 2320+ 795 867 755 1388 1387 535
## [85] 1653+ 1904+ 1868+ 1767+ 855 1157 2380+ 1679 498 2138+ 2175+ 563
## [97] 46+ 2144+ 344 945+ 1905+ 2659+ 1977 2401+ 1499+ 1856+ 595 2148+
## [109] 2126+ 1975 2438+ 2233+ 286 2170+ 729 1449 991 481 1814 2018
## [121] 712 1869+ 307 983 120 1525 1680+ 1730 805 2388+ 559 1977+
## [133] 308 1965+ 548 293 2017+ 1152+ 1401+ 2128+ 1965+ 2161+ 1183 1108
## [145] 2065+ 918+ 745 1283+ 1527+ 285 1306 797 1441+ 2612+ 956 1637+
## [157] 2456+ 1641 1717+ 1858+ 2049+ 2388+ 2296+ 1884+ 1059 564 2239+ 2237+
## [169] 476 1514+ 1617+ 1094 784 1760+ 1013+ 779+ 1807 772 2456 2205+
## [181] 544 336 2057+ 575 2011+ 2010+ 2009+ 1984+ 2030 1002 1280 338
## [193] 533 1182+ 71+ 114+ 63+ 1655+ 857 369 1627+ 855 2370+ 853+
## [205] 692+ 475 1861+ 1080 1521 1693+ 859+ 1109+ 1192 1806 500 1814
## [217] 890 1114+ 2271+ 17+ 964 540 747 650 168+ 1169+ 1675 1862+
## [229] 629+ 181 343 936+ 195+ 503 827 1427+ 1722+ 1692+ 177+ 679
## [241] 1164 350 578 1598+ 491 1366 1225 338 2227+ 1601 1841+ 2177+
## [253] 2052+ 1140 799 1105 548 227 448 2172+ 2161+ 471 2014+ 554
## [265] 1246 1926+ 1207 1852+ 1459 2237+ 933+ 1481 1807+ 542 1441+ 1277+
## [277] 1486+ 1502 1922+ 2048+ 600 1765+ 491 305 2195+ 758+ 648 761+
## [289] 596+ 195 473 1528 169 272 731 2059+ 415 1120 316 637
## [301] 186+ 769 727 1701 2015 1956+ 945 2153+ 838 113 1833+ 2056+
## [313] 1729+ 2024+ 2039 2027+ 2007+ 973+ 2156+ 1499+ 424+ 859 180 1625+
## [325] 1938+ 1343 646 2192+ 502 1675+ 42+ 410 624 1560+ 455 1629+
## [337] 529 1820+ 1756+ 375 1323+ 1233+ 986+ 650+ 577 184 1840+ 273+
## [349] 177 545 1185+ 631+ 1853+ 1854+ 1645+ 544 1666+ 353 1167+ 495
## [361] 967+ 1720+ 598 392 1502+ 229+ 1838+ 1833+ 550 426 1834+ 2030+
## [373] 573 1666+ 1979+ 1703+ 670 1791+ 1685+ 191 370 173 242 420
## [385] 247 888+ 622 806+ 1163+ 1721+ 741+ 160 970+ 892+ 753+ 1703+
## [397] 1720+ 1355+ 1603+ 476 1350+ 1341+ 1093+ 792+ 586 1434+ 67+ 623+
## [409] 1582+ 1771+ 960 571 747+ 1578+ 732 460 1208+ 730 722+ 717+
## [421] 1093 2051+ 370 861 1587 1589 1463 1826+ 1231+ 1117+ 836 1222+
## [433] 1981+ 624 742 1818+ 1493 1432+ 801 1786+ 1847+ 2009+ 1926+ 1818+
## [445] 1100+ 1499+ 1253 1789+ 1707+ 1714+ 1717+ 329 854+ 177 281 205
## [457] 751+ 629 1722+ 241 1352 1702+ 1222+ 1089+ 1243+ 995+ 1088+ 1842+
## [469] 1821+ 1371 707 1743+ 1781+ 865 57+ 1152+ 1460 1434+ 1604+ 772+
## [481] 1146 371 883 1735+ 554 1174 1250+ 530 1502+ 1364+ 1170 1729+
## [493] 1642+ 877+ 515 272 891 1356+ 1352+ 1077+ 675 359 1645+ 1356+
## [505] 1632+ 967+ 1091+ 918 557 1219 438 1624+ 1036 359 171 1490+
## [517] 233 1240+ 1751+ 1878+ 1684 1701+ 1701+ 1693+ 379 1105 548 1296
## [529] 1483+ 675+ 285 1472+ 1363 420 982 1459+ 1192+ 1264+ 1095+ 1078+
## [541] 1624+ 1600+ 385 1475+ 1171+ 1751+ 1756+ 798+ 940+ 766+ 790 1340+
## [553] 490 1557+ 981 1094+ 1730+ 1483+ 714 372 1331+ 394 652+ 657+
## [565] 567+ 429+ 566+ 15+ 310+ 1296+ 488+ 1505+ 776 1570+ 1469+ 1472+
## [577] 1342+ 1349+ 1162 1342+ 797 1232+ 959 1351+ 486 733+ 526+ 463+
## [589] 529+ 623+ 546+ 213+ 276+ 942+ 570+ 1177+ 1113+ 288 723+ 403
## [601] 1443+ 1317+ 1218 1358+ 360 550 1435+ 541+ 1329+ 1357+ 552 870+
## [613] 859 734+ 687 308 98 368+ 432+ 319+ 65+ 1343+ 748 737+
## [625] 461+ 465 842 918+ 374 1089+ 223 1212+ 1119+ 974+ 1230+ 1205+
## [637] 1090+ 1095+ 449 972+ 825+ 525 762 175 855+ 740+ 1090 1219+
## [649] 1195+ 338 628+ 1125+ 916+ 972+ 553+ 662 594 828+ 651+ 637+
## [661] 615+ 740+ 1062+ 8+ 936+ 867+ 249 281 758+ 377 969+ 974+
## [673] 866 504 721+ 594 841+ 695+ 16+ 29+ 18+ 17+ 857+ 768+
## [685] 858+ 770+#converting to survival time
#Estimation of Survival Function Method Used: Kaplan-Meier Method
model <- survfit(with(GBCD, Surv(rectime, censrec)) ~ 1)
summary(model)## Call: survfit(formula = with(GBCD, Surv(rectime, censrec)) ~ 1)
##
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 72 672 1 0.999 0.00149 0.996 1.000
## 98 671 1 0.997 0.00210 0.993 1.000
## 113 670 1 0.996 0.00257 0.991 1.000
## 120 668 1 0.994 0.00297 0.988 1.000
## 160 666 1 0.993 0.00332 0.986 0.999
## 169 664 1 0.991 0.00363 0.984 0.998
## 171 663 1 0.990 0.00392 0.982 0.997
## 173 662 1 0.988 0.00419 0.980 0.996
## 175 661 1 0.987 0.00445 0.978 0.995
## 177 660 2 0.984 0.00491 0.974 0.993
## 180 657 1 0.982 0.00512 0.972 0.992
## 181 656 1 0.981 0.00533 0.970 0.991
## 184 655 1 0.979 0.00553 0.968 0.990
## 191 653 1 0.978 0.00572 0.966 0.989
## 195 652 1 0.976 0.00590 0.965 0.988
## 205 650 1 0.975 0.00608 0.963 0.987
## 223 648 1 0.973 0.00626 0.961 0.985
## 227 647 1 0.972 0.00643 0.959 0.984
## 233 645 1 0.970 0.00659 0.957 0.983
## 238 644 1 0.969 0.00675 0.955 0.982
## 241 643 1 0.967 0.00691 0.954 0.981
## 242 642 1 0.966 0.00706 0.952 0.979
## 247 641 1 0.964 0.00720 0.950 0.978
## 249 640 1 0.963 0.00735 0.948 0.977
## 251 639 1 0.961 0.00749 0.946 0.976
## 272 638 2 0.958 0.00776 0.943 0.973
## 275 635 1 0.957 0.00790 0.941 0.972
## 281 633 2 0.953 0.00816 0.938 0.970
## 285 631 2 0.950 0.00841 0.934 0.967
## 286 629 1 0.949 0.00853 0.932 0.966
## 288 628 1 0.947 0.00865 0.931 0.965
## 293 627 1 0.946 0.00876 0.929 0.963
## 305 625 1 0.944 0.00888 0.927 0.962
## 307 624 1 0.943 0.00899 0.925 0.961
## 308 623 2 0.940 0.00922 0.922 0.958
## 316 620 1 0.938 0.00932 0.920 0.957
## 329 617 1 0.937 0.00943 0.919 0.956
## 336 616 1 0.935 0.00954 0.917 0.954
## 338 615 3 0.931 0.00985 0.912 0.950
## 343 612 1 0.929 0.00995 0.910 0.949
## 344 611 1 0.928 0.01005 0.908 0.948
## 348 610 1 0.926 0.01015 0.907 0.946
## 350 609 1 0.925 0.01024 0.905 0.945
## 353 608 1 0.923 0.01034 0.903 0.944
## 357 607 1 0.922 0.01043 0.901 0.942
## 358 606 1 0.920 0.01053 0.900 0.941
## 359 605 2 0.917 0.01071 0.896 0.938
## 360 603 1 0.916 0.01080 0.895 0.937
## 369 601 1 0.914 0.01089 0.893 0.936
## 370 600 2 0.911 0.01106 0.890 0.933
## 371 598 1 0.909 0.01115 0.888 0.932
## 372 597 1 0.908 0.01123 0.886 0.930
## 374 596 1 0.906 0.01132 0.885 0.929
## 375 595 1 0.905 0.01140 0.883 0.928
## 377 594 1 0.903 0.01148 0.881 0.926
## 379 593 1 0.902 0.01156 0.879 0.925
## 385 592 1 0.900 0.01164 0.878 0.923
## 392 591 1 0.899 0.01172 0.876 0.922
## 394 590 1 0.897 0.01180 0.874 0.921
## 403 589 1 0.896 0.01188 0.873 0.919
## 410 588 1 0.894 0.01196 0.871 0.918
## 415 587 1 0.893 0.01203 0.869 0.917
## 417 586 1 0.891 0.01211 0.868 0.915
## 420 585 2 0.888 0.01226 0.864 0.912
## 426 582 2 0.885 0.01240 0.861 0.910
## 436 578 1 0.884 0.01248 0.859 0.908
## 438 577 1 0.882 0.01255 0.858 0.907
## 446 576 1 0.880 0.01262 0.856 0.906
## 448 575 1 0.879 0.01269 0.854 0.904
## 449 574 1 0.877 0.01276 0.853 0.903
## 455 573 1 0.876 0.01283 0.851 0.901
## 456 572 1 0.874 0.01290 0.849 0.900
## 460 571 1 0.873 0.01297 0.848 0.899
## 465 568 1 0.871 0.01303 0.846 0.897
## 471 567 1 0.870 0.01310 0.844 0.896
## 473 566 1 0.868 0.01317 0.843 0.894
## 475 565 1 0.867 0.01323 0.841 0.893
## 476 564 2 0.864 0.01337 0.838 0.890
## 481 562 1 0.862 0.01343 0.836 0.889
## 486 561 1 0.861 0.01349 0.834 0.887
## 490 559 1 0.859 0.01356 0.833 0.886
## 491 558 3 0.854 0.01374 0.828 0.882
## 495 555 1 0.853 0.01380 0.826 0.880
## 498 554 1 0.851 0.01387 0.825 0.879
## 500 553 1 0.850 0.01393 0.823 0.878
## 502 552 1 0.848 0.01398 0.821 0.876
## 503 551 1 0.847 0.01404 0.820 0.875
## 504 550 1 0.845 0.01410 0.818 0.873
## 515 549 1 0.844 0.01416 0.816 0.872
## 518 548 1 0.842 0.01422 0.815 0.870
## 525 547 1 0.841 0.01428 0.813 0.869
## 529 545 1 0.839 0.01433 0.811 0.868
## 530 543 1 0.837 0.01439 0.810 0.866
## 533 542 1 0.836 0.01445 0.808 0.865
## 535 541 1 0.834 0.01450 0.806 0.863
## 536 540 1 0.833 0.01456 0.805 0.862
## 537 539 1 0.831 0.01461 0.803 0.860
## 540 538 1 0.830 0.01467 0.801 0.859
## 542 536 1 0.828 0.01472 0.800 0.858
## 544 535 2 0.825 0.01483 0.797 0.855
## 545 533 1 0.824 0.01488 0.795 0.853
## 547 531 1 0.822 0.01493 0.793 0.852
## 548 530 3 0.817 0.01509 0.788 0.847
## 550 527 2 0.814 0.01519 0.785 0.845
## 552 525 2 0.811 0.01529 0.782 0.842
## 554 522 2 0.808 0.01539 0.778 0.839
## 557 520 1 0.806 0.01543 0.777 0.837
## 559 519 1 0.805 0.01548 0.775 0.836
## 563 518 1 0.803 0.01553 0.773 0.834
## 564 517 1 0.802 0.01558 0.772 0.833
## 571 513 1 0.800 0.01563 0.770 0.831
## 573 512 1 0.799 0.01567 0.769 0.830
## 575 511 1 0.797 0.01572 0.767 0.829
## 577 510 1 0.796 0.01577 0.765 0.827
## 578 509 1 0.794 0.01581 0.764 0.826
## 579 508 1 0.792 0.01586 0.762 0.824
## 586 507 1 0.791 0.01591 0.760 0.823
## 594 506 2 0.788 0.01600 0.757 0.820
## 595 504 1 0.786 0.01604 0.755 0.818
## 598 502 1 0.785 0.01608 0.754 0.817
## 600 501 1 0.783 0.01613 0.752 0.815
## 612 500 1 0.781 0.01617 0.750 0.814
## 622 498 1 0.780 0.01622 0.749 0.812
## 624 495 2 0.777 0.01630 0.745 0.809
## 629 492 1 0.775 0.01635 0.744 0.808
## 632 489 1 0.774 0.01639 0.742 0.806
## 637 488 1 0.772 0.01643 0.740 0.805
## 646 486 1 0.770 0.01647 0.739 0.803
## 648 485 1 0.769 0.01652 0.737 0.802
## 650 484 1 0.767 0.01656 0.735 0.800
## 662 479 1 0.766 0.01660 0.734 0.799
## 670 477 1 0.764 0.01664 0.732 0.797
## 675 476 1 0.762 0.01669 0.730 0.796
## 679 474 1 0.761 0.01673 0.729 0.794
## 687 473 1 0.759 0.01677 0.727 0.793
## 698 470 1 0.758 0.01681 0.725 0.791
## 707 469 1 0.756 0.01685 0.724 0.790
## 712 468 1 0.754 0.01689 0.722 0.788
## 714 467 1 0.753 0.01694 0.720 0.787
## 722 464 1 0.751 0.01698 0.719 0.785
## 727 461 1 0.749 0.01702 0.717 0.784
## 729 460 1 0.748 0.01706 0.715 0.782
## 730 459 1 0.746 0.01710 0.713 0.781
## 731 458 1 0.745 0.01714 0.712 0.779
## 732 457 1 0.743 0.01718 0.710 0.777
## 742 449 1 0.741 0.01722 0.708 0.776
## 745 448 1 0.740 0.01726 0.707 0.774
## 747 447 1 0.738 0.01730 0.705 0.773
## 748 445 1 0.736 0.01734 0.703 0.771
## 754 442 1 0.735 0.01738 0.701 0.770
## 755 441 1 0.733 0.01742 0.700 0.768
## 762 437 1 0.731 0.01746 0.698 0.766
## 769 434 1 0.730 0.01750 0.696 0.765
## 772 432 1 0.728 0.01755 0.694 0.763
## 776 430 1 0.726 0.01759 0.693 0.762
## 784 428 1 0.725 0.01763 0.691 0.760
## 790 427 1 0.723 0.01767 0.689 0.758
## 795 425 1 0.721 0.01771 0.687 0.757
## 797 424 2 0.718 0.01779 0.684 0.753
## 799 421 1 0.716 0.01783 0.682 0.752
## 801 420 1 0.714 0.01786 0.680 0.750
## 805 419 1 0.713 0.01790 0.678 0.749
## 819 417 1 0.711 0.01794 0.677 0.747
## 827 415 1 0.709 0.01798 0.675 0.745
## 836 413 1 0.708 0.01802 0.673 0.744
## 838 412 1 0.706 0.01806 0.671 0.742
## 842 410 1 0.704 0.01809 0.669 0.740
## 855 407 2 0.701 0.01817 0.666 0.737
## 857 404 1 0.699 0.01821 0.664 0.736
## 859 401 2 0.695 0.01828 0.660 0.732
## 861 398 1 0.694 0.01832 0.659 0.731
## 865 397 1 0.692 0.01836 0.657 0.729
## 866 396 1 0.690 0.01839 0.655 0.727
## 867 395 1 0.688 0.01843 0.653 0.726
## 876 392 1 0.687 0.01847 0.651 0.724
## 883 390 1 0.685 0.01850 0.650 0.722
## 889 388 1 0.683 0.01854 0.648 0.720
## 890 387 1 0.681 0.01857 0.646 0.719
## 891 386 1 0.680 0.01861 0.644 0.717
## 893 384 1 0.678 0.01865 0.642 0.715
## 918 382 1 0.676 0.01868 0.640 0.714
## 945 374 1 0.674 0.01872 0.639 0.712
## 956 372 2 0.671 0.01879 0.635 0.708
## 959 370 1 0.669 0.01883 0.633 0.707
## 960 369 1 0.667 0.01886 0.631 0.705
## 964 368 1 0.665 0.01890 0.629 0.703
## 981 358 1 0.663 0.01894 0.627 0.702
## 982 357 1 0.661 0.01898 0.625 0.700
## 983 356 1 0.660 0.01901 0.623 0.698
## 991 354 1 0.658 0.01905 0.621 0.696
## 1002 352 1 0.656 0.01909 0.620 0.694
## 1036 350 1 0.654 0.01913 0.618 0.693
## 1043 349 1 0.652 0.01916 0.616 0.691
## 1059 348 1 0.650 0.01920 0.614 0.689
## 1080 344 1 0.648 0.01924 0.612 0.687
## 1090 340 1 0.646 0.01927 0.610 0.685
## 1093 337 1 0.645 0.01931 0.608 0.684
## 1094 335 1 0.643 0.01935 0.606 0.682
## 1105 330 2 0.639 0.01943 0.602 0.678
## 1108 328 1 0.637 0.01947 0.600 0.676
## 1120 322 1 0.635 0.01951 0.598 0.674
## 1140 320 1 0.633 0.01955 0.596 0.672
## 1146 319 1 0.631 0.01958 0.594 0.670
## 1150 318 1 0.629 0.01962 0.592 0.669
## 1157 315 1 0.627 0.01966 0.589 0.667
## 1162 314 1 0.625 0.01970 0.587 0.665
## 1164 312 1 0.623 0.01974 0.585 0.663
## 1170 309 1 0.621 0.01978 0.583 0.661
## 1174 307 1 0.619 0.01982 0.581 0.659
## 1183 304 1 0.617 0.01986 0.579 0.657
## 1192 302 1 0.615 0.01989 0.577 0.655
## 1193 300 1 0.613 0.01993 0.575 0.653
## 1207 297 1 0.611 0.01997 0.573 0.651
## 1218 294 1 0.609 0.02001 0.571 0.649
## 1219 293 1 0.606 0.02005 0.568 0.647
## 1225 289 1 0.604 0.02009 0.566 0.645
## 1246 282 1 0.602 0.02013 0.564 0.643
## 1253 280 1 0.600 0.02018 0.562 0.641
## 1279 277 1 0.598 0.02022 0.560 0.639
## 1280 276 2 0.594 0.02030 0.555 0.635
## 1296 273 1 0.591 0.02035 0.553 0.633
## 1306 271 1 0.589 0.02039 0.551 0.631
## 1329 268 1 0.587 0.02043 0.548 0.628
## 1337 265 1 0.585 0.02047 0.546 0.626
## 1343 260 1 0.583 0.02052 0.544 0.624
## 1352 255 1 0.580 0.02056 0.541 0.622
## 1363 248 1 0.578 0.02061 0.539 0.620
## 1366 246 1 0.576 0.02066 0.536 0.618
## 1371 245 1 0.573 0.02071 0.534 0.615
## 1387 244 1 0.571 0.02076 0.532 0.613
## 1388 243 1 0.569 0.02081 0.529 0.611
## 1420 241 1 0.566 0.02085 0.527 0.609
## 1449 232 1 0.564 0.02091 0.524 0.606
## 1459 231 1 0.561 0.02096 0.522 0.604
## 1460 229 1 0.559 0.02101 0.519 0.602
## 1463 228 1 0.556 0.02106 0.517 0.599
## 1481 223 1 0.554 0.02111 0.514 0.597
## 1493 218 1 0.551 0.02117 0.511 0.594
## 1502 214 1 0.549 0.02122 0.509 0.592
## 1521 209 1 0.546 0.02128 0.506 0.590
## 1525 208 1 0.544 0.02134 0.503 0.587
## 1528 206 1 0.541 0.02140 0.501 0.585
## 1587 200 1 0.538 0.02147 0.498 0.582
## 1589 199 1 0.535 0.02153 0.495 0.579
## 1601 196 1 0.533 0.02159 0.492 0.577
## 1641 185 1 0.530 0.02167 0.489 0.574
## 1675 177 1 0.527 0.02175 0.486 0.571
## 1679 175 1 0.524 0.02183 0.483 0.568
## 1684 173 1 0.521 0.02191 0.480 0.566
## 1701 168 1 0.518 0.02200 0.476 0.563
## 1730 150 1 0.514 0.02212 0.473 0.560
## 1753 144 1 0.511 0.02226 0.469 0.556
## 1763 140 1 0.507 0.02240 0.465 0.553
## 1806 132 1 0.503 0.02255 0.461 0.549
## 1807 131 1 0.499 0.02271 0.457 0.546
## 1814 129 2 0.492 0.02300 0.449 0.539
## 1918 94 1 0.486 0.02335 0.443 0.534
## 1975 84 1 0.481 0.02378 0.436 0.530
## 1977 82 1 0.475 0.02420 0.430 0.525
## 1989 77 1 0.469 0.02466 0.423 0.520
## 1990 76 1 0.462 0.02509 0.416 0.514
## 2015 68 1 0.456 0.02563 0.408 0.509
## 2018 66 1 0.449 0.02615 0.400 0.503
## 2030 63 1 0.442 0.02669 0.392 0.497
## 2034 61 1 0.434 0.02722 0.384 0.491
## 2039 60 1 0.427 0.02771 0.376 0.485
## 2093 51 1 0.419 0.02840 0.367 0.478
## 2286 25 1 0.402 0.03182 0.344 0.469
## 2372 19 1 0.381 0.03651 0.316 0.460
## 2456 10 1 0.343 0.04884 0.259 0.453#Plot model
plot(model)
abline(c(0.5,0), lty = 3, col = "red")
title("Survival Probabilities of patient: Kaplan-mier method")
The above graph indicates that there is an estimated 99.7% probability of disease recurrence for longer than 100 days, wile 85% for longer than 500%, 65.8% for longer than 1000 days, 42.6% for longer than 2,000 days, while there is about 34.3% probability for longer than 2456 days. In addition, there is about 1087 days for the median time to event of breast cancer patients.
2. For each of the comparison below, identify the most appropriate test and do the analysis.
Tests:
Two-group Logrank Tests
Two-group Wilcoxon-Gehan Tests
Multi-group Logrank Tests
Logrank Tests for Ordered Groups
Stratified Logrank Tests
a. survival times between menopause status
#LogRank Test
M <- survfit(Surv(rectime, censrec) ~ meno, data = GBCD)
M## Call: survfit(formula = Surv(rectime, censrec) ~ meno, data = GBCD)
##
## n events median 0.95LCL 0.95UCL
## meno=1 290 119 2015 1587 NA
## meno=2 396 180 1701 1481 1990plot(survfit(Surv(rectime,censrec) ~ meno,data = GBCD),
col = c("blue", "black"),
lwd = c(2,2),
mark.time = FALSE)
legend("bottomleft",
legend = c("1 - No", "2 - Yes"),
col = c("blue", "black"),
lwd = c(2,2),
bty = "n")
abline(c(0.5,0), lty = 3, col = "red")
title("survival times between menopause status")
#test
survdiff(Surv(rectime,censrec) ~ meno,data = GBCD )## Call:
## survdiff(formula = Surv(rectime, censrec) ~ meno, data = GBCD)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## meno=1 290 119 124 0.164 0.28
## meno=2 396 180 175 0.115 0.28
##
## Chisq= 0.3 on 1 degrees of freedom, p= 0.6In this case, the two-group log-rank test was used to identify if there is a difference between the survival times of menopause status over time. The result indicates that with Chisq= 0.3 on 1 degrees of freedom, p= 0.6 greater than \(a=0.05\) level of significance. Thus, this means that the true survival function in both status are not different (or similar) since we cannot reject the null hypotheses.Therefore, we state that at 5% level of significance the survival times of the menopause status are equal.
b. survival times between progesterone status
#LogRank Test
P <- survfit(Surv(rectime, censrec) ~ prog, data = GBCD)
P## Call: survfit(formula = Surv(rectime, censrec) ~ prog, data = GBCD)
##
## n events median 0.95LCL 0.95UCL
## prog=0 277 154 1140 842 1387
## prog=1 409 145 2286 1989 NAplot(survfit(Surv(rectime,censrec) ~ prog,data = GBCD),
col = c("blue", "black"),
lwd = c(2,2),
mark.time = FALSE)
legend("bottomleft",
legend = c("1 - Positive", "0 - Negative"),
col = c("blue", "black"),
lwd = c(2,2),
bty = "n")
abline(c(0.5,0), lty = 3, col = "red")
title("survival times between progesterone status")
#test
survdiff(Surv(rectime,censrec) ~ prog,data = GBCD )## Call:
## survdiff(formula = Surv(rectime, censrec) ~ prog, data = GBCD)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## prog=0 277 154 97.3 33.0 49.3
## prog=1 409 145 201.7 15.9 49.3
##
## Chisq= 49.3 on 1 degrees of freedom, p= 2e-12In this case, we used log rank test since it is a powerful test for detecting differences in survival curves, specifically when the proportional hazards assumption is met. Since the curves do not diverge nor converge over time, then this may be evidence that the proportional hazards assumption is met.The result shows a computed chi-square of 49.3 on 1 degrees of freedom with a p-value = 2e−12 < \(a=0.05\). Thus, the null hypotheses will be rejected and state that the true survival function in both groups are different. Therefore, we conclude that the true survival times between progesterone status are different.
c. survival times across cancer cells grade
G <- survfit(Surv(rectime, censrec) ~ grade, data = GBCD, conf.type = "log") #Greenwood's CI
G## Call: survfit(formula = Surv(rectime, censrec) ~ grade, data = GBCD,
## conf.type = "log")
##
## n events median 0.95LCL 0.95UCL
## grade=1 81 18 NA 1990 NA
## grade=2 444 202 1730 1493 2030
## grade=3 161 79 1337 960 NAplot(survfit(Surv(rectime, censrec) ~ grade, data = GBCD),
col = c("blue", "black", "red"),
lwd = c(2,2,2,2),
mark.time = FALSE)
legend("topright",
legend = c("3 - low", "2 - medium","1 - high"),
col = c("blue", "black", "red"),
lwd = c(2,2,2,2),
cex = 0.80,
bty = "n")
abline(c(0.5,0), lty = 3, col = "red")
title("survival times across cancer cells grade")
#multigroup logrank test
survdiff(Surv(rectime, censrec) ~ grade,data = GBCD )## Call:
## survdiff(formula = Surv(rectime, censrec) ~ grade, data = GBCD)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## grade=1 81 18 42.2 13.8469 16.159
## grade=2 444 202 198.2 0.0725 0.215
## grade=3 161 79 58.6 7.0788 8.848
##
## Chisq= 21.1 on 2 degrees of freedom, p= 3e-05In this case, we used the logrank test for ordered groups since it is
used to test comparison between three or more ordered groups of
categorical variables (1-high, 2-medium, 3-low).
The result shows a Chisq= 21.1 on 2 degrees of freedom with p-value =
3e−05 < \(a=0.05\). Therefore, null
hypotheses will be rejected which means that the survival times for
different groups of cancer cells grade is the equal. Therefore, we can
conclude that there is at least one pair of survival times for different
groups of cancer cells grade that are different.
d. survival times between estrogen status based on the treatment assignment (hormone or adjuvant)
EH <- survfit(Surv(rectime,censrec) ~ estr + strata(horm), data = GBCD, conf.type = "log") #Greenwood's CI
EH## Call: survfit(formula = Surv(rectime, censrec) ~ estr + strata(horm),
## data = GBCD, conf.type = "log")
##
## n events median 0.95LCL 0.95UCL
## estr=0, strata(horm)=horm=1 181 97 1296 960 1587
## estr=0, strata(horm)=horm=2 91 40 1918 1140 NA
## estr=1, strata(horm)=horm=1 259 108 1814 1528 NA
## estr=1, strata(horm)=horm=2 155 54 2030 1989 NAplot(EH,
col = c("blue", "red", "orange", "black"),
lwd = c(2,2,2,2),
mark.time = FALSE)
legend("topright",
legend = c(" Negative ~ No"," Negative ~ Yes",
" Positive ~ No"," Positive ~ Yes"),
col = c("blue", "red", "orange", "black"),
lwd = c(2,2,2,2),
cex = 0.80,
bty = "n")
abline(c(0.5,0), lty = 3, col = "red")
title("survival times between estrogen status based on hormone therapy")
#stratified
survdiff(Surv(rectime,censrec) ~ estr + strata(horm), data = GBCD)## Call:
## survdiff(formula = Surv(rectime, censrec) ~ estr + strata(horm),
## data = GBCD)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## estr=0 272 137 108 7.68 12.2
## estr=1 414 162 191 4.35 12.2
##
## Chisq= 12.2 on 1 degrees of freedom, p= 5e-04In this case, we used stratified log rank test since we are comparing survival curves of the estrogen status based on the treatment assignment, (hormone or adjuvant).
The results shows that the calculated chi-square value is 12.2 with 1 degrees of freedom, and the p-value is 0.0005 < 0.05 level of significance. Thus, the null hypothesis is rejected, which means that there is difference on the survival times between the estrogen status based on the treatment assignment (hormone or adjuvant). Therefore, we can conclude that the survival times between estrogen status differ in the type of treatments (hormone or adjuvant).
e. survival times based on tumor size
#install.packages("PHInfiniteEstimates")
library(PHInfiniteEstimates)
#trend
GBCD["size"] = cut(GBCD$size, c(0,20, 40, 60, 80,100, 120),
include.lowest=TRUE)
table(GBCD$size)##
## [0,20] (20,40] (40,60] (60,80] (80,100] (100,120]
## 180 408 78 16 3 1attach(GBCD)
survdiff(Surv(rectime,censrec) ~ size, data = GBCD)## Call:
## survdiff(formula = Surv(rectime, censrec) ~ size, data = GBCD)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## size=[0,20] 180 65 8.80e+01 6.0266 8.6418
## size=(20,40] 408 186 1.73e+02 1.0444 2.4854
## size=(40,60] 78 34 3.16e+01 0.1819 0.2040
## size=(60,80] 16 11 5.02e+00 7.1354 7.2702
## size=(80,100] 3 2 1.77e+00 0.0313 0.0315
## size=(100,120] 1 1 7.47e-03 131.8861 132.0834
##
## Chisq= 147 on 5 degrees of freedom, p= <2e-16plot(survfit(Surv(rectime, censrec)~size),
col = c("gray","blue","black","green","brown","orange"),
lwd = c(2,2,2,2,2,2),
mark.time = F)
legend("topright",
legend = unique(size),
col = c("gray","blue","black","green","brown","orange"),
lwd = c(2,2,2,2,2,2,2),
bty = "n")
abline(c(0.5,0), lty = 3, col = "red")
title("survival times based on tumor size")
fit1 <- coxph(Surv(rectime,censrec) ~ size, data = GBCD)
fit1## Call:
## coxph(formula = Surv(rectime, censrec) ~ size, data = GBCD)
##
## coef exp(coef) se(coef) z p
## size(20,40] 0.3837 1.4676 0.1447 2.651 0.008026
## size(40,60] 0.3823 1.4656 0.2123 1.801 0.071686
## size(60,80] 1.0982 2.9987 0.3270 3.358 0.000784
## size(80,100] 0.4255 1.5304 0.7183 0.592 0.553620
## size(100,120] 5.4410 230.6754 1.1239 4.841 1.29e-06
##
## Likelihood ratio test=21.35 on 5 df, p=0.0006959
## n= 686, number of events= 299w <- c(-2,-1,0,1,2)
U_t <- w %*% coef(fit1)
V_t <- w %*% fit1$var %*% w
U_t^2/V_t## [,1]
## [1,] 18.6459pchisq(U_t^2/V_t, df=1, lower.tail=FALSE)## [,1]
## [1,] 1.57385e-05In this case, we use trend test since we want to compare the survival curves of more than three ordered groups of tumor size which can be arrange or ordered, with that the magnitude of the differences is relevant. Hence, there will be six group arrangement for the tumor size.
The results show that p-value of 1.57385e-05 or 0.0000157385 < \(a=0.05\) , thus null hypotheses will be rejected which means that the survival times for different groups are not equal. Therefore, we can conclude that there is at least one pairing of survival times for different groups of tumor size which are not equal.
f. survival times based on the number of axillary lymph nodes with metastases
library(survival)
library(survminer)
#trend
GBCD["nodes"] = cut(GBCD$nodes, c(0,20,40, 60),
include.lowest=TRUE)
table(GBCD$nodes)##
## [0,20] (20,40] (40,60]
## 675 10 1attach(GBCD)
survdiff(Surv(rectime,censrec) ~ nodes, data = GBCD)## Call:
## survdiff(formula = Surv(rectime, censrec) ~ nodes, data = GBCD)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## nodes=[0,20] 675 290 296.103 0.126 13.031
## nodes=(20,40] 10 9 2.585 15.924 16.117
## nodes=(40,60] 1 0 0.313 0.313 0.313
##
## Chisq= 16.4 on 2 degrees of freedom, p= 3e-04plot(survfit(Surv(rectime, censrec)~nodes),
col = c("red","blue","brown"),
lwd = c(2,2,2,2,2,2),
mark.time = F)
legend("topright",
legend = unique(nodes),
col = c("red","blue","brown"),
lwd = c(2,2,2,2,2,2,2),
bty = "n")
abline(c(0.5,0), lty = 3, col = "red")
title("survival times based on the number of axillary lymph nodes with metastases")
fit2 <- coxph(Surv(rectime,censrec) ~ nodes, data = GBCD)
fit2## Call:
## coxph(formula = Surv(rectime, censrec) ~ nodes, data = GBCD)
##
## coef exp(coef) se(coef) z p
## nodes(20,40] 1.276e+00 3.581e+00 3.402e-01 3.750 0.000177
## nodes(40,60] -1.349e+01 1.389e-06 1.539e+03 -0.009 0.993008
##
## Likelihood ratio test=10.42 on 2 df, p=0.005452
## n= 686, number of events= 299obs_n = c(290, 9,0)
exp_n = c(296.103,2.585, 0.313)
w_n = c(-1,0,1)
u_n = sum(w_n*(obs_n-exp_n ))
v_n = sum(w_n^2*exp_n)- ((sum(w_n*exp_n))^2/(sum(exp_n)))
x_n = u_n^2/v_n
x_n## [1] 8.816292pchisq(x_n, df=1, lower.tail=F)## [1] 0.002985527In this case, we also use trend test since we want to compare the survival curves of more than three ordered groups of number of axillary lymph nodes with metastases which can be arrange or ordered, with that the magnitude of the differences is relevant. Hence, there will be three group arrangement for the axillary lymph nodes with metastases.
The results show that p-value of 0.002985527 < \(a=0.05\) , thus null hypotheses will be rejected which means that the survival times for different groups are not the same. Therefore, we can conclude that at least one survival times for different groups of axillary lymph nodes with metastases are not equal.