R4CR
Day2 - 회귀 / 생존 분석 | 2023-06-15
Jinseob Kim
Executive Summary
| Dafault | Repeated measure | Survey | |
|---|---|---|---|
| Continuous | linear regression | GEE | Survey GLM |
| Event | GLM (logistic) | GEE | Survey GLM |
| Time & Event | Cox | marginal Cox | Survey Cox |
| 0,1,2,3 (rare event) | GLM (poisson) | GEE | Survey GLM |
실습데이터
Linear regression
Continuous
Simple
\[Y = \beta_0 + \beta_1 X + \epsilon\]
오차제곱합을 최소로하는 \(\beta_0, \beta_1\) 구한다.
\(Y\) 정규분포하는 연속변수, \(X\) 는 연속, 범주형 다 가능
- \(X\) 연속변수일 땐 상관분석 과 동일
- \(X\) 2범주일 땐
t.testwith 등분산 과 동일
Simple 2
Pearson's product-moment correlation
data: colon$age and colon$nodes
t = -3.6597, df = 1774, p-value = 0.0002599
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.13254350 -0.04021182
sample estimates:
cor
-0.08656354 Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.20164796 0.428601525 12.136326 1.279575e-32
age -0.02572043 0.007028022 -3.659696 2.598993e-04 Estimate Std. Error t value Pr(>|t|)
(Intercept) 60.8780528 0.40542964 150.156886 0.0000000000
nodes -0.2913344 0.07960617 -3.659696 0.0002598993Simple 3
Two Sample t-test
data: time by sex
t = -0.65063, df = 1774, p-value = 0.5154
alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
95 percent confidence interval:
-117.4506 58.9366
sample estimates:
mean in group 0 mean in group 1
1527.400 1556.657 Estimate Std. Error t value Pr(>|t|)
(Intercept) 1527.39953 32.36421 47.1940973 1.189988e-315
sex 29.25699 44.96686 0.6506345 5.153667e-013범주 이상?
rx: 치료법 3개
더미변수로 자동으로 바뀐 후 회귀식에 포함. 실제로는 변수 2개가 들어감
(Intercept) rxLev rxLev+5FU
1853 1 1 0
1854 1 1 0
1855 1 0 1
1856 1 0 1
1857 1 1 0
1858 1 1 0둘다 0 이면 Obs (reference)
3범주 이상 2
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1439.33770 38.03569 37.8417658 3.280940e-230
rxLev 39.62148 54.29140 0.7297929 4.656132e-01
rxLev+5FU 276.84569 54.53001 5.0769414 4.238516e-07Obs와 Lev+5FU 군이 유의한 차이가 있음. ANOVA 형태로도 볼 수 있다 (등분산 가정).
Analysis of Variance Table
Response: time
Df Sum Sq Mean Sq F value Pr(>F)
rx 2 26301809 13150904 14.902 3.819e-07 ***
Residuals 1773 1564664354 882495
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ANOVA p-value 는 rx 중 튀는 것이 있는지? 를 통합평가.
Multiple
여러 변수들을 포함
\[Y = \beta_0 + \beta_1 X_{1} + \beta_2 X_{2} + \cdots + \epsilon\]
- \(\beta_1\) 해석: \(X_2, X_3 \cdots\) 를 보정한다면, \(X_1\) 이 1 증가할 때 \(Y\) 가 \(\beta_1\) 만큼 증가한다.
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1425.2773011 119.689429 11.90813019 1.652628e-31
sex 45.8552044 44.755366 1.02457444 3.057040e-01
age -0.1706051 1.873203 -0.09107668 9.274420e-01
rxLev 38.3536023 54.334331 0.70588157 4.803546e-01
rxLev+5FU 279.7044350 54.618879 5.12102117 3.369819e-07논문용 테이블은 보정 전후 결과를 같이 보여주는 것이 대세
Multiple 2 +
Logistic regression
0 or 1
logit
\[ P(Y = 1) = \frac{\exp{(X)}}{1 + \exp{(X)}}\]
Odds Ratio
\[ \begin{aligned} P(Y = 1) &= \frac{\exp{(\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots)}}{1 + \exp{(\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots)}} \\\\ \ln(\frac{p}{1-p}) &= \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots \end{aligned} \]
\(\beta_1\) 해석: \(X_2, X_3 \cdots\) 들 보정한다면, \(X_1\)이 1 증가할 때, \(\ln(\frac{p}{1-p})\) 가 \(\beta_1\) 만큼 증가한다.
\(\frac{p}{1-p}\) 가 \(\exp(\beta_1)\) 배 증가한다. 즉 Odd Ratio = \(\exp(\beta_1)\)
Odds Ratio 2
Call:
glm(formula = status ~ sex + age + rx, family = binomial, data = colon)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.442008 0.257400 1.717 0.0859 .
sex -0.107210 0.096230 -1.114 0.2652
age -0.003047 0.004026 -0.757 0.4491
rxLev -0.090039 0.116087 -0.776 0.4380
rxLev+5FU -0.623946 0.117929 -5.291 1.22e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 2461.7 on 1775 degrees of freedom
Residual deviance: 2427.9 on 1771 degrees of freedom
AIC: 2437.9
Number of Fisher Scoring iterations: 4Odds Ratio 3 +
Cox proportional hazard
Time & Event
Time to event data
대부분 Right censored: XX 일에 사망 or XX 일 까지 생존
Time to event 를 하나의 변수로
Kaplan-meier plot
생존분석에서 table 1의 의미
- 보통 logrank test p-value 를 같이 보여줌.
계산: time 순서로 정렬
\[ \begin{aligned} P(t) &= \frac{t \text{ 구간 생존수}}{t \text{ 시점 관찰대상 수}} : \text{구간 생존율}\\\\ S(t) & = S(t-1) \times P(t) \end{aligned} \]
출처: https://dermabae.tistory.com/180
plot
중도절단 marks 는 보통 생략.
summary
Call: survfit(formula = Surv(time, status) ~ rx, data = colon)
rx=Obs
time n.risk n.event survival std.err lower 95% CI upper 95% CI
20 610 1 0.998 0.00164 0.995 1.000
36 609 1 0.997 0.00231 0.992 1.000
43 608 1 0.995 0.00283 0.990 1.000
45 607 1 0.993 0.00327 0.987 1.000
59 606 1 0.992 0.00365 0.985 0.999
72 605 1 0.990 0.00400 0.982 0.998
77 604 1 0.989 0.00431 0.980 0.997
79 603 1 0.987 0.00461 0.978 0.996
80 602 2 0.984 0.00514 0.974 0.994
85 600 1 0.982 0.00539 0.971 0.993
86 599 1 0.980 0.00562 0.969 0.991
88 598 1 0.979 0.00585 0.967 0.990
94 597 1 0.977 0.00606 0.965 0.989
98 596 1 0.975 0.00627 0.963 0.988
99 595 2 0.972 0.00666 0.959 0.985
101 593 1 0.970 0.00685 0.957 0.984
102 592 1 0.969 0.00703 0.955 0.983
103 591 1 0.967 0.00721 0.953 0.981
108 590 1 0.966 0.00738 0.951 0.980
109 589 1 0.964 0.00755 0.949 0.979
113 588 2 0.961 0.00787 0.945 0.976
121 586 1 0.959 0.00803 0.943 0.975
122 585 1 0.957 0.00818 0.941 0.974
125 584 1 0.956 0.00833 0.940 0.972
127 583 1 0.954 0.00847 0.938 0.971
131 582 1 0.952 0.00862 0.936 0.969
139 581 1 0.951 0.00876 0.934 0.968
143 580 1 0.949 0.00889 0.932 0.967
145 579 1 0.948 0.00903 0.930 0.965
154 578 1 0.946 0.00916 0.928 0.964
157 577 1 0.944 0.00929 0.926 0.963
161 576 1 0.943 0.00942 0.924 0.961
164 575 1 0.941 0.00954 0.922 0.960
165 574 2 0.938 0.00979 0.919 0.957
166 572 2 0.934 0.01002 0.915 0.954
167 570 1 0.933 0.01014 0.913 0.953
173 569 3 0.928 0.01047 0.908 0.949
185 566 2 0.925 0.01069 0.904 0.946
187 564 1 0.923 0.01080 0.902 0.944
188 563 1 0.921 0.01090 0.900 0.943
189 562 1 0.920 0.01100 0.898 0.941
201 561 1 0.918 0.01111 0.897 0.940
203 560 1 0.916 0.01121 0.895 0.939
208 559 2 0.913 0.01140 0.891 0.936
215 557 2 0.910 0.01160 0.887 0.933
218 555 2 0.907 0.01178 0.884 0.930
221 553 1 0.905 0.01188 0.882 0.928
223 552 1 0.903 0.01197 0.880 0.927
227 551 1 0.902 0.01206 0.878 0.926
228 550 1 0.900 0.01215 0.877 0.924
229 549 1 0.898 0.01223 0.875 0.923
230 548 3 0.893 0.01249 0.869 0.918
237 545 1 0.892 0.01258 0.867 0.917
238 544 3 0.887 0.01282 0.862 0.912
241 541 1 0.885 0.01290 0.860 0.911
242 540 1 0.884 0.01298 0.859 0.909
243 539 1 0.882 0.01306 0.857 0.908
245 538 1 0.880 0.01314 0.855 0.906
253 537 1 0.879 0.01322 0.853 0.905
256 536 1 0.877 0.01330 0.851 0.903
257 535 1 0.875 0.01337 0.850 0.902
259 534 2 0.872 0.01352 0.846 0.899
263 532 1 0.870 0.01359 0.844 0.898
264 531 2 0.867 0.01374 0.841 0.895
271 529 1 0.866 0.01381 0.839 0.893
273 528 1 0.864 0.01388 0.837 0.892
275 527 1 0.862 0.01395 0.835 0.890
276 526 1 0.861 0.01402 0.834 0.889
279 525 1 0.859 0.01409 0.832 0.887
280 524 1 0.857 0.01416 0.830 0.886
286 523 1 0.856 0.01423 0.828 0.884
289 522 1 0.854 0.01429 0.827 0.883
291 521 1 0.852 0.01436 0.825 0.881
294 520 1 0.851 0.01442 0.823 0.880
296 519 1 0.849 0.01449 0.821 0.878
304 518 1 0.848 0.01455 0.819 0.877
308 517 1 0.846 0.01462 0.818 0.875
311 516 1 0.844 0.01468 0.816 0.874
313 515 1 0.843 0.01474 0.814 0.872
315 514 1 0.841 0.01481 0.812 0.871
322 513 1 0.839 0.01487 0.811 0.869
331 512 1 0.838 0.01493 0.809 0.867
334 511 1 0.836 0.01499 0.807 0.866
337 510 1 0.834 0.01505 0.805 0.864
344 509 1 0.833 0.01511 0.804 0.863
349 508 1 0.831 0.01517 0.802 0.861
352 507 1 0.830 0.01523 0.800 0.860
354 506 1 0.828 0.01528 0.798 0.858
360 505 1 0.826 0.01534 0.797 0.857
362 504 1 0.825 0.01540 0.795 0.855
365 503 1 0.823 0.01545 0.793 0.854
372 502 1 0.821 0.01551 0.791 0.852
374 501 1 0.820 0.01557 0.790 0.851
378 500 1 0.818 0.01562 0.788 0.849
379 499 1 0.816 0.01568 0.786 0.848
381 498 1 0.815 0.01573 0.785 0.846
382 497 1 0.813 0.01578 0.783 0.845
384 496 3 0.808 0.01594 0.778 0.840
390 493 1 0.807 0.01599 0.776 0.839
398 492 1 0.805 0.01604 0.774 0.837
401 491 1 0.803 0.01610 0.772 0.835
402 490 1 0.802 0.01615 0.771 0.834
406 489 1 0.800 0.01620 0.769 0.832
409 488 1 0.798 0.01625 0.767 0.831
411 487 2 0.795 0.01634 0.764 0.828
413 485 2 0.792 0.01644 0.760 0.825
417 483 1 0.790 0.01649 0.759 0.823
421 482 1 0.789 0.01653 0.757 0.822
433 480 2 0.785 0.01663 0.753 0.819
435 478 1 0.784 0.01667 0.752 0.817
437 477 2 0.780 0.01676 0.748 0.814
438 475 2 0.777 0.01685 0.745 0.811
459 471 1 0.775 0.01690 0.743 0.809
461 470 1 0.774 0.01694 0.741 0.808
462 469 1 0.772 0.01699 0.739 0.806
464 468 1 0.770 0.01703 0.738 0.805
465 467 2 0.767 0.01712 0.734 0.801
474 465 1 0.765 0.01716 0.733 0.800
480 464 1 0.764 0.01720 0.731 0.798
485 463 1 0.762 0.01724 0.729 0.797
489 461 1 0.761 0.01729 0.727 0.795
493 460 1 0.759 0.01733 0.726 0.794
495 459 1 0.757 0.01737 0.724 0.792
496 458 1 0.756 0.01741 0.722 0.790
499 457 2 0.752 0.01749 0.719 0.787
506 455 1 0.751 0.01753 0.717 0.786
510 454 1 0.749 0.01757 0.715 0.784
523 453 1 0.747 0.01761 0.714 0.783
532 452 1 0.746 0.01764 0.712 0.781
534 451 1 0.744 0.01768 0.710 0.779
537 450 1 0.742 0.01772 0.708 0.778
540 449 1 0.741 0.01776 0.707 0.776
542 448 1 0.739 0.01779 0.705 0.775
543 447 1 0.737 0.01783 0.703 0.773
547 446 1 0.736 0.01787 0.702 0.772
555 445 1 0.734 0.01790 0.700 0.770
561 444 1 0.732 0.01794 0.698 0.768
563 443 2 0.729 0.01801 0.695 0.765
570 441 1 0.727 0.01805 0.693 0.764
576 440 1 0.726 0.01808 0.691 0.762
577 439 1 0.724 0.01811 0.690 0.761
581 438 1 0.722 0.01815 0.688 0.759
587 437 1 0.721 0.01818 0.686 0.757
591 436 1 0.719 0.01822 0.684 0.756
593 435 1 0.718 0.01825 0.683 0.754
594 434 1 0.716 0.01828 0.681 0.753
595 433 1 0.714 0.01831 0.679 0.751
599 432 1 0.713 0.01835 0.678 0.749
608 430 1 0.711 0.01838 0.676 0.748
612 429 1 0.709 0.01841 0.674 0.746
622 428 1 0.708 0.01844 0.672 0.745
625 427 1 0.706 0.01847 0.671 0.743
632 426 1 0.704 0.01850 0.669 0.742
659 425 2 0.701 0.01856 0.666 0.738
663 423 2 0.698 0.01862 0.662 0.735
665 421 1 0.696 0.01865 0.660 0.734
670 419 1 0.694 0.01868 0.659 0.732
673 418 1 0.693 0.01871 0.657 0.730
685 417 1 0.691 0.01874 0.655 0.729
686 416 1 0.689 0.01877 0.654 0.727
687 415 1 0.688 0.01880 0.652 0.726
692 414 1 0.686 0.01882 0.650 0.724
700 413 1 0.684 0.01885 0.648 0.722
702 412 2 0.681 0.01891 0.645 0.719
709 410 1 0.679 0.01893 0.643 0.718
712 409 1 0.678 0.01896 0.642 0.716
716 408 1 0.676 0.01899 0.640 0.714
717 407 1 0.674 0.01901 0.638 0.713
718 406 1 0.673 0.01904 0.636 0.711
721 405 1 0.671 0.01906 0.635 0.710
726 404 1 0.669 0.01909 0.633 0.708
730 403 1 0.668 0.01911 0.631 0.706
731 402 1 0.666 0.01914 0.630 0.705
735 401 1 0.664 0.01916 0.628 0.703
743 400 1 0.663 0.01918 0.626 0.701
748 399 1 0.661 0.01921 0.625 0.700
752 398 1 0.659 0.01923 0.623 0.698
753 397 1 0.658 0.01925 0.621 0.697
758 396 1 0.656 0.01928 0.619 0.695
760 395 1 0.654 0.01930 0.618 0.693
761 394 1 0.653 0.01932 0.616 0.692
770 393 1 0.651 0.01934 0.614 0.690
772 392 1 0.649 0.01937 0.613 0.689
774 391 2 0.646 0.01941 0.609 0.685
775 389 1 0.645 0.01943 0.608 0.684
803 388 1 0.643 0.01945 0.606 0.682
832 387 1 0.641 0.01947 0.604 0.681
833 386 1 0.640 0.01949 0.602 0.679
835 385 1 0.638 0.01951 0.601 0.677
840 384 1 0.636 0.01953 0.599 0.676
845 383 1 0.635 0.01955 0.597 0.674
854 381 1 0.633 0.01957 0.596 0.672
855 380 1 0.631 0.01959 0.594 0.671
863 379 1 0.630 0.01961 0.592 0.669
871 378 1 0.628 0.01963 0.591 0.668
874 377 1 0.626 0.01965 0.589 0.666
883 376 1 0.625 0.01966 0.587 0.664
887 375 1 0.623 0.01968 0.585 0.663
901 373 1 0.621 0.01970 0.584 0.661
912 372 1 0.620 0.01972 0.582 0.659
924 371 1 0.618 0.01974 0.580 0.658
928 370 1 0.616 0.01975 0.579 0.656
929 369 1 0.615 0.01977 0.577 0.655
930 368 1 0.613 0.01979 0.575 0.653
936 367 1 0.611 0.01980 0.574 0.651
949 366 1 0.610 0.01982 0.572 0.650
957 365 1 0.608 0.01984 0.570 0.648
961 364 1 0.606 0.01985 0.569 0.646
963 363 1 0.605 0.01987 0.567 0.645
966 362 1 0.603 0.01988 0.565 0.643
975 361 1 0.601 0.01990 0.563 0.641
976 360 1 0.600 0.01991 0.562 0.640
1020 359 1 0.598 0.01993 0.560 0.638
1021 358 1 0.596 0.01994 0.558 0.637
1031 357 1 0.594 0.01995 0.557 0.635
1042 356 1 0.593 0.01997 0.555 0.633
1048 355 1 0.591 0.01998 0.553 0.632
1057 354 2 0.588 0.02001 0.550 0.628
1070 352 1 0.586 0.02002 0.548 0.627
1079 351 1 0.584 0.02003 0.547 0.625
1081 350 1 0.583 0.02004 0.545 0.623
1083 349 1 0.581 0.02006 0.543 0.622
1089 348 1 0.579 0.02007 0.541 0.620
1101 347 1 0.578 0.02008 0.540 0.619
1106 346 1 0.576 0.02009 0.538 0.617
1130 345 1 0.574 0.02010 0.536 0.615
1133 344 1 0.573 0.02011 0.535 0.614
1134 343 1 0.571 0.02012 0.533 0.612
1136 342 1 0.569 0.02013 0.531 0.610
1139 341 2 0.566 0.02015 0.528 0.607
1159 339 1 0.564 0.02016 0.526 0.605
1166 338 1 0.563 0.02017 0.525 0.604
1178 337 1 0.561 0.02018 0.523 0.602
1195 336 1 0.559 0.02019 0.521 0.600
1198 335 1 0.558 0.02020 0.520 0.599
1209 333 1 0.556 0.02021 0.518 0.597
1216 332 1 0.554 0.02022 0.516 0.595
1230 331 1 0.553 0.02022 0.514 0.594
1236 330 1 0.551 0.02023 0.513 0.592
1237 329 1 0.549 0.02024 0.511 0.591
1246 328 1 0.548 0.02025 0.509 0.589
1262 327 1 0.546 0.02026 0.508 0.587
1272 326 1 0.544 0.02026 0.506 0.586
1274 325 1 0.543 0.02027 0.504 0.584
1290 324 1 0.541 0.02028 0.503 0.582
1295 323 1 0.539 0.02028 0.501 0.581
1304 322 1 0.538 0.02029 0.499 0.579
1313 321 1 0.536 0.02029 0.498 0.577
1314 320 1 0.534 0.02030 0.496 0.576
1323 318 1 0.533 0.02030 0.494 0.574
1327 317 1 0.531 0.02031 0.493 0.572
1353 316 1 0.529 0.02032 0.491 0.571
1363 315 1 0.528 0.02032 0.489 0.569
1375 313 1 0.526 0.02032 0.488 0.567
1432 312 1 0.524 0.02033 0.486 0.566
1434 311 1 0.523 0.02033 0.484 0.564
1436 310 1 0.521 0.02034 0.482 0.562
1437 309 1 0.519 0.02034 0.481 0.561
1446 308 1 0.517 0.02035 0.479 0.559
1447 307 1 0.516 0.02035 0.477 0.557
1455 306 1 0.514 0.02035 0.476 0.556
1466 305 1 0.512 0.02036 0.474 0.554
1475 304 1 0.511 0.02036 0.472 0.552
1482 303 1 0.509 0.02036 0.471 0.551
1530 302 1 0.507 0.02036 0.469 0.549
1535 301 1 0.506 0.02036 0.467 0.547
1548 300 1 0.504 0.02037 0.466 0.546
1606 299 1 0.502 0.02037 0.464 0.544
1656 298 1 0.501 0.02037 0.462 0.542
1679 297 1 0.499 0.02037 0.461 0.541
1692 296 1 0.497 0.02037 0.459 0.539
1723 295 1 0.496 0.02037 0.457 0.537
1745 294 1 0.494 0.02037 0.456 0.535
1749 293 1 0.492 0.02037 0.454 0.534
1759 292 1 0.491 0.02037 0.452 0.532
1772 291 1 0.489 0.02037 0.450 0.530
1788 290 1 0.487 0.02037 0.449 0.529
1790 289 1 0.485 0.02037 0.447 0.527
1818 284 1 0.484 0.02037 0.445 0.525
1875 270 1 0.482 0.02037 0.444 0.524
1896 267 1 0.480 0.02038 0.442 0.522
1907 263 1 0.478 0.02038 0.440 0.520
1915 262 1 0.476 0.02038 0.438 0.518
1950 257 1 0.475 0.02039 0.436 0.516
1981 252 1 0.473 0.02040 0.434 0.514
2035 241 1 0.471 0.02040 0.432 0.513
2036 240 1 0.469 0.02041 0.430 0.511
2077 235 1 0.467 0.02042 0.428 0.509
2083 234 1 0.465 0.02043 0.426 0.507
2085 233 1 0.463 0.02044 0.424 0.505
2133 211 1 0.461 0.02046 0.422 0.503
2148 207 1 0.458 0.02049 0.420 0.500
2171 189 1 0.456 0.02052 0.418 0.498
2213 165 1 0.453 0.02058 0.415 0.495
2257 144 1 0.450 0.02068 0.411 0.492
2284 136 1 0.447 0.02079 0.408 0.489
2287 135 1 0.443 0.02090 0.404 0.486
2288 134 1 0.440 0.02100 0.401 0.483
2351 118 1 0.436 0.02115 0.397 0.480
2527 84 1 0.431 0.02153 0.391 0.476
2552 76 1 0.426 0.02198 0.385 0.471
2695 50 1 0.417 0.02313 0.374 0.465
2789 28 1 0.402 0.02667 0.353 0.458
rx=Lev
time n.risk n.event survival std.err lower 95% CI upper 95% CI
19 588 1 0.998 0.00170 0.995 1.000
24 587 1 0.997 0.00240 0.992 1.000
28 585 1 0.995 0.00294 0.989 1.000
35 584 1 0.993 0.00339 0.987 1.000
38 583 1 0.991 0.00379 0.984 0.999
56 582 1 0.990 0.00415 0.982 0.998
62 580 2 0.986 0.00479 0.977 0.996
72 578 1 0.985 0.00507 0.975 0.995
77 577 1 0.983 0.00534 0.973 0.993
78 576 1 0.981 0.00560 0.970 0.992
80 575 1 0.980 0.00584 0.968 0.991
85 574 1 0.978 0.00608 0.966 0.990
91 573 1 0.976 0.00630 0.964 0.989
93 572 1 0.974 0.00652 0.962 0.987
98 571 2 0.971 0.00693 0.958 0.985
100 569 1 0.969 0.00712 0.955 0.983
105 568 1 0.968 0.00731 0.953 0.982
111 567 1 0.966 0.00750 0.951 0.981
113 566 2 0.962 0.00785 0.947 0.978
116 564 2 0.959 0.00818 0.943 0.975
119 562 1 0.957 0.00834 0.941 0.974
121 561 1 0.956 0.00850 0.939 0.972
122 560 1 0.954 0.00866 0.937 0.971
129 559 1 0.952 0.00881 0.935 0.970
133 558 1 0.951 0.00896 0.933 0.968
136 557 1 0.949 0.00910 0.931 0.967
141 556 1 0.947 0.00924 0.929 0.965
145 555 1 0.945 0.00938 0.927 0.964
146 554 1 0.944 0.00952 0.925 0.963
147 553 1 0.942 0.00965 0.923 0.961
150 552 1 0.940 0.00979 0.921 0.960
157 551 1 0.939 0.00992 0.919 0.958
165 550 1 0.937 0.01004 0.917 0.957
169 549 1 0.935 0.01017 0.915 0.955
171 548 2 0.932 0.01041 0.912 0.952
174 546 1 0.930 0.01053 0.910 0.951
175 545 1 0.928 0.01065 0.908 0.949
176 544 2 0.925 0.01088 0.904 0.947
179 542 1 0.923 0.01100 0.902 0.945
181 541 1 0.922 0.01111 0.900 0.944
183 540 1 0.920 0.01122 0.898 0.942
185 539 2 0.916 0.01143 0.894 0.939
189 537 1 0.915 0.01154 0.892 0.938
191 536 3 0.910 0.01185 0.887 0.933
196 533 1 0.908 0.01195 0.885 0.932
204 532 1 0.906 0.01204 0.883 0.930
206 531 1 0.904 0.01214 0.881 0.929
216 530 1 0.903 0.01224 0.879 0.927
218 529 1 0.901 0.01233 0.877 0.926
219 528 2 0.898 0.01252 0.873 0.923
222 525 1 0.896 0.01261 0.872 0.921
224 524 1 0.894 0.01270 0.870 0.919
226 523 1 0.893 0.01279 0.868 0.918
229 522 1 0.891 0.01288 0.866 0.916
230 521 2 0.887 0.01306 0.862 0.913
232 519 1 0.886 0.01315 0.860 0.912
235 518 1 0.884 0.01323 0.858 0.910
246 517 1 0.882 0.01332 0.857 0.909
250 516 1 0.881 0.01340 0.855 0.907
257 515 1 0.879 0.01348 0.853 0.906
258 514 1 0.877 0.01356 0.851 0.904
260 513 1 0.875 0.01364 0.849 0.903
262 512 1 0.874 0.01372 0.847 0.901
263 511 1 0.872 0.01380 0.845 0.899
274 510 1 0.870 0.01388 0.843 0.898
276 509 1 0.869 0.01396 0.842 0.896
279 508 2 0.865 0.01411 0.838 0.893
283 506 1 0.863 0.01419 0.836 0.892
286 505 2 0.860 0.01434 0.832 0.889
290 503 2 0.857 0.01448 0.829 0.885
294 501 1 0.855 0.01455 0.827 0.884
300 500 1 0.853 0.01462 0.825 0.882
313 499 1 0.851 0.01469 0.823 0.881
314 498 2 0.848 0.01483 0.819 0.878
316 496 1 0.846 0.01490 0.818 0.876
323 495 1 0.845 0.01497 0.816 0.874
325 494 1 0.843 0.01504 0.814 0.873
330 493 1 0.841 0.01510 0.812 0.871
333 492 1 0.839 0.01517 0.810 0.870
335 491 1 0.838 0.01523 0.808 0.868
336 490 1 0.836 0.01530 0.807 0.867
337 489 2 0.833 0.01543 0.803 0.863
342 487 1 0.831 0.01549 0.801 0.862
343 486 1 0.829 0.01555 0.799 0.860
348 485 2 0.826 0.01567 0.796 0.857
349 483 1 0.824 0.01573 0.794 0.856
352 482 1 0.822 0.01579 0.792 0.854
355 481 1 0.821 0.01585 0.790 0.852
356 480 3 0.816 0.01603 0.785 0.848
362 477 1 0.814 0.01609 0.783 0.846
366 476 2 0.810 0.01620 0.779 0.843
370 474 1 0.809 0.01625 0.777 0.841
372 473 1 0.807 0.01631 0.776 0.840
376 472 1 0.805 0.01636 0.774 0.838
382 471 1 0.804 0.01642 0.772 0.836
386 470 1 0.802 0.01647 0.770 0.835
389 469 1 0.800 0.01653 0.768 0.833
402 468 1 0.798 0.01658 0.767 0.832
406 467 1 0.797 0.01663 0.765 0.830
413 466 1 0.795 0.01668 0.763 0.828
415 465 1 0.793 0.01674 0.761 0.827
420 464 1 0.792 0.01679 0.759 0.825
422 463 1 0.790 0.01684 0.758 0.824
429 462 1 0.788 0.01689 0.756 0.822
430 461 1 0.786 0.01694 0.754 0.820
438 460 1 0.785 0.01699 0.752 0.819
439 459 2 0.781 0.01708 0.749 0.816
440 457 1 0.780 0.01713 0.747 0.814
443 456 1 0.778 0.01718 0.745 0.812
454 455 1 0.776 0.01723 0.743 0.811
458 454 1 0.775 0.01727 0.741 0.809
465 453 1 0.773 0.01732 0.740 0.808
472 452 1 0.771 0.01736 0.738 0.806
474 451 1 0.769 0.01741 0.736 0.804
475 450 1 0.768 0.01746 0.734 0.803
476 449 1 0.766 0.01750 0.732 0.801
482 448 1 0.764 0.01754 0.731 0.799
486 447 1 0.763 0.01759 0.729 0.798
490 445 1 0.761 0.01763 0.727 0.796
491 444 1 0.759 0.01768 0.725 0.795
498 443 1 0.757 0.01772 0.723 0.793
499 442 1 0.756 0.01776 0.722 0.791
504 441 1 0.754 0.01780 0.720 0.790
505 440 1 0.752 0.01784 0.718 0.788
511 439 1 0.751 0.01789 0.716 0.786
512 438 1 0.749 0.01793 0.715 0.785
513 437 1 0.747 0.01797 0.713 0.783
522 436 1 0.745 0.01801 0.711 0.782
525 435 1 0.744 0.01805 0.709 0.780
527 434 1 0.742 0.01809 0.707 0.778
532 433 1 0.740 0.01813 0.706 0.777
546 432 1 0.739 0.01817 0.704 0.775
548 431 1 0.737 0.01820 0.702 0.773
553 430 1 0.735 0.01824 0.700 0.772
559 429 1 0.733 0.01828 0.698 0.770
560 428 1 0.732 0.01832 0.697 0.769
565 427 1 0.730 0.01835 0.695 0.767
573 426 3 0.725 0.01846 0.690 0.762
578 423 1 0.723 0.01850 0.688 0.760
580 422 1 0.721 0.01853 0.686 0.759
582 421 1 0.720 0.01857 0.684 0.757
583 420 1 0.718 0.01860 0.682 0.755
589 419 1 0.716 0.01864 0.681 0.754
593 418 1 0.715 0.01867 0.679 0.752
599 417 1 0.713 0.01871 0.677 0.750
602 416 1 0.711 0.01874 0.675 0.749
608 415 1 0.709 0.01877 0.674 0.747
613 414 1 0.708 0.01881 0.672 0.746
615 413 1 0.706 0.01884 0.670 0.744
628 412 1 0.704 0.01887 0.668 0.742
629 410 1 0.703 0.01890 0.666 0.741
638 409 1 0.701 0.01893 0.665 0.739
642 408 1 0.699 0.01897 0.663 0.737
643 407 1 0.697 0.01900 0.661 0.736
647 406 1 0.696 0.01903 0.659 0.734
654 405 1 0.694 0.01906 0.658 0.732
663 404 1 0.692 0.01909 0.656 0.731
664 403 1 0.691 0.01912 0.654 0.729
668 402 1 0.689 0.01915 0.652 0.727
669 401 1 0.687 0.01918 0.651 0.726
672 400 1 0.685 0.01920 0.649 0.724
675 399 1 0.684 0.01923 0.647 0.722
678 398 1 0.682 0.01926 0.645 0.721
680 397 1 0.680 0.01929 0.643 0.719
684 396 1 0.679 0.01932 0.642 0.717
697 395 1 0.677 0.01934 0.640 0.716
706 394 1 0.675 0.01937 0.638 0.714
708 393 1 0.673 0.01940 0.636 0.712
709 392 1 0.672 0.01942 0.635 0.711
717 391 1 0.670 0.01945 0.633 0.709
720 390 1 0.668 0.01948 0.631 0.708
723 389 1 0.667 0.01950 0.629 0.706
729 388 1 0.665 0.01953 0.628 0.704
730 387 1 0.663 0.01955 0.626 0.703
739 386 1 0.661 0.01958 0.624 0.701
742 385 1 0.660 0.01960 0.622 0.699
743 384 1 0.658 0.01963 0.621 0.698
751 383 1 0.656 0.01965 0.619 0.696
755 382 1 0.654 0.01967 0.617 0.694
759 381 2 0.651 0.01972 0.614 0.691
764 379 1 0.649 0.01974 0.612 0.689
766 378 1 0.648 0.01976 0.610 0.688
797 377 1 0.646 0.01979 0.608 0.686
806 376 1 0.644 0.01981 0.606 0.684
828 375 1 0.642 0.01983 0.605 0.683
833 374 1 0.641 0.01985 0.603 0.681
846 373 1 0.639 0.01987 0.601 0.679
851 372 1 0.637 0.01989 0.599 0.678
858 371 1 0.636 0.01991 0.598 0.676
875 370 1 0.634 0.01993 0.596 0.674
883 369 1 0.632 0.01995 0.594 0.672
890 368 1 0.630 0.01997 0.592 0.671
891 367 1 0.629 0.01999 0.591 0.669
900 366 1 0.627 0.02001 0.589 0.667
902 365 1 0.625 0.02003 0.587 0.666
905 364 1 0.624 0.02005 0.585 0.664
909 363 1 0.622 0.02007 0.584 0.662
922 362 1 0.620 0.02008 0.582 0.661
931 361 1 0.618 0.02010 0.580 0.659
938 360 1 0.617 0.02012 0.578 0.657
939 358 1 0.615 0.02014 0.577 0.656
940 357 1 0.613 0.02015 0.575 0.654
942 356 1 0.612 0.02017 0.573 0.652
944 355 1 0.610 0.02019 0.571 0.651
952 354 1 0.608 0.02020 0.570 0.649
959 353 1 0.606 0.02022 0.568 0.647
960 352 1 0.605 0.02024 0.566 0.646
961 351 2 0.601 0.02027 0.563 0.642
968 349 1 0.599 0.02028 0.561 0.641
969 348 1 0.598 0.02030 0.559 0.639
986 347 1 0.596 0.02031 0.558 0.637
997 346 1 0.594 0.02032 0.556 0.635
1013 345 1 0.593 0.02034 0.554 0.634
1018 344 1 0.591 0.02035 0.552 0.632
1026 343 1 0.589 0.02037 0.551 0.630
1029 342 1 0.587 0.02038 0.549 0.629
1034 341 1 0.586 0.02039 0.547 0.627
1037 340 1 0.584 0.02040 0.545 0.625
1041 339 1 0.582 0.02042 0.544 0.624
1046 338 1 0.581 0.02043 0.542 0.622
1052 337 1 0.579 0.02044 0.540 0.620
1055 336 1 0.577 0.02045 0.538 0.619
1092 335 1 0.575 0.02046 0.537 0.617
1103 334 1 0.574 0.02047 0.535 0.615
1105 332 1 0.572 0.02049 0.533 0.613
1108 331 1 0.570 0.02050 0.531 0.612
1112 330 1 0.568 0.02051 0.530 0.610
1114 329 1 0.567 0.02052 0.528 0.608
1117 328 1 0.565 0.02053 0.526 0.607
1122 327 1 0.563 0.02054 0.524 0.605
1145 326 1 0.562 0.02055 0.523 0.603
1154 325 1 0.560 0.02056 0.521 0.602
1161 324 1 0.558 0.02057 0.519 0.600
1178 323 1 0.556 0.02057 0.517 0.598
1183 322 1 0.555 0.02058 0.516 0.596
1186 321 1 0.553 0.02059 0.514 0.595
1191 320 1 0.551 0.02060 0.512 0.593
1207 319 1 0.549 0.02061 0.510 0.591
1211 318 1 0.548 0.02061 0.509 0.590
1215 317 1 0.546 0.02062 0.507 0.588
1219 316 1 0.544 0.02063 0.505 0.586
1262 315 1 0.543 0.02064 0.504 0.585
1275 314 1 0.541 0.02064 0.502 0.583
1295 313 1 0.539 0.02065 0.500 0.581
1298 312 1 0.537 0.02065 0.498 0.579
1325 311 1 0.536 0.02066 0.497 0.578
1399 310 1 0.534 0.02067 0.495 0.576
1405 309 1 0.532 0.02067 0.493 0.574
1434 308 1 0.530 0.02068 0.491 0.573
1471 307 1 0.529 0.02068 0.490 0.571
1509 306 1 0.527 0.02068 0.488 0.569
1551 303 1 0.525 0.02069 0.486 0.567
1561 302 1 0.523 0.02069 0.484 0.566
1564 301 1 0.522 0.02070 0.483 0.564
1568 300 1 0.520 0.02070 0.481 0.562
1589 299 1 0.518 0.02071 0.479 0.560
1606 298 1 0.517 0.02071 0.478 0.559
1647 297 1 0.515 0.02071 0.476 0.557
1652 296 1 0.513 0.02072 0.474 0.555
1687 295 1 0.511 0.02072 0.472 0.554
1709 294 1 0.510 0.02072 0.471 0.552
1768 293 1 0.508 0.02072 0.469 0.550
1829 288 1 0.506 0.02073 0.467 0.548
1839 285 1 0.504 0.02073 0.465 0.547
1850 284 1 0.503 0.02073 0.463 0.545
1851 283 1 0.501 0.02073 0.462 0.543
1885 278 1 0.499 0.02074 0.460 0.541
1895 277 1 0.497 0.02074 0.458 0.540
1918 274 1 0.495 0.02074 0.456 0.538
1932 272 1 0.494 0.02075 0.454 0.536
1976 265 1 0.492 0.02075 0.453 0.534
2012 256 1 0.490 0.02076 0.451 0.532
2018 255 1 0.488 0.02077 0.449 0.530
2023 254 1 0.486 0.02077 0.447 0.528
2067 245 1 0.484 0.02078 0.445 0.526
2079 242 1 0.482 0.02079 0.443 0.524
2128 228 1 0.480 0.02081 0.441 0.522
2152 213 1 0.478 0.02083 0.438 0.520
2171 206 1 0.475 0.02086 0.436 0.518
2231 180 1 0.473 0.02091 0.433 0.515
2458 115 1 0.468 0.02113 0.429 0.512
2593 72 1 0.462 0.02182 0.421 0.507
2683 58 1 0.454 0.02285 0.411 0.501
2718 45 1 0.444 0.02447 0.398 0.495
2910 16 1 0.416 0.03532 0.352 0.491
rx=Lev+5FU
time n.risk n.event survival std.err lower 95% CI upper 95% CI
8 578 1 0.998 0.00173 0.995 1.000
9 577 1 0.997 0.00244 0.992 1.000
23 576 1 0.995 0.00299 0.989 1.000
34 574 1 0.993 0.00345 0.986 1.000
40 573 1 0.991 0.00385 0.984 0.999
45 572 1 0.990 0.00422 0.981 0.998
49 570 1 0.988 0.00456 0.979 0.997
52 569 1 0.986 0.00487 0.977 0.996
63 568 1 0.984 0.00516 0.974 0.995
68 567 1 0.983 0.00543 0.972 0.993
79 566 1 0.981 0.00569 0.970 0.992
86 565 1 0.979 0.00594 0.968 0.991
91 564 1 0.977 0.00618 0.965 0.990
101 563 1 0.976 0.00641 0.963 0.988
116 562 1 0.974 0.00663 0.961 0.987
127 561 1 0.972 0.00684 0.959 0.986
132 560 1 0.971 0.00705 0.957 0.984
138 559 1 0.969 0.00724 0.955 0.983
141 558 1 0.967 0.00744 0.953 0.982
144 557 1 0.965 0.00762 0.950 0.980
146 556 1 0.964 0.00780 0.948 0.979
154 555 1 0.962 0.00798 0.946 0.978
157 554 1 0.960 0.00815 0.944 0.976
160 553 1 0.958 0.00832 0.942 0.975
161 552 1 0.957 0.00849 0.940 0.973
165 551 1 0.955 0.00865 0.938 0.972
185 550 1 0.953 0.00880 0.936 0.971
198 549 1 0.951 0.00896 0.934 0.969
199 548 1 0.950 0.00911 0.932 0.968
205 547 1 0.948 0.00925 0.930 0.966
208 546 1 0.946 0.00940 0.928 0.965
215 545 1 0.944 0.00954 0.926 0.963
218 544 1 0.943 0.00968 0.924 0.962
237 543 1 0.941 0.00982 0.922 0.960
242 542 1 0.939 0.00995 0.920 0.959
245 541 1 0.938 0.01008 0.918 0.957
248 540 1 0.936 0.01021 0.916 0.956
251 539 1 0.934 0.01034 0.914 0.955
252 538 1 0.932 0.01046 0.912 0.953
255 537 1 0.931 0.01059 0.910 0.952
256 536 1 0.929 0.01071 0.908 0.950
260 535 1 0.927 0.01083 0.906 0.949
261 534 1 0.925 0.01095 0.904 0.947
269 533 1 0.924 0.01106 0.902 0.946
271 532 1 0.922 0.01118 0.900 0.944
274 531 1 0.920 0.01129 0.898 0.943
276 530 1 0.918 0.01140 0.896 0.941
279 529 1 0.917 0.01151 0.894 0.940
283 528 1 0.915 0.01162 0.892 0.938
285 527 1 0.913 0.01173 0.891 0.936
293 526 1 0.911 0.01183 0.889 0.935
296 525 1 0.910 0.01194 0.887 0.933
302 524 1 0.908 0.01204 0.885 0.932
303 523 1 0.906 0.01214 0.883 0.930
304 522 1 0.905 0.01224 0.881 0.929
315 521 1 0.903 0.01234 0.879 0.927
322 520 2 0.899 0.01254 0.875 0.924
324 518 1 0.898 0.01263 0.873 0.923
326 517 1 0.896 0.01273 0.871 0.921
328 515 1 0.894 0.01282 0.869 0.920
329 514 1 0.892 0.01291 0.867 0.918
336 513 1 0.891 0.01300 0.866 0.916
340 512 1 0.889 0.01309 0.864 0.915
355 510 1 0.887 0.01318 0.862 0.913
360 508 1 0.885 0.01327 0.860 0.912
363 507 1 0.884 0.01336 0.858 0.910
380 506 1 0.882 0.01345 0.856 0.909
386 505 1 0.880 0.01353 0.854 0.907
389 504 1 0.878 0.01362 0.852 0.906
392 503 1 0.877 0.01370 0.850 0.904
393 502 1 0.875 0.01379 0.848 0.902
400 501 1 0.873 0.01387 0.846 0.901
405 500 1 0.871 0.01395 0.845 0.899
408 499 1 0.870 0.01403 0.843 0.898
415 498 1 0.868 0.01411 0.841 0.896
422 497 1 0.866 0.01419 0.839 0.894
428 496 1 0.864 0.01427 0.837 0.893
430 495 1 0.863 0.01435 0.835 0.891
431 494 1 0.861 0.01443 0.833 0.890
434 493 1 0.859 0.01450 0.831 0.888
441 492 1 0.857 0.01458 0.829 0.887
443 491 1 0.856 0.01465 0.827 0.885
448 490 2 0.852 0.01480 0.824 0.882
449 488 1 0.850 0.01487 0.822 0.880
454 487 2 0.847 0.01501 0.818 0.877
458 485 1 0.845 0.01508 0.816 0.875
460 484 1 0.843 0.01515 0.814 0.874
466 483 2 0.840 0.01529 0.811 0.871
484 481 1 0.838 0.01536 0.809 0.869
485 480 1 0.837 0.01542 0.807 0.867
491 479 1 0.835 0.01549 0.805 0.866
497 478 1 0.833 0.01556 0.803 0.864
498 477 1 0.831 0.01562 0.801 0.862
503 476 1 0.830 0.01569 0.799 0.861
510 475 1 0.828 0.01575 0.797 0.859
526 474 1 0.826 0.01581 0.796 0.858
529 473 1 0.824 0.01588 0.794 0.856
536 472 1 0.823 0.01594 0.792 0.854
543 471 1 0.821 0.01600 0.790 0.853
550 470 1 0.819 0.01606 0.788 0.851
554 469 2 0.816 0.01618 0.784 0.848
576 467 1 0.814 0.01624 0.783 0.846
578 466 2 0.810 0.01636 0.779 0.843
580 464 1 0.809 0.01642 0.777 0.841
591 463 1 0.807 0.01647 0.775 0.840
592 462 1 0.805 0.01653 0.773 0.838
593 461 1 0.803 0.01659 0.771 0.837
594 460 1 0.802 0.01664 0.770 0.835
601 459 1 0.800 0.01670 0.768 0.833
602 458 1 0.798 0.01675 0.766 0.832
603 457 1 0.796 0.01680 0.764 0.830
604 456 1 0.795 0.01686 0.762 0.828
609 455 1 0.793 0.01691 0.760 0.827
614 454 1 0.791 0.01696 0.759 0.825
616 453 2 0.788 0.01707 0.755 0.822
617 451 1 0.786 0.01712 0.753 0.820
622 450 1 0.784 0.01717 0.751 0.819
636 449 1 0.782 0.01722 0.749 0.817
641 448 1 0.781 0.01727 0.748 0.815
642 447 1 0.779 0.01732 0.746 0.814
649 446 1 0.777 0.01737 0.744 0.812
657 445 1 0.775 0.01742 0.742 0.810
666 444 1 0.774 0.01747 0.740 0.809
674 443 1 0.772 0.01751 0.738 0.807
683 442 1 0.770 0.01756 0.736 0.805
692 441 2 0.767 0.01765 0.733 0.802
693 439 1 0.765 0.01770 0.731 0.800
696 438 1 0.763 0.01775 0.729 0.799
701 437 1 0.761 0.01779 0.727 0.797
711 436 1 0.760 0.01783 0.726 0.795
712 435 1 0.758 0.01788 0.724 0.794
736 434 1 0.756 0.01792 0.722 0.792
765 433 1 0.754 0.01797 0.720 0.790
802 432 2 0.751 0.01805 0.716 0.787
805 430 1 0.749 0.01809 0.715 0.786
806 429 1 0.747 0.01814 0.713 0.784
811 428 1 0.746 0.01818 0.711 0.782
827 427 1 0.744 0.01822 0.709 0.781
844 426 1 0.742 0.01826 0.707 0.779
849 424 1 0.740 0.01830 0.705 0.777
853 423 1 0.739 0.01834 0.704 0.776
862 422 1 0.737 0.01838 0.702 0.774
884 421 1 0.735 0.01842 0.700 0.772
887 419 2 0.732 0.01850 0.696 0.769
904 417 1 0.730 0.01854 0.694 0.767
905 416 1 0.728 0.01858 0.693 0.766
911 415 1 0.726 0.01861 0.691 0.764
916 414 1 0.725 0.01865 0.689 0.762
918 413 1 0.723 0.01869 0.687 0.760
934 412 1 0.721 0.01873 0.685 0.759
936 411 1 0.719 0.01876 0.684 0.757
961 410 1 0.718 0.01880 0.682 0.755
968 409 1 0.716 0.01883 0.680 0.754
977 408 1 0.714 0.01887 0.678 0.752
993 407 1 0.712 0.01890 0.676 0.750
1022 406 1 0.711 0.01894 0.674 0.749
1024 405 1 0.709 0.01897 0.673 0.747
1025 404 1 0.707 0.01901 0.671 0.745
1032 403 1 0.705 0.01904 0.669 0.744
1037 402 1 0.704 0.01907 0.667 0.742
1122 401 1 0.702 0.01911 0.665 0.740
1138 400 1 0.700 0.01914 0.664 0.739
1142 399 1 0.698 0.01917 0.662 0.737
1145 398 1 0.697 0.01920 0.660 0.735
1151 397 1 0.695 0.01924 0.658 0.734
1159 396 1 0.693 0.01927 0.656 0.732
1193 395 1 0.691 0.01930 0.655 0.730
1201 394 1 0.690 0.01933 0.653 0.729
1212 393 1 0.688 0.01936 0.651 0.727
1233 392 1 0.686 0.01939 0.649 0.725
1246 391 1 0.684 0.01942 0.647 0.723
1276 390 2 0.681 0.01948 0.644 0.720
1277 388 1 0.679 0.01951 0.642 0.718
1279 387 1 0.677 0.01953 0.640 0.717
1302 384 1 0.676 0.01956 0.638 0.715
1306 383 1 0.674 0.01959 0.636 0.713
1329 382 1 0.672 0.01962 0.635 0.712
1365 381 1 0.670 0.01965 0.633 0.710
1387 380 1 0.668 0.01967 0.631 0.708
1388 379 1 0.667 0.01970 0.629 0.706
1424 376 1 0.665 0.01973 0.627 0.705
1439 375 1 0.663 0.01975 0.626 0.703
1446 374 1 0.661 0.01978 0.624 0.701
1488 371 1 0.660 0.01981 0.622 0.700
1495 370 1 0.658 0.01983 0.620 0.698
1511 369 1 0.656 0.01986 0.618 0.696
1521 368 1 0.654 0.01989 0.616 0.694
1550 367 1 0.652 0.01991 0.615 0.693
1607 365 1 0.651 0.01994 0.613 0.691
1620 363 1 0.649 0.01996 0.611 0.689
1637 362 1 0.647 0.01999 0.609 0.687
1644 361 1 0.645 0.02001 0.607 0.686
1668 360 2 0.642 0.02006 0.604 0.682
1671 358 1 0.640 0.02009 0.602 0.681
1723 357 1 0.638 0.02011 0.600 0.679
1743 356 1 0.636 0.02013 0.598 0.677
1752 355 1 0.635 0.02016 0.596 0.675
1767 354 1 0.633 0.02018 0.594 0.674
1783 353 1 0.631 0.02020 0.593 0.672
1786 352 1 0.629 0.02022 0.591 0.670
1798 351 1 0.627 0.02024 0.589 0.668
1812 348 1 0.626 0.02027 0.587 0.667
1831 339 1 0.624 0.02029 0.585 0.665
1856 335 1 0.622 0.02031 0.583 0.663
1876 329 1 0.620 0.02034 0.581 0.661
1995 313 1 0.618 0.02037 0.579 0.659
2021 304 1 0.616 0.02041 0.577 0.657
2028 301 1 0.614 0.02044 0.575 0.655
2031 298 1 0.612 0.02048 0.573 0.653
2052 291 1 0.610 0.02051 0.571 0.651
2074 283 1 0.608 0.02055 0.569 0.649
2127 265 1 0.605 0.02060 0.566 0.647
2174 247 1 0.603 0.02067 0.564 0.645
2197 229 1 0.600 0.02074 0.561 0.642
2318 184 1 0.597 0.02088 0.557 0.639
2482 135 1 0.593 0.02119 0.552 0.636
2542 99 1 0.587 0.02181 0.545 0.631
2725 64 1 0.577 0.02331 0.533 0.625Logrank test
구간별로 예상/기대 발생 수 계산 후 합쳐서 카이제곱검정
Call:
survdiff(formula = Surv(time, status) ~ rx, data = colon)
N Observed Expected (O-E)^2/E (O-E)^2/V
rx=Obs 610 336 288 8.12 12.11
rx=Lev 588 310 279 3.37 4.95
rx=Lev+5FU 578 230 309 20.20 31.28
Chisq= 31.8 on 2 degrees of freedom, p= 1e-07 각 구간들의 결과를 합친다?
- 구간별 발생 양상이 비슷하다는 가정 (비례위험(proportional hazards) 가정)
Cox model
Hazard function: \(h(t)\)
- \(t\) 까진 생존하고 \(t\) 직후에 사망할 가능성
Cox model: Hazard Ratio(HR) 을 평가
\[ \begin{aligned} h(t) &= \exp({\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots}) \\\\ &= h_0(t) \exp({\beta_1 X_1 + \beta_2 X_2 + \cdots}) \end{aligned} \] \(X_1\) 1 증가할 때 \(h(t)\) 는 \(\exp(\beta_1)\) 배 증가. 즉
\[\text{HR} = \exp{(\beta_1)}\]
특징
Kaplan-meier 와 마찬가지로 구간별로 통계량을 계산.
- 구간별 양상 비슷하다는 비례위험가정
Time independent HR: 시간은 \(h_0(t)\) 에만.
- 모형이 심플: HR 값이 시간에 상관없이 일정함
- Time dependent cox 도 가능.
\(h_0(t)\) 는 구하지 않는다. 계산 간단해지는 장점
- Cox 가 준모수(semi-parametric) 방법이라고 불리는 이유
- 예측모형 만들땐 문제. \(h_0(t)\) 를 따로 얻어야 함.
cox 2
Call:
coxph(formula = Surv(time, status) ~ sex + age + rx, data = colon)
n= 1776, number of events= 876
coef exp(coef) se(coef) z Pr(>|z|)
sex -0.072247 0.930301 0.067646 -1.068 0.286
age -0.001144 0.998857 0.002876 -0.398 0.691
rxLev -0.049102 0.952084 0.078788 -0.623 0.533
rxLev+5FU -0.452209 0.636221 0.085709 -5.276 1.32e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
sex 0.9303 1.075 0.8148 1.0622
age 0.9989 1.001 0.9932 1.0045
rxLev 0.9521 1.050 0.8159 1.1111
rxLev+5FU 0.6362 1.572 0.5378 0.7526
Concordance= 0.55 (se = 0.01 )
Likelihood ratio test= 34.51 on 4 df, p=6e-07
Wald test = 32.59 on 4 df, p=1e-06
Score (logrank) test = 33.07 on 4 df, p=1e-06cox 3
생존분석: 비례위험가정
트렌드 일정하다는 가정: 생존곡선 겹치면 안됨
- 비례위험가정 test까진 필요없음: 그림으로 확인
Landmark-analysis
시간 나눠 따로 분석
Time-dependent cox
Time-dependent cox 2
Call:
coxph(formula = Surv(tstart, time, status) ~ trt + prior + karno:strata(tgroup),
data = vet2)
n= 225, number of events= 128
coef exp(coef) se(coef) z Pr(>|z|)
trt -0.011025 0.989035 0.189062 -0.058 0.953
prior -0.006107 0.993912 0.020355 -0.300 0.764
karno:strata(tgroup)tgroup=1 -0.048755 0.952414 0.006222 -7.836 4.64e-15 ***
karno:strata(tgroup)tgroup=2 0.008050 1.008083 0.012823 0.628 0.530
karno:strata(tgroup)tgroup=3 -0.008349 0.991686 0.014620 -0.571 0.568
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
trt 0.9890 1.011 0.6828 1.4327
prior 0.9939 1.006 0.9550 1.0344
karno:strata(tgroup)tgroup=1 0.9524 1.050 0.9409 0.9641
karno:strata(tgroup)tgroup=2 1.0081 0.992 0.9831 1.0337
karno:strata(tgroup)tgroup=3 0.9917 1.008 0.9637 1.0205
Concordance= 0.725 (se = 0.024 )
Likelihood ratio test= 63.04 on 5 df, p=3e-12
Wald test = 63.7 on 5 df, p=2e-12
Score (logrank) test = 71.33 on 5 df, p=5e-14Time-dependent covariate
생존분석 모든 변수는 Index date 이전에 측정해야
- 예) F/U lab, medication
Time-dependent covariate 다루려면 이를 고려한 cox 필요
Executive Summary
| Dafault | Repeated measure | Survey | |
|---|---|---|---|
| Continuous | linear regression | GEE | Survey GLM |
| Event | GLM (logistic) | GEE | Survey GLM |
| Time & Event | Cox | marginal Cox | Survey Cox |
| 0,1,2,3 (rare event) | GLM (poisson) | GEE | Survey GLM |