???? ?????COX??

1
?????????COX??
???????? ???????
2
??? ????

• ???,??????,??????? (Survival
  Analysis)?????????????????????,???????,??????,????
  ????????????,?????????
• ?????(survival time)?????????(???,??,?????????
  ???)??????????????

3
??????????????? 1.?????(censored data)
????????????????,???????????????? (1)?????????????
---?? (2)??????????????---?? ???????t,? 4
??????4?? ????????????????,?????,?????????
?,????????
4
2. ???????????????? ??????,??????????,????
??????? ???????????????????????????????
5

• ????????????????,????(????)????????????????????
  ?????????????????????????????????????????????????
  ?????????????????????????????????????,??????????
  ????????????????????????
• ???????????(?????????), ??????????????????,???
  ??,????????????????Cox??????,??,Cox
  ????????????????

6
???????

• ??
• (1)??????,??????---????
• (2)??(??????????????)
• ????---????
• ?????????---????
• ?????????2??????????????????(??1????,?0????4
  ?4,1??4?4,0??)?
• (3)???---?????????????

7
??? ?????????????

• ?. ???(Survival Rate)
• ????????????,????????????????t???,?S(t) ??
  s(t)P(T?t)
• ?5???? s(5)P(T?5)
• ???t????,S(t)????????????????,
  ?????????,???????,??????????????,??????????

8
(No Transcript)
9

• 1.2 ??????
• (Probability Density Function)
• ???????,??f(t),????
• f(t)lim (???????(t,t?t)?????/?t)
• ?????????????t???,f(t) ???????????????,???????
  ??????????????????????,???????,?????????,?????????
  ,?????,???????

10

• 1.3 ????(Hazard Function)
• ?h(t)??,????
• h(t)lim(???t?????????(t,?t)???/?t)
• ????h(t)?,????????t,????,?????????????????????
  h(t)???????t??????t????????????????????????t????,h
  (t)?????????,???,???????????????,??????,??????????
  ?(???)??????

11
????????? ??, ???????? ??,
?????? ??, ?????????? ???,??????
12
?????? (1)??????? (2)?????????? (3)??????
???????
13
??? ????????

• ???S(t)?????????????????????,?????????,?,?????
  ?????,?????????????????????,?????????????????,????
  ??????????????,??????????????????
  1?????,????????????????????

14
????????????????S(t)???t???,??????????????
?? ????????????????????,???????,??????????
???
15
?. ?????(Product-Limit Method)

• ???????PL?,???????Kaplan?Meier?1958??????,
  ?????Kaplan-Meier?, ???????????????????????????
• ?S(t)??t?????,s(ti/ti-1)????ti-1????
  ti??????,??s(1),s(2)??????,??????,?s(2/1)???????,?
  ????????,?????????
• S(2)S(1)S(2/1),????
• S(ti)S(ti-1)S(ti/ti-1)

16

• ?22.1 ???????(???)???(???)??????????,
  ????????????,??""???????,?????,?""
  ???????,?????,??????????????
• ??? 10,2,12,13,18,6,19,26,9,8,6,43,9,4,31
  ,24
• ??? 2,13,7,11,6,1,11,3,17,7

17
????????????? ------------------------------------
- ?? ?? ???? ???? ????? ??? di
?di/ni(ni-di)???? ti si ni di
(ni-di)/ni ??S(ti)ni(ni-di) ???? ?
? ? ? ? ? ? ?
??v? ------------------------------------- 2
? 4 ? 15 1 0.9333 0.9333
0.004762 0.004762 0.0644 6 ? 6 ? 8
? 9 ? 11 1 0.9090 0.8485
0.009091 0.013853 0.0999 9 ? 10 ? 9
1 0.8889 0.7542 0.013889 0.027742
0.1256 12 ? 13 ? 7 1
0.8571 0.6465 0.023810 0.051551 0.1468 18
? 6 1 0.8333 0.5387 0.033333
0.084885 0.1570 19 ? 24 ? 4 1
0.7500 0.4040 0.083333 0.168218 0.1657
26 ? 3 1 0.6667 0.2694
0.166667 0.334885 0.1559 31 ? 2
1 0.5000 0.1347 0.500000 0.834885
0.1231 43 ? -----------------------------------
--
18
(No Transcript)
19
?. ????(Life Table Method)

• ???????????, ???????????,????????????
  ,????????????,?????????????

20
???1946?1?1??1951?12?31????126?????,??????22.
2,???????????? ?22.2 126??????????????
-------------------------------------------------
---- ??(?) ???? ???? ???? ???? ???? ????? ?????
di ?di/ni(ni-di)???? ti n'i di
ui wi ni S(ti/ti-1) S(ti)
ni(ni-di) ???? ? ? ?
? ? ? ? ?
? ? ??v? --------------------------
--------------------------- 0- 126
47 4 15 116.5 0.5966
0.5966 5.80510-3 5.80510-3 0.0455 1-
60 5 6 11 51.5
0.9029 0.5386 2.08810-3 7.89310-3 0.0479
2- 38 2 0 15 30.5
0.9344 0.5033 2.30110-3 0.0102
0.0508 3- 21 2 2 7
16.5 0.8788 0.4423 8.35910-3
0.0186 0.0602 4- 10 0 0
6 7.0 1.0000 0.4423 0
0.0186 0.0602 5- 4 0
0 4 2.0 1.0000 0.4423
0 0.0186 0.0602 ----------------------
-------------------------------
21
????????????? 1.?????????(?)0-,1-,2-,??,????
????,??,??, ???????22.2?1-5?????,?????????,????ni
?????? n'in'i-1-di-ui-wi
?????,????1-,? n'2126-47-4-1560
2.??????????????,(??????)ni nin'i-ui/2-wi/2
????????????????,??????????,?????
?? ????,n1126-4/2-15/2116.5
3.???(22.5)(22.6)(22.7)????????S(ti/ti-1),?????s(
ti)?????? ????????22.2?,?7,8,11?,??9,10??????1
1????? ?????0--? S(ti/ti-1)(116.5-47)/11
6.50.5966 S(ti)10.59660.5966
SE(S(ti))0.5966v5.80510-30.0455
??????????0.59660.0455 ???????????0.53860.04
79 ??????????????????????????????,??,?????????
?????????1?????,??????????
22
??? ??????

• ???????????????,??????????????????,????????
• H0???????????????
• H1????????????????
• ?????????Logrank?,
• ??Wilcoxon?,?Cox-Mantel??????H0?,????????????

23
????????,??????????????????????? (1)?????
?? ???S1?S2,????V1?V2 ?????
?2 (S1-S2)2 / (V1V2) df1
24
(2)??????????? ???S1,S2?S3,???V1,V2?V3
????? ??W11/V1, W21/V2,W31/V3
??????? S(W1S1W2S2W3S3) /(W1W2W3)
?2 W1(S1-S)2 W2(S2-S)2 W3(S3-S)2
df3-1
25
(No Transcript)
26
3.1 Logrank??(Log Rank Test)
????????????,????????,??????Savage???
Logrank????????? 1.???????????,??????,?????i1,
???????????????,???????????????????m1,m2,???nm1m
2? ?22.1?????????????????22.3??1,2?? 2.???????
?????????i2(???????),???22.3??3?? 3.????????i3(??2
2.3??4?)? 4.???????(????)?????????????r,??????????
?????r???i3???i1?????? rn-i11
(22.10) ??????7(?)???r??? r26-9118 ??,????5??
27
5.???i1?logrank??,???logrank???W,???? ???i1
????i3??????W?? i3 W? 1/rj-1
(22.11) j1 ???i1?????,??????????????
??,??????i3?i31???W? i3 W? 1/rj
(22.12) j1 ???,?????????????????,?W
0,?????????????????????,???????W?? ???22.3??1????
???(???)??(22.11)?? W1/26-1-0.96 ?2,3?????
???(??),?????i31?2???(22.12)?,?
W1/260.04 ?4???????,??(22.11)?
W1/261/23-1-0.92 ???(??22.3?6?)
28
6.????????(??????,???i2)?logrank?????T T?W
i2 ??????????i2??????????????? E(T)m1/n
?W V(T)m1m2/n(n-1)?(W-E(T)/m1)2 (22.14)????
??????????,m1???????????,(??????m116),n?????
ZT-E(T)/vV(T) Z????????,???Z0.051.96,Z0.01
2.58??????? ?????T3.822,E(T)0.440210-6, V(T
)3.1755,Z2.145,?Plt0.05, ??H0,??????????????
29
3.2 Cox-Mantel??(Cox-Mantel Test)

• ????Savage??(Generalized Savage
  Test),???????????????????22.1?????????????,???????
  ?????A?,?????B??????H0,H1???

30
(No Transcript)
31
????? 1.???????????????,????????????????,?????????
???????????(???)?????(??22.4?1-3?) 2.??A,B????????
?????????????n1i,d1i,n2i,d 2i??(??22.4??4-7?)? 3.?
????????????????Pi Pi(d1id2i)/(n1in2i)
(22.16) 4.????????(???B?)??
??????????????,????????????????? E(d2i)n2iPi
(22.17) ???22.4?8.9??? 5.??????(???
B?)???????????? E(?d2i)?n2iPi
(22.18) V(?d2i)?n1in2i/(n1in2i-1)Pi(1-Pi)
(22.19) ??????????????Z
Z?d2i-E(?d2i)/vV(?d2i)
(22.20) ??Z0.051.96,Z0.012.58,??????? ??????? ?d
2i7, E(?d2i)3.212284, V(?d2i)1.916190,
Z2.7363 ? Plt0.01,??H0,??????????????
32
3.3 ??Wilcoxon??(Generalized Wilcoxon Test)

• ???Breslow???(Breslow Test),?????????????????,
  ?H0,H1????????22.1??????????

33
(No Transcript)
34
1.???????????????,???????????????,????????????????
?????(???)?????(??22.5??1-3?)? 2.?????????????????
?(??????)S(ti),?????4? 3.????????????Ui? Ui
S(ti-1)S(ti)-1 ???????? S(ti)-1
???????? (22.21) ??S(0)1 ?????
???Ti1?????(??),?U110.9615-10.9615, ?2,3??????
??? U2U30.9615-1-0.0385 ?4?????????
U40.96150.9197-10.8812 ???? 4.???????????????,
????A?,????????6??, T?Ui(A)
(22.22) T?????0,??? V(T)m1m2?U2/(m1m2)(m1m2
-1) (22.23) (22.23)???U2??????????U????,
ZT/vV(T) (22.24) Z????????,???Z0.05
1.96,Z0.012.58??????? ?????T2.8712,?U26.6559,V
(T)1.6384,Z2.243,Plt0.05,??H0, ????????????????
35
??? ??????????SAS?? ?LIFETEST??
36
??? COX??
COX??????????(?????,??????)??????????,????
???? h(t,x)h0(t)exp(?1x1 ?2x2
?mxm ) X1,X2,,Xm????
?1 ,?2,,?m?????,???????? ?I
gt0???????????,????????? ?I lt0???????????,?????
????
37
h(t,x)h0(t)exp(?1x1 ?2x2 ?mxm
) h0(t)???????,???????X1,X2,,Xm??0??????
?????,??????? h(t,x)????????X????????,??h0(t
)???,??????????????(proportional hazard model)
COX????????????,?????????
38
COX????? ?LOGISTIC???? (1)????
??????(???)?????????????
?????????????,???,??????????????,????????????????
39
(2)???????????????,??????????(????)
????Xi???????bi, ????Xi?????????exp(bi)
?Xi??????,???(1??,0???)
exp(bi)?????????????? ?????(????)
40
?Xi??????,??????,?0,1,2,3,4?????
exp(bi)????????,????????,???3?????0??????????
exp(3bi) ?Xi??????,???(?)
exp(bi)???????,???????? ?60????35??????????
?exp(25bi)
41
(3)?????????????????? ???????????bi
??????,????????????????? (4) ???????????
???XL?XK???????????,???????XLK XLXK
,???????bLK??,?XL?XK??????????
42
?18.4
?? -----------------------------------------------
--------------------------------- ??
???? P?
????? --------------------------------------------
------------------------------------ ??????
-0.7169 0.0469
0.488 ?????? -1.0077 0.0068
0.365 ??
0.3585 0.0007 1.431
TNM?? 0.1603 0.0003
1.174 ???? 0.7019
0.0385 2.018 ??????
0.2703 0.0001 1.310
-------------------------------------------------
-------------------------------
43
??????,TNM????,????????,??????
?????????????? ??????????????????
COX????? h(t,x)h0(t)exp(-0.7169X3b -
1.0077X3c
0.3585X40.1603X50.7019X8c
0.2703X9)