Survival Analysis for Randomized Clinical Trials II Cox Regression. Ziad Taib Biostatistics AstraZeneca February 26, 2008

Transcription

1 Survval alyss for Raomz Clcal rals II Cox Rgrsso a ab osascs sraca Fbruary 6, 8

2 la Irouco o proporoal azar mol H aral lkloo Comparg wo groups umrcal xampl Comparso w log-rak s

3 mol xp z + + k k z Ursag basl azar xp[ z z + + z k k k z ]

4 Cox Rgrsso Mol roporoal azar. No spcfc srbuoal assumpos bu clus svral mpora paramrc mols as spcal cass. aral lkloo smao Sm-paramrc aur. Easy mplmao SS procur HREG. aramrc approacs ar a alrav, bu y rqur srogr assumpos abou.

5 Cox proporoal azars mol, cou Ca al bo couous a cagorcal prcor varabls k: logsc, lar rgrsso Wou kowg basl azar o, ca sll calcula coffcs for ac covara, a rfor azar rao ssums mulplcav rsk s s proporoal azar assumpo Ca b compsa par w raco rms

6 Cox Rgrsso I 97 Cox suggs amol for survval aa a woul mak possbl o ak covaras o accou. Up o was cusomary o scrs couos varabls a bul subgroups. Cox a was o mol azar ra fuco < + wr s o b ursoo as a sy.. a probably by m u. Mulpl by m w g a probably. k of aalogy w sp as sac by m u. Mulpl by m w g sac.

7 Cox suggso s o mol usg,..., ; paramr vcor;,..., covara vcor. k k wr ac paramr s a masur of mporac of corrspog varabl. cosquc of s s a wo vuals w ffr covara valus wll av azar ra fucos wc ffr by a mulplcav rm wc s sam for all valus of.

8 covaras ca b scr, couous or v cagorcal. I s also possbl o grals mol o allow m varyg covaras. C C. ;,; ; -

9 Exampl ssum w av a suao w o covara a aks wo ffr valus a. s s cas w w ws o compar wo rams k ; ; ; ; ;.

10 From fo w s mmaly a S [ S ] u u u u u u Comms O of avaags of H mol s a w ca sma paramrs a mak comparsos bw rams wou avg o sma basl. basl ca b rpr as cas corrspog o a vual w covara valus zro. pa sm-paramrc s u o fac a w o o mol xplcly. Usual lkloo ory os o apply.

11 aral lkloo ky o ursag paral lkloo s followg cooal probably: pa s ur o pa s urg pa s ur /o pa s urg pa s ur / Σ o pa s ur pa s urg pa s urg pa s ur / Σ pa s urg

12 rfor pa s ur o pa s urg * L R # of pas a rsk a m. alv a ur obsrvao a. L furr RR. + < pa s ur

13 From s follows a w ca bas our frc o lkloo fuco **, L L L R R R pa s ur o pa s urg

14 Comms If w ra s xprsso as a orary lkloo fuco w ca g paramr smas. Du o o-lary w cao op fg xplc formulas for sams. W av o rly o umrcal maxmsao mos suc as Nwo-Rapso. O op of usual scor fucos, w ca cosr sco rvav o compu formao a us s o sma covarac marx of paramr vcor ˆ. s for orary ML mo, smas ar asympocally ubas a a mulvara ormal w ma a covarac marx a ca b sma usg a vrs of covarac marx.

15 Maxmum Lkloo smas L L sa for lkloo fuco a for som paramr vcor of rs., maxmum lkloo smas ar fou by solvg s of quaos log L,,,..., Df formao marx, I o av lms log L I E. I ca b sow a sma o av a asympocally ormal srbuo w mas a varac/covarac marx [ ] I

16 W ak logl l R l Log LogL LogL

17 Scor a formao k k R R k R R R R k k k k l l U wo rvavs ca b us o g paramr smas a formao. O ffculy w avo so far s occurc of s.

18 s # of falurs a. s sum of covara valus for pas wo a m. Noc a sum arbrary orr mus b mpos o s. R l Log LogL s k k R R k R R R R k k k l s l

19 umrcal Exampl IME O RELIEF OF ICH SYMOMS FOR IENS USING SNDRD ND EXERIMENL CREM

20 Wa abou a -s? ma ffrc m o cur of. ays s o sascally sgfca bw wo groups.

21 Usg SS ROC HREG ; MODEL RELIEFIME * SUS DRUG ;

22 No a sma of DRUG varabl s w a p valu of.346. gav sg cas a gav assocao bw azar of bg cur a DRUG varabl. u varabl DRUG s co for w rug a co for saar rug. rfor azar of bg cur s lowr group gv saar rug. s s a awkwar bu accura way of sayg a w rug s o prouc a cur mor quckly a saar rug. ma m o cur s lowr group gv w rug. r s a vrs rlaosp bw avrag m o a v a azar of a v.

23 rao of azars s gv by ˆ ˆ.3396 ˆ.6 ac m po cur ra of saar rug s abou 5% of a of w rug. u mor posvly w mg sa a cur ra s.8 ms gr group gv xprmal cram compar o group gv saar cram.

24 Comparg wo groups gral L b umbr a rsk group a m b umbr a rsk group a m I s U ; χ I U Q scor s

25 umrcal xampl ssum a a suy amg a comparg wo groups rsul followg aa U-.55 I ˆ.599;. ;,; ; 599 ˆ.. ram rucs azar ra o almos ¼ compar w placbo.

26 Log-rak vs scor s HREG wo mos gv vry smlar aswrs, wy? O o a scor s ca b bas o sasc ; χ I U Q ; # a rsk bo groups; # vs bo groups; ; ; s I U + +

27 CURE NO CURE DRUG - CONROL - OL -

28 Compar s o Log-rak sasc [ ] χ U Var U Q H M [ ] [ ] - D ar ; N M D V D N E σ µ k U µ [ ] [ ] ar ; - V σ E µ ; χ H M - Q

29 ; χ Q ; χ H M - Q LOG-RNK ES SCORE ES.. o s a wo formulas ar cal. I gral wo formulas ar approxmaly sam w s larg. For xampl a cas w s r s a slg ffrc - Q QM-H6.793

30 Gralzaos of Cox rgrsso. m p covaras. Srafcao 3. Gral lk fuco 4. Lkloo rao ss 5. Sampl sz rmao 6. Gooss of f 7. SS

31 Rfrcs Cox & Oaks 984 alyss of survval aa. Capma & Hall. Flmg & Harrgo 99 Coug procsss a survval aalyss. Wly & Sos. llso 995. Survval aalyss usg SS Sysm. SS Isu. rau & Grambsc Molg Survval Daa. Sprgr. Hougaar alyss of Mulvara survval aa. Sprgr.

32 Qusos or Comms?