A primary objective of a phase II trial is to screen for antitumor activity; agents which are found to have substantial antitumor activity and an

Transcription

1 SURVIVAL ANALYSIS

2 A prmary objctv of a phas II tral s to scrn for anttumor actvty; agnts whch ar found to hav substantal anttumor actvty and an approprat spctrum of toxcty ar lkly ncorporatd nto combnatons to b valuatd for patnt bnft n controlld phas III trals. Th scondary am s oftn concrnd patnt survvorshp.

3 Thrfor, n addton to stmatng th rspons rat and prformng th Ch-squar tst (on-arm or two-arm trals); othr data analyss for phas II trals may nvolv: (1) Logstc Rgrsson, & (2) Kaplan-Mr Survval Curv & mayb (3) Comparson of 2 survval curvs, and (4) Cox s Proportonal Hazards Rgrsson.

4 Th rvws of Logstc Rgrsson wr n ssson #2; thrfor, ths rvw ssson #3 s dvotd to Survval Analyss. Th mor advancd part, comptng rsks, s n th last analyss rvw #4.

5 If survval s of hgh prorty, ovrall survval (ndpont s dath by any caus) s rcommndd as th ndpont. Us of tm to dath du to th dsas may b problmatc. Evn f caus of dath nformaton s rlabl, on can nvr safly assum th ndpndnc of causs of dath; In addton, caus-spcfc ndpont dos not nclud all aspcts of tratmnt on survval.

6 Dsas-fr survval (ndpont s dath or rlaps) s oftn anothr good choc. Us of progrsson-fr survval s also basd bcaus follow-up on progrsson s oftn dscontnud aftr a patnt s takn off th study.

7 To focus on ovrall survval or dsas-fr survval, th most common task s stmatng th survval curv, calld th Kaplan-Mr Curv and som typcal survval rats such as on-yar survval rat.

8 Basc Outcom: SURVIVAL TIME

9 Th Survval Tm T s from Enrollmnt (Startng Pont) to Dath (f focusng on Ovrall Survval or to Dath or Rlaps (f focusng on Dsas-fr Survval (th Endng Evnt of Intrst) masurd n Yars or Months (masurmnt scal).

10 SURVIVAL FUNCTION Basc Functons to charactrz a Survval Tm T ar: (1) th Survval Functon S(t), and (2) th Hazard Functon. Survval Functon s dfnd as: S(t) = Pr(T>t) ; for t>0 = 1 - cdf F(t) (Emphasz th good nws ). It s rlatvly asy to stmat vn n th prsnc of cnsorng.

11 HAZARD (OR RISK) FUNCTION Hazard, or Rsk, Functon gvs th Instantanous Falur Rat at tm t assumng survval at t : ( t) lm 0 Pr[ t T t t T ] For a small tm ncrmnt, th Probablty that an vnt occurs durng th tm ntrval [t,t+) s, approxmatly: Pr[ t T t t T ] { ( t)}{ }

12 HAZARD FUNCTION Th Hazard, Functon masurs th pronnss to falur as a functon of ag It may ncras or dcras wth tm for long-trm or short-trm rsks, or rmans constant- vn follows a mor complcatd procss. For xampl, hazard assocatd wth acut lukma ncrass, hazard for organ transplant & AIDS dcrass; mayb non-monotonc (.g. Hodgkn s dsas). Bathtub hazard curv for human lf.

13 DIFFERENCES Survval Functon: How far to vnt (for th populaton) Hazard Functon: How fast (approachng th ndng vnt)- lk vlocty. Also calld Rsk Functon. Hazard Functon s dfnd smlar to th Dnsty Functon but condtonal ; th Hazard Functon can tll you mor, but mor dffcult to stmat- compard to S(t).

14 Survval Functon & Survval Curv Th Survval Functon d dfnd as: S(t) = Pr( T>t ); f t s n yars, S(t) s calld t-yar survval rat. For xampl, S(2) s th 2-yar survval rat (%); S(5) or 5-yar survval rat s a convntonal bnchmark n Cancr Survval.

15 BONE-MARROW TRANSPLANT: ALLOGENIC Vrsus AUTOLOGOUS Allognc: From a Matchd Donor Autologous: From Slf

16 For Phas II trals, whch ar oftn n shortr trm lastng just a yar or two, on-yar survval rat s oftn usd a bnch mark to masur tratmnt succss (.g. vrsus hstorcal data).

17 A DISCRETE MODEL A dscrt modl for survval tm T s dfnd as on takng valus x 1 <x 2 < <x k wth assocatd (postv) probablty mass: f(x ) = Pr(T = x ) > 0; and f(x) = 0 lswhr Th graph s a Stp Curv wth jumps at ths support ponts: x 1 <x 2 < <x k. Not that: S(x ) = Pr(T>x ) & S(x - ) = S(x -1 ); at ach vrtcal drop n th graph, th pont s at lowr lvl

18 SURVIVAL CURVE FOR A DISCRETE MODEL Not: At ach vrtcal drop n th graph, th pont s at lowr lvl Valu at ach drop s qual to th valu of th dnsty functon at that pont.

19 DISCRETE MODEL ) ( ) ( ] Pr[ x S x f x T x T ) (1 ) ( 1 1 j j x f ) (1 ) ( 1 j j x S

20 SURVIVAL DATA: An Exampl Forty-two (42) lukma pts n rmsson, aftr chmothrapy, wr randomzd to rcv thr Placbo or 6-mrcaptopurn (6-MP) n a 52-wk tral w/o follow-up. Endng vnt s rlaps ; tms n wks ar: Placbo:1,2,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,15,17, 22,23 (all 21 patnts hav rlapsd). 6-MP: 6,6,6,7,10,13,16,22,23,6+,9+,10+,11+, 17+,19+,20+,25+,32+,32+,34+,35+.

21 ABOUT DATA IN THE EXAMPLE Patnts wth tm t+ hav had no rlaps yt!.g., 10+ was from a patnt who nrolld on wk 42 and was stll n rmsson. Ths phnomnon s typcal of survval data, calld cnsorng ; th study was trmnatd bfor som subjcts hav thr ndng vnt (cnsord obsrvaton). All mthods larnd, ncludng sampl man & t-tsts, do not apply bcaus som subjcts only provd partal nformaton (.g. tm s mor than 10 wks)

22 A CLINICAL TRIAL Study subjcts wthout vnts whn tral s trmnatd wr cnsord

23 THE CONCEPT At Rsk Consdr th sampl: {7, 9+, 10, 12, 25+}. At t = 10, for xampl, 3 subjcts ar at rsk (thos wth duraton qual 10, 12, & 25; th on wth duraton 7 dd bfor 10 and th on wth duraton 9 was cnsord bfor 10 so thy ar not at rsk at t=10. In othr words, subjcts at rsk at t ar thos havng bn followd t unts of tm or longr. Actually, thy ar at rsk at tm t -. In th sampl: {7,9+,10,10+,12,25+}; thr ar four (4) subjcts at rsk at t=10 bcaus w count daths and cnsord cass at t among thos at rsk at t -.

24 Th rsk st R at tm t conssts of subjcts at rsk at t - ; th numbr of subjcts s n, ths ar undr obsrvaton at tm t -. Th dath st or vnt st D conssts of thos wth vnts at tm t (t s a subst of th rsk st); th numbr of subjcts s d, ths dd at t.

25 KAPLAN-MEIER, or PRODUCT-LIMIT METHOD Th Kaplan-Mr or also calld Product-Lmt mthod s th mthod that gnralzs th Emprcal Estmat for us wth Cnsord Data. Approach: To assum th Dscrt Modl f ( t ) 1 j1 (1 ) j S( (1 ) Thn us MLE mthod to stmat th s whr s ar th hazards at tms wth vnt(s) & t ) j1 j

26 cnsord 0 f 1for vnt and whr ) (1 )] ( [ )] ( [ Th Lklhood Functon s : d n m d n t S t f

27 KAPLAN-MEIER ESTIMATE Rsults: Nots: d n ^ S(t) Th Product s ovr all t t; Stp-functon t (1 At ach vrtcal drop on th graph, th pont s at lowr lvl; rght-contnuous t d n )

28 EXAMPLE #1 Consdr a small sampl of tms (n yars): {3,4,4,9,9}; n=5 and no cnsorng. t n d 1-(d / n ) S(t ) =(.8000)(.500) =(.4000)(0) Kaplan-Mr Mthod appls to data wthout cnsorng as wll (why not?); Rsults ar dntcal to mprcal stmats.

29 EXAMPL #2: Lukma Data 6MP:6,6,6,7,10,13,16,22,23,6+,9+,10+,11+, 17+,19+,20+,25+,32+,32+,34+,35+(n=21,d=9) t n d 1-(d / n ) S(t ) =(.7529)(.9167) (Exampl: Mdan s btwn 22 & 23)

30 D: Drug 6-MP P: Placbo RESULTS: Mdans ar 23 and 8 for 6-MP and Placbo groups rspctvly.

31 BONE-MARROW TRANSPLANT: **ALLOGENIC Vrsus AUTOLOGOUS** Allognc: From a Matchd Donor Autologous: From Slf Evnt: Dath; Sam Pattrn of Curvs for Lukma Rlaps: Evdnc of Cur!

32 COMPARISON OF TWO SURVIVAL DISTRIBUTIONS Suppos w hav n 1 and n 2 ndvduals rcvng tratmnts 1 and 2, rspctvly, n a clncal tral. Th study provds 2 sampls of survval data: {t 1, 1 }, from 1 to n 1 and {t 2j, 2j }, j from 1 to n 2. Th sampl szs n 1 & n 2 may or may not b qual Th Null Hypothss to b tstd s: H 0 : S 1 (t) = S 2 (t) Goal: to gnralz th Wlcoxon tst for us wth cnsord data (Surprs: W ll gt mor!)

33 STARTING STEP At ordrd tm t, 1 m, th data may b summarzd nto a 2-by-2 tabl as follows: Status Sampl: Dad Alv Total 1 d 1 a 1 n 1 2 d 2 a 2 n 2 Total d a n n 1, for xampl, s th numbr of subjcts n group 1 who wr at rsk at tm t (.. alv at t - ); subjcts who dd or wr cnsord at t ar ncludd.

34 EXAMPLE #1A Sampl #1:{3,5+,10} & Sampl #2:{3,6,9+,15+}. W hav: N=7, d=# of vnts=4, and m=# of tm ponts = 3 at t = 3 (2 vnts) and 6, 10 (1 vnt ach) At t=6, for xampl, w form a 2x2 tabl: Status Sampl: Dad Alv Total {10} {6,9+,15+} Total d 1

35 In ths arrangmnt, BASIC APPROACH Status Sampl: Dad Alv Total 1 d 1 a 1 n 1 2 d a 2 2 n 2 Total d a n Th Null Hypothss of qual survval dstrbutons mpls th ndpndnc of Status and Sampl n ach crossclassfd 2-by-2 tabl. In othr words, th frst row (sampl 1) can b consdrd as a random sampl from th last row (Total, whch srvs as a fnt populaton).

36 BASIC APPROACH Status Sampl: Dad Alv Total 1 d 1 a 1 n 1 2 d a 2 2 n 2 Total d a n Whn th frst row (sampl 1) can b consdrd as a random sampl from th last row (Total, acts as fnt populaton), w hav from th Hyprgomtrc Dstrbuton : E 0 n d n n ( d ) 1 ; Var ( d ) n n 2 ( n 2 a d 1)

37 TARONE-WARE FAMILY Th Tst Statstc: rprsnts not a Tst but a Famly of Tsts ach ndxd by a wght rprsntng th rlatv mportanc of tm pont (rlatv to othr tm ponts). Ths famly s calld th Taron-War famly of tsts. ) ( ; 1) ( ) ( 0 ) ( ; Var z n n a d n n w Var E m ] n d n w [d θ 1 1 m 1

38 CHOICES OF WEIGHTS FOR THE TARONE-WARE FAMILY How to choos th wght? (1) Th Gnralzd Wlcoxon, wth w = n, puts mor wght on th arly obsrvatons and bcaus of that ts us s mor powrful n dtctng th ffcts of short-trm rsk. (2) Th Log-Rank tst, wth w = 1, puts qual wght on ach obsrvaton and, by dfault, s mor snstv to latr dffrncs (as compard to th Gnralzd Wlcoxon); n fact, t s most powrful undr PHM (constant, long-trm rsks).

39 EXAMPLE #1B Sampl #1:{3,5+,10} & Sampl #2:{3,6,9+,15+}. W hav: m=# of tm ponts = 3 at t = 3, 6, 10. t n d 1 E 0 (d 1 ) Var 0 (d 1 ) Gnralzd Wlcoxon: Log-Rank: z W z L (7)(1.857) (4)(0.250) (2)(1.500) (49)(.408) (16)(.188) (4)(.250) (1.857) (0.250) (1.500) (.408) (.188) (.250)

40 CHOICES OF WEIGHTS FOR THE TARONE-WARE FAMILY Whch tst should w us wthout apror mphass? May b t s not a bad da to try both; th rsults would tll whthr th two groups ar dffrnt and, n addton, whr th dffrncs ar: () f th gnralzd Wlcoxon s mor sgnfcant, thr ar mor dffrncs at arly tms, () f th Log-Rank tst s mor sgnfcant, thr ar mor dffrncs at latr tms.

41 TARONE-WARE FAMILY Bcaus of th way th tst statstc s formulatd (trms n th sum ar not squard ): m 1 [ d1 1 n w th tsts n ths famly, rgardlss of th chocs of wghts, ar only powrful whn on group s bttr than th othr all th tms; othrws, som trms n th sum ar postv and othrs ar ngatv, and som of thm ar canclld out. For xampl, th tsts ar vrtually powrlss aganst crossng curvs altrnatv. n d ]

42 EXAMPLE #2 Forty-two (42) lukma pts n rmsson, aftr chmothrapy, wr randomzd to rcv thr Placbo or 6-mrcaptopurn (6-MP) n a 52-wk tral w/o follow-up. Endng vnt s rlaps ; tms n wks ar: Placbo:1,2,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,1 5,17, 22,23. 6MP: 6,6,6,7,10,13,16,22,23,6+,9+,10+,11+, 17+,19+,20+,25+,32+,32+,34+,35+.

43 D: Drug 6-MP P: Placbo RESULTS: (1)Log-Rank X 2 =16.793, p-valu=.0001 (2)Wlcoxon X 2 =13.458, p-valu=.0002

44 BONE-MARROW TRANSPLANT: ALLOGENIC Vrsus AUTOLOGOUS Allognc: From a Matchd Donor Autologous: From Slf Evnt: Dath; Sam Pattrn of Curvs for Lukma Rlaps: Evdnc of Cur! Not: Th Cox-Mantl tst (.. Log-Rank tst) sn undr th graph.

45 Nonparamtrc mthods ar popular. Howvr, thr ar nstancs whr w know th dstrbuton; n som spcal applcatons.g. n th sampl sz dtrmnaton - w may assum t. If w know th dstrbuton of tm to vnt, or w may rasonably assum t, say xponntal, t would mor ffcnt or powrful to us ths dstrbuton to form (paramtrc) tsts of sgnfcanc than to apply non-paramtrc tsts.

46 REGRESSION ANALYSIS Prmary focus s on stmatng th rspons rat but somtms w also want to s f som factor or factors mght affct th survval tm ; such as ag or gndr of th patnt, or othr factors from th patnt slcton procss dctatd by ncluson crtra (and xcluson crtra ). It s mportant to dntfy and undrstand th nflunc of such factors n ordr to optmz condtons for usng th tratmnt.

47 Rspons rats gnrally dcras as th xtnt of pror thrapy ncrass. Patnts who hav fald svral pror rgmns ar mor lkly to hav tumors composd larg numbrs of rsstant clls, and such patnts ar also lss lkly to b abl to tolrat full doss of th nvstgatonal drug. Th prsnc or absnc of pror thrapy s an mportant factor.

48 OTHER POSSIBLE GAINS It s mportant to dntfy both optmal and sub-optmal condtons. Insghts can lad to modfcatons of ncluson/xcluson crtra. Evn f w fnd, n rsarch, that a factor dos not nflunc th tratmnt s rsult, w may b abl to rlax th condtons undr whch th tst s appld. (so,.. ngatv fndngs ar good/bnfcal as wll)

49 REGRESSION MODEL Tradtonally, n Rgrsson, w modl th man - In our survval data: ara undr th Survval Curv - as a functon of covarat(s). Th usual Rgrsson Modl taks th form: Obsrvaton = a functon of covarat(s) + Error. Th problm s that som of th obsrvatons wr cnsord! Ths approach dos not work unlss w assum a Modl for th Error Trm - such as Exponntal or Wbull.

50 NEWER APPROACH W know that th Survval Functon s rlatd to th Hazard Functon. If an Exposur has som ffct on th Man of Survval Tm (whch s rlatd to th Survval Functon), t would hav som ffcts on th Hazard Functon as wll. That mans w can modl th Hazard nstad of modlng th Man.

51 WEIBULL MODELS Wbull Modl (wth 2 paramtrs:p>0 & >0; s th locaton and p th shap ): ( t) S( t) p( t) p1 xp[ ( t) Wbull Rgrsson: for all p ] t 0 p p ( a constant ), λ α βx

52 TERMINOLOGY Rlatv Rsk s also calld Rsks Rato (smlar to Odds Rato ) or, smply, Hazards Rato. Usd to masur th ffct of xposur, xposd vs. non-xposd ; th strngth of th rlatonshp, vry much lk th coffcnt of corrlaton.

53 AN EXAMPLE OF PROPORTIONAL HAZARDS For two Wbull s: So that th Hazards ar proportonal f and only f p 1 = p 0 = p (sam shap paramtr ); Undr PHM: So, Proportonal Hazards xst! (ths s smlar to th qual varanc modl for 2-sampl t-tst) ) ( ) ( ) ( ) ( p p t p t p t t p t t } { ) ( ) (

54 PHM:POSSIBLE GENERALIZATION Th PHM for a bnary rsk: RR(t) can b x wrttn as follows: ( t) 0( t) whr 0( t) ( t; E') th basln hazard and th covarat X s dfnd as (=1 f xposd) and (=0 f not xposd); = Frst, w can xtnd th modl, th Proportonal Hazard Modl (PHM) from Bnary Rsk to covr any rsk factor, dscrt or contnuous. Thn w ll xtnd to mak t multvarat: t) ( t) 1 ( 0 k x

55 REGRESSION COEFFICIENT Consdr a bnary rsk factor: (X=1 f xposd, X=0 f not xposd), and consdr 2 subjcts: (1) Subjct A: X=0, hs/hr hazard s A (t) = 0 (t). (2) Subjct B: X=1, hs/hr hazard s B (t) = 0 (t). Ths rato rprsnts th Rlatv Rsk of B vrsus A, or th Rlatv Rsk du to xposur : RR( t) B ( t) ( t) A ( t) ( t) 0 0 constant It xplans th trm Proportonal Hazards and th rgrsson coffcnt rprsnts th Rlatv Rsk on th log scal.

56 REGRESSION COEFFICIENT Consdr a contnuous rsk factor X & 2 subjcts: (1) Subjct A: X=x, hs/hr hazard s A (t)= 0 (t) x. (2) Subjct B: X=x+1, hs/hr hazard s: B (t) = 0 (t) (x+1). Ths rato rprsnts th Rlatv Rsk of B vrsus A, th Rlatv Rsk du 1 unt ncras n X : RR( t) B ( t) ( t) A ( t) ( t) ( x1) 0 x 0 constant It xplans th trm Proportonal Hazards and coffcnt rprsnts th Rlatv Rsk, on th log scal, du to on unt ncras n X.

57 ASSUMPTIONS OF PHM Thr actually ar two (2) assumptons n PHM: (1) Proportonal Hazards: Rato of hazards, or Rlatv Rsk, s ndpndnt of tm (2) Lnarty: For a contnuous factor, th coffcnt rprsnts th Rlatv Rsk, on th log scal, du to on unt ncras n X : RR( t) B ( t) ( t) A ( t) ( t) ( x1) 0 x 0 constant Not that ths RR s ndpndnt of X=x, a rsult of th lnarty assumpton whch may not b tru.

58 CONDITIONAL APPROACH Dnot th ordrd dstnct dath tms by t s: and lt R b th rsk st just bfor tm t, n th numbr of subjcts n R, D th dad st at t, d th numbr of subjcts (.. daths) n D. Lt C b th collcton of all possbl combnatons of subjcts from R, ach combnaton - a subst of R -has d mmbrs; D s tslf on of ths combnatons. Exampl: R = {A,B,C} and D = {A,B}; Thn n = 3, d = 2, and C = {{A,B}= D, {A,C}, {B,C}}. Th collcton C has 3 mmbrs; n gnral: N n d

59 APPROACH: COX S ARGUMENT Cox wrts (1972): "Suppos thn that 0 (t) s arbtrary. No nformaton can b contrbutd about by tm ntrvals n whch no falurs occur... W thrfor argu condtonally on th st of nstants at whch falurs occur. Onc w rqur a mthod of analyss holdng for all 0 (t), consdraton of ths condtonal dstrbuton sms nvtabl."

60 LIKELIHOOD FUNCTION Wth hs condtonal argumnt, Cox proposd th followng Partal Lklhood Functon : L m 1 Pr( D R, d ) W ll consdr sparatly th cass wthout and wth ts and llustrat usng th followng small xampl wth 4 subjcts: {A: (t = 4,x = 2, = 1), B: (t = 8,x = 5, = 1), C: (t = 8,x = 7, = 1), and D: (t=11,x=18, =0).

61 LIKELIHOOD FUNCTION Wthout ts, th componnt of th Partal Lklhood at t s: For our xampl {A:(t=4,x=2,=1), B:(t=8,x=5,=1), C:(t=8,x=7,=1), and D:(t=11,x=18,=0), at t=4, th componnt of th Partal Lklhood s: j j n j x x n j x x t t d R D L ) ( ) ( ), Pr( L t

62 LIKELIHOOD FUNCTION Wth ts, componnt of th Partal Lklhood at t s: For our xampl {A:(t=4,x=2,=1), B:(t=8,x=5,=1), C:(t=8,x=7,=1), and D:(t=11,x=18,=0), at t=8, th componnt of th Partal Lklhood s: j D u j D u j C s s C x x C D x x D t t L ) ( ) ( L t

63 EXAMPLE For th small xampl wth 4 subjcts: {A: (t = 4,x = 2, = 1), B: (t = 8,x = 5, = 1), C: (t = 8,x = 7, = 1), and D: (t=11,x=18, =0), Th Partal Lklhood Functon s: 2 12 ( L ) { }{ W can : () maxmz t to obtan MLE for, Tsats (1981) gvs a proof of th asymptotc normalty of ;and () us th scor tst, for xampl, to tst Null Hypothss: H 0 : =0 }

64 ESTIMATION From th Partal Lklhood: Estmaton, by traton, s carrd out usng: } ln {,or,lnl 1 1 j j C s m m C s s s L m s s j s j s s s j m j j j j j j s s L s s L d d } { ] [ ln d d },and { ln

65 EXERCISES T10.1 Undr th logstc Rgrsson modl wth a contnuous covarat X, fnd th Odds Rato assocatd wth m unts ncras n X T10.2 Us ths smpl data st (of Exampl #1), say, tm s n months: {2,3+,5,5,6+,8,11,11+,12,15+}. Run SAS to form th Kaplan Mr curv and calculat th 95% confdnc ntrval for on-yar survval rat.