Monitoring event times in early phase clinical trials: some practical issues

Transcription

1 ARTICLE Cliical Trials 2005; 2: Mitrig evet times i early phase cliical trials: sme practical issues eter F Thall a Leik H Wte a ad Nizar M Tair b Backgrud I may early phase cliical trials it is scietifically iapprpriate r lgistically ifeasible t characterize patiet utcme as a biary variable. I such settigs it fte is mre atural t cstruct early stppig rules based timet-evet variables. This type f desig may ivlve a variety f cmplicatis hwever. urpse The purpse f this paper is t illustrate by example hw e may deal with varius cmplicatis that may arise whe mitrig time-t-evet utcmes i a early phase cliical trial. Methds We preset a series f Bayesia desigs fr a phase II cliical trial i kidey cacer. Each desig icludes a prcedure fr mitrig the times t a severe adverse evet disease prgressi ad death. The first desig which is the simplest is based the time t failure defied as ay f the three evets assumig expetially distributed failure times with a iverse gamma prir the mea. This desig is cmpared by simulati t the CMA desig (Cheug ad Thall Bimetrics 2002; 58: 89 97). The mdel ad mitrig prcedure are the exteded successively t accmmdate several cmm practical cmplicatis ad we als study the methd s rbustess. Results Our simulatis shw that ) e may apply the mitrig rule peridically rather tha ctiuusly withut a substative degradati f the desig s reliability; 2) it is very imprtat t accut fr iterval cesrig due t peridic evaluati f disease status; 3) it is imprtat t accut fr the effect f disease prgressi the subsequet death rate; 4) cductig a radmized trial presets little additial difficulty ad prvides ubiased cmpariss; ad 5) the expetial-iverse gamma mdel is surprisigly rbust i mst cases. Limitatis The methds discussed here d t accut fr patiet hetergeeity. This is a imprtat but cmplex issue that may be dealt with by extedig the mdels ad methds give here t accmmdate patiet cvariates ad treatmetcvariate iteracti. Cclusis Bayesia prcedures fr mitrig time-t-evet utcmes ffer a practical way t cduct a variety f early phase trials. Csiderable care must be give hwever t mdelig the imprtat aspects f the trial at had ad t calibratig the prir ad the desig parameters t esure that the desig will have gd peratig characteristics. Cliical Trials 2005; 2: Itrducti I may cliical trials the primary therapeutic utcmes are evets that ccur at radm times. Examples f such evets iclude a give amut f tumr shrikage disease prgressi r regimerelated death i clgy; egraftmet r graftversus-hst disease i be marrw trasplatati; a epartmet f Bistatistics ad Applied Mathematics The Uiversity f Texas M.. Aders Cacer Ceter Hust TX USA b epartmet f Geituriary Medical Oclgy The Uiversity f Texas M.. Aders Cacer Ceter Hust TX USA Authr fr crrespdece: eter F Thall epartmet f Bistatistics ad Applied Mathematics The Uiversity f Texas M.. Aders Cacer Ceter 55 Hlcmbe Bulevard Hust TX USA. rex@mdaders.rg Sciety fr Cliical Trials / c2a

2 468 F Thall LH Wte ad NM Tair resluti f a ifecti with a atibitic; ad lwerig systlic bld pressure by a specified amut whe studyig a ati-hypertesi aget. May statistical desigs fr early phase trials are based the prbability f a cmpsite utcme defied i terms f e r mre evet times ccurrig withi a specified time perid frm the start f treatmet. Fr example i clgy respse may be defied as the evet that the patiet survives seve mths withut sufferig disease prgressi. etig the times t prgressi ad death by T ad T respse is R {7 mi(t T )}. Hwever if e wishes t make iterim decisis based the idicatr X f R ad the prbability R r(r) r(x ) the several lgistical prblems arise. The mst severe prblem created by this apprach is that a failure (prgressi r death) may be bserved at ay time up t seve mths whereas a respse ca ly be bserved if the patiet is fllwed fr seve mths t esure that failure has t ccurred. Csequetly fr the first seve mths f the trial ly values f X 0 may be bserved ad this prblem persists thereafter sice the bserved prprti f patiets fr whm X 0 still ver-represets the actual value f R. This reders ay early stppig rule based X very ureliable. While i thery this bias culd be avided by ly scrig X fr each patiet at seve mths such a apprach is very impractical. Ather pssible alterative might be t use a much shrter iterval fr defiig respse say R { mi(t T )}. This wuld have the udesirable effect f declarig patiets fr whm T 7 t be respders ctrary t the actual gal f the trial. I geeral the use f a idicatr such as X is ly feasible if it ca be bserved very quickly ad it prvides a reasable summary f patiet utcme. Ather pssible way t vercme this prblem might be t first apply the iterim mitrig rule ly after a sufficietly lg time iterval has elapsed s that a reasable umber f patiets have bee fllwed fr seve mths r lger. Hwever this defeats the purpse f safety mitrig sice igrig early failures that ccur befre first applyig the stppig rule fails t prtect agaist the case where the failure rate is uacceptably high. The lss f ifrmati resultig frm discretizig the time-t-evet (TTE) variables T ad T by usig X t characterize the patiet s utcme is illustrated by a patiet wh at a give time durig the trial whe a iterim decisi must be made is alive ad has bee prgressi-free fr six mths. Althugh X is t yet kw fr this patiet the bserved evet {6 mi(t T )} clearly prvides useful ifrmati abut R. Similarly if tw patiets bth survive 2 mths with T 6.5 fr e ad T 7.5 fr the ther the X 0 fr the first patiet while X fr the ther despite the fact that their actual utcmes are very similar. The usual mtivati fr usig discretized utcmes such as X t evaluate treatmets i early phase trials is that it is impractical r udesirable t wait t bserve the evet times. The uderlyig scietific ratiale which usually is t stated explicitly is that the early utcme is a reasable surrgate fr a TTE variable f primary iterest [ 2]. Hwever surrgacy typically is difficult t verify i practice [3 5]. Rather tha basig cliical trial desig a biary utcme that is a surrgate fr e r mre TTE variables a umber f authrs have advcated the mre direct apprach f frmulatig the uderlyig statistical mdel ad iterim decisi rules i terms f the TTE variables themselves. Fllma ad Albert [6] prpse mitrig the rate f a adverse evet by usig a irichlet prcess prir fr the prbabilities f the evet a large set f discretized evet times. They cmpute a apprximate psterir that is a mixture f irichlet prcesses by usig a data augmetati algrithm. Rser [7] takes a similar apprach but uses Gibbs samplig t geerate psterirs. Cheug ad Thall [8] prpse a methd ctiuus mitrig based a apprximate psterir (CMA) fr cstructig futility mitrig rules based e r mre evet times used t defie a cmpsite evet R i phase II. Thall et al. [9] use a hierarchical Bayesia mdel t accut fr multiple disease subtypes i a phase II trial based evet times. A geeral discussi f Bayesia methds i cliical trials is give by Spiegelhalter et al. [0]. I this paper we preset a series f Bayesia desigs fr a phase II trial based three TTE variables. The desigs are develped i the ctext f a trial t evaluate a ew treatmet fr kidey cacer ad are based the times t death a severe adverse evet (SAE) ad disease prgressi. The first desig is based the time t failure defied as ay f these three evets assumig that failure time is expetially distributed with mea fllwig a iverse gamma prir. The perfrmace f this desig is evaluated via simulati ad cmpared t the CMA desig. Oce the iitial TTE desig based the expetial-iverse gamma mdel is established we successively refie the mdel ad desig t icrprate additial elemets f cmplexity that may arise i practice. These refiemets accut i tur fr the cmplicatis that disease prgressi is evaluated fr each patiet at a sequece f times rather tha ctiuusly the hazard f death icreases at the time f disease prgressi ad e may wish t radmize patiets rather tha cduct a siglearm trial. I each case we derive a geeral likelihd but i rder t fcus these particular Cliical Trials 2005; 2:

3 Evet times i early phase cliical trials 469 issues we deal with the simple case where each evet time is expetially distributed with iverse gamma prir the mea. Next t accut fr the effect f prgressi survival we exted the survival time distributi t a piecewise expetial with hazard chagig at prgressi. Sice the assumpti f expetially distributed evet times may be iadequate i may settigs we study the rbustess f the iitial desig ad f CMA t departures frm expetiality ad we als iclude a mre geeral versi f the iitial desig based the assumpti that failure time fllws a geeralized gamma distributi. We clse with a brief discussi f sme additial issues icludig patiet hetergeeity ad the use f multiple stppig rules. Xelda Gemzar fr kidey cacer A sigle-arm phase II trial f the experimetal (E) cmbiati Xelda Gemzar (X G) was cducted t btai a prelimiary evaluati f this cmbiati fr treatmet f advaced kidey cacer. The trial was limited t patiets wh previusly had received immutherapy ad either did t respd r achieved at least a partial disease remissi but subsequetly relapsed. Fr these patiets disease prgressi r death ccur average i abut six t eight mths with stadard therapy (S) which csists f 5-flururacil Gemzar (5-FU G). While a imprvemet i prgressi-free survival (FS) time was desired ather mtivati fr the trial was the fact that Xelda is chemically similar t 5-FU but Xelda is give rally rather tha itraveusly as is required with 5-FU. Each patiet s disease status was evaluated at the time f trial etry (baselie) ad thereafter at eight week itervals util treatmet failure up t 48 weeks defied as prgressive disease cmpared t the baselie evaluati a regime-related SAE at a level f severity precludig further treatmet r death. At this writig the trial has accrued the maximum f 84 patiets ad fllw up curretly is gig. A simple desig fr the Xelda Gemzar trial Our first desig mitrs the time t treatmet failure T mi{t T } which we assume is expetially distributed with mea fllwig a iverse gamma (IG) prir. Frmally T has pdf f (t ) e t/ ad has prir p ( a b) e b/ b a (a) /(a) where ( ) is the gamma fucti ad a b 0 are fixed hyperparameters. We will dete this by T ~ Exp() ad a b ~ IG(a b) ad refer t this as the expetial-iverse gamma (E-IG) mdel. Sice the IG(a b) distributi has mea b /(a ) ad variace b 2 /{(a ) 2 (a 2)} we require a 2. etig the hazard by assumig that ab~ig(a b) is equivalet t assumig that a b~gam(a b) a gamma distributi with pdf f() e b b a a /(a) which has mea a/b ad variace a/b 2. Sice media(t) lg(2) ad i geeral if X~IG(a b) the cx~ig(a cb) fr ay c 0 the abve prirs i terms f ad are equivalet t specifyig ~IG(a lg(2) b). It will be cveiet t use these three equivalet frms f the distributi ad we will mve freely betwee them. This mdel is especially tractable sice the IG is a cjugate prir fr the expetial. Give the abve structure we will idex the histrical stadard ad experimetal treatmets by j S E s that j j j are the parameters ad a j b j are the hyperparameters fr treatmet j. The time scale f all evet times ad their crrespdig parameters will be i mths. Wheever mitrig r bservati itervals are give i terms f weeks this will be stated explicitly. We will prvide details f prir elicitati later i the ctext f a mre cmplex mdel ctaiig the mdels csidered i this secti ad the ext as special cases. The histrical stadard treatmet media failure time has prir S ~ IG( ) btaied frm the elicited mea 4.0 fr S ad 95% credible iterval (CI) r(3.0 S 5.2) Equivaletly sice S S / lg(2) the histrical mea time t failure has prir S ~ IG( ). O the evet rate dmai S has mea a S /b S 0.77 ad variace a S /b S We calibrated the prir f E t have the same mea but var( E ) 0var( S ) t reflect the much greater prir ucertaity abut E. Thus a E /b E 0.77 ad a E /b E which implies that E ~ IG( ). With the excepti f the radmized trial t be discussed later i each f the differet cases that we will csider the distributis f ay parameters crrespdig t S d t chage durig the sigle-arm trial f E. Let dete the umber f patiets wh have bee erlled at the time f ay give iterim decisi. Fr the ith patiet i... let T i be the bserved time f failure r admiistrative right cesrig ad let i I(T i T i ) idicate that T i is a failure time. ete the survivr fucti (sf) by F(t) r(t t) which takes the frm F(t) e t/ uder the Exp() mdel. Fr data (T... T ) detig the umber f failures by N i i ad the ttal bservati time by T i T i the likelihd is the well kw expressi E E i i = E i L( data ) = f ( T ) F ( T ) N + E = Exp( T / ). E E i i E () Cliical Trials 2005; 2:

4 470 F Thall LH Wte ad NM Tair sterir cmputatis are facilitated by cjugacy sice E ~ IG(a E b E ) implies that [ E N T ] ~ IG(a E N b E T ). The desig specified that a maximum f 84 patiets were t be accrued subject t the futility mitrig rule that the trial shuld be stpped early if based the curret data r( + 3 < data ) < p. s E L Thus the trial is stpped early if give the curret data it is ulikely that the media failure time with E is at least a three mth imprvemet ver the histrical media with S. This rule is similar t the futility stppig rule give by Thall ad Sim [] wh deal with respse prbabilities S r S (R) ad E r E (R) fr biary utcmes rather tha media evet times. Sice S ad E are idepedet parameters fllwig IG distributis (2) may be cmputed by umerical itegrati usig a package such as S-LUS r sftware freely available at I ur simulatis we will use the superscript t idetify fixed parameter values that determie the prbability distributis used t geerate patiet utcmes t distiguish them frm the radm parameters i the Bayesia mdel. The cutff p L i (2) was calibrated t btai prbability f early termiati (ET) 0.0 if the media failure time with E is E 7 mths the prir mea E( S) 4 plus the desired 3 mth imprvemet. This gave p L T btai the umerical value f p L yieldig a desig with ET 0.0 fr a give E we first evaluated the criteri prbability r( S 3 E) uder the prir ad used this as a upper limit p L fr p L. Next we evaluated the rule fr E usig cut-ff p L/2. If the resultig ET 0.0 the p L/4 was used ext; if ET 0.0 the 3p L/4 was used ext. This methd f bisecti was iterated util ET 0.0 was btaied withi three decimal places f accuracy. Additially after the secd bisecti the methd was refied by liearly iterplatig r extraplatig. Fr each cut-ff studied the trial was simulated 00 times with this icreased t 2000 t esure the desired (2) accuracy f the fial value f p L. I mst cases this required five t 0 iteratis. I the simulatis the rule (2) was applied ctiuusly. Each time a ew patiet became available fr erllmet based the mst recet data at that time the psterir stppig criteri r( S 3 E data ) was updated ad applied. The maximum sample size f 84 was chse assumig a accrual rate f six patiets per mth t esure a maximum trial durati f abut 4 mths t btai desirable early stppig prbabilities ad t btai a reliable psterir fr E. Fr example N failures ad ttal bservati time T wuld give empirical mea failure time 706.3/ mths which is what wuld be expected if E seve mths ad these data wuld give a psterir 95% CI fr E f ( ). This desig s peratig characteristics (OCs) are summarized i the prti f Table labeled expetial-iverse gamma. Fr the simulati results summarized i Tables ad 7 each case was simulated 2000 times ad the distributis f the umber f patiets ad trial durati are summarized by their 25th 50th ad 75th percetiles. As a basis fr cmparis we simulated the trial usig CMA which als cstructs stppig rules usig right-cesred evet times. I this case CMA is based a prbability f the frm E E (t* T) fr a fixed time t* ad it relies the decmpsiti E {A (t)} E {A (t) t* T} E E {A (t) t* T } ( E ) where A (t) is the patiet s bserved data at time t i the trial. CMA stps the trial if r( S E data) p L with this rule applied ctiuusly usig a apprximate psterir fr E btaied by treatig E {A (t) t* T} ad E {A (t) t* T} as uisace parameters ad estimatig them empirically. etails are give i Cheug ad Thall [8]. T make the tw methds cmparable we cstructed prirs ad the mitrig rule fr CMA as fllws. We used t* 7 t defie r(7 T) ad derived a beta(a S b S ) prir S by equatig its mea m S a S /(a S b S ) ad variace m S ( m S )/(a S b S ) t the mea ad variace f S exp(7/ S ) uder the IG( ) prir Table Operatig characteristics f the expetial-iverse gamma mdel-based desig ad CMA fr the Xelda Gemzar trial based expetially distributed failure times Mitrig based right-cesred evet times Expetial-iverse gamma mdel Apprximate psterirs (CMA) E ET N. pats. Trial durati ET N. pats. Trial durati Cliical Trials 2005; 2:

5 Evet times i early phase cliical trials 47 S. This yielded a beta( ) prir fr S. We used a beta( ) prir fr E which has the same mea as S but effective sample size a E b E 2. Sice E 7 implies that E exp [7/{7/lg(2)}] 0.50 we used i the CMA stppig rule. As with (2) we calibrated the cut-ff f the CMA rule t btai ET 0.0 at E 7 equivaletly if E 0.50 which yielded p L The simulati results are summarized i Table. Sice bth methds have ET 0.0 at E seve mths differeces ca be see as decreases frm 7 t less desirable values with the ET icreasig much less rapidly fr CMA. I the udesirable case where E 4 CMA has ET 0.87 ad media sample size 53 cmpared t ET 0.96 ad media sample size 33 fr the E-IG mdel-based methd. Thus i this case CMA has substatially less desirable OCs cmpared t the mdel-based methd. This cmparis is t etirely fair hwever sice the E-IG mdel-based methd is beig evaluated assumig that the uderlyig mdel is crrect. I the secti Rbustess we will evaluate the rbustess f these desigs whe the failure times are t expetially distributed. It is useful t csider hw a cmparable desig based the biary discretized utcme X I(7 T) rather tha T wuld behave i this case. Suppse e assumes the same prirs E ad S ad applies the same stppig rule as used fr the CMA desig but w assumig the likelihd i X i E ( E ) X i ad scrig the X i s whe they are first bserved s that X i at seve mths if T i 7 ad X i 0 at T i if T i 7. The psterir f E wuld be (0.587 i X i.43 i ( X i )) s cmputig the early stppig criteri prbability r( S E data) is straightfrward. Hwever iitially ly values f X i 0 may be bserved s the data fr estimatig E are heavily biased. A csequece f this is that if E 7 s that E 0.50 a desirable value the the early stppig prbability with this apprach exceeds That is the rule is virtually certai t stp the trial i a desirable case where e wuld t wat t stp. The targeted three-mth imprvemet f E ver S i the stppig rule (2) is a subjective value determied by the cliicia. I the illustrative trial it is arguable that sice Xelda is give rally ad has a lwer risk f adverse effects it may be reasable t use the stppig criteri r( S E data ) with targeted imprvemet i media failure time. This may be csidered a phase II equivalece trial as defied by Thall ad Sug fr discrete utcmes ([2] Secti III). I this case the histrical mea f 4.0 mths fr S is csidered a desirable value f E ad ly very small values such as E.0 r 2.0 mths are udesirable. Calibratig this versi f the rule t have ET 0.0 fr E 4.0 yields a desig with p L Fr E 2 ad 3 the respective early stppig prbabilities are ET ad 0.64 with media sample sizes 3 23 ad 59. Sice it may t be feasible t mitr the data ctiuusly i sme trials it is wrthwhile t examie the behavir f the desig if the early stppig rule is applied peridically. T d this we repeated the simulatis but with (2) applied every k weeks fr k r 24. The results summarized i Table 2 shw that there is a gradual declie i ET as the mitrig iterval is icreased but eve mitrig every eight weeks still maitais ET 0.93 whe E 4. It thus appears that if e accuts fr the evet times i this way the applyig the stppig rule peridically still prvides a safe desig while impsig less f a practical burde durig trial cduct. Mrever if peridic mitrig is plaed iitially the the value f p L i the stppig rule (2) may be calibrated s that ET equals a give small value whe E equals a desirable target. Accutig fr iterval cesred prgressi times Because each patiet s disease status is evaluated at eight-week itervals the actual time f ay patiets disease prgressi is t available. Rather it is ly kw whether prgressi ccurred durig each time iterval betwee successive examiatis. Fr example if prgressi is first discvered at the week 24 examiati the it is ly kw that prgressi ccurred betwee weeks 6 ad 24. This srt f iterval cesrig f the time f trasiti betwee disease states is cmm i medical settigs where the patiet s disease status is evaluated peridically by tests such as magetic resace imagig r cmputed axial tmgraphy sca. The previus prbability mdel igres this cmplicati. T accut fr iterval cesrig first csider a sigle patiet ad temprarily suppress bth the treatmet ad patiet idices. Let T dete the time f disease prgressi T A mi{t T A } Table 2 Effect f peridic rather tha ctiuus mitrig. The clum labeled 0 crrespds t ctiuus mitrig. Each etry is the prbability f early termiati Mitrig perid i weeks E Cliical Trials 2005; 2:

6 472 F Thall LH Wte ad NM Tair the time t death r a SAE ad idetify the crrespdig parameters ad prbability fuctis by the subscripts ad A. Thus we w accut fr tw evet times rather tha ly e. Fr w we will assume that T ad T A are idepedet ad expetially distributed with parameters ad A. Let dete the successive times whe the patiet s disease status is evaluated allwig the pssibility that a patiet s actual evaluati times may deviate frm the scheduled times. We will assume that like death a SAE is a termiatig evet i that fllw-up eds at the time f a SAE but patiets may be fllwed fr sme perid f time after prgressi. Each patiet s likelihd ctributi may take e f the fllwig pssible frms which are a csequece f the fact that while the actual value f T A is bserved ly the iterval durig which T ccurs ca be kw. Let T dete the time f the last bserved evet r fllw up with A I(T A T ) the idicatr that the patiet s time f death r a SAE is bserved. ete the last disease evaluati time by k let I( k T k ) idicate that prgressi is discvered at k ad dete the prbability that prgressi ccurs betwee the k st ad kth disease evaluati times by k r( k T k ) F ( k )F ( k ). If the patiet dies r has a SAE withut previus bservati f disease prgressi the i additi t bservig T A it is kw that k T s i this case the likelihd ctributi is f A (T A )F ( k ). If the evaluati at k is egative the patiet survives t k withut a SAE ad the evaluati at k shws prgressi the k k ad it is kw that k T k T A. Sice i ay case k T the likelihd ctributi f this patiet is either k F A (T ) if T A is admiistratively cesred at T r k f A (T A ) if the patiet dies r has a SAE i.e. if T T A. If T A is admiistratively cesred at T ad prgressi was t bserved at k the the likelihd ctributi is F A (T )F ( k ). Sice it is ly kw if a patiet prgresses durig e f the itervals ( k k ] ad als survives t k it fllws that if either the last fllw time is k T T A r if prgressi is discvered at k ad the patiet later dies r has a SAE i which case k T T A. Accutig fr all f this additial structure due t csiderig ad A as separate evets ad accutig fr the iterval cesrig f T ad w reitrducig the patiet idices the likelihd is give geerally by A A i i = i A i A L( data f f ) = f ( T ) F ( T ) F ik p ik A ( ). ip i i (3) Fr the expetial case dete the umber f deaths r SAEs by N A i ia ad T i ik ( i ). The geeral likelihd (3) w takes the specific frm L( data ) = Sice L(data A ) L(data A ) L(data ) the psterirs f A ad may be cmputed separately ad these tw parameters als are idepedet a psteriri. Fr this exteded mdel prirs the fur parameters S ( S SA ) ad E ( E E A ) are required. We assume idepedet prirs with jr ~ IG(a jr b jr ) fr each treatmet j S E ad utcme r A. Sme care must be take whe specifyig the fur hyperparameters (a j b j a ja b ja ) fr each j sice the fact that the hazards f idepedet expetials are additive impses sme cstraits. Specifically T j mi(t j T ja ) implies that j j ja fr each j E S. This i tur implies that E( j ) E( j ) E( ja ) ad due t the idepedece f j ad ja var( j ) var( j ) var( j ). Here (a S b S a SA b SA ) ( ) which implies that E( S ) 7 ad E( SA ) As befre we assumed that the meas f the hazards fr E were the same as thse see histrically but we iflated the variaces by multiplyig by 0 s that E( Er ) E( Sr ) ad var( Er ) 0 var( Sr ) fr r ad A. This yielded (a E b E a EA b EA ) ( ). T accut fr iterval cesrig f T we write (2) i the frm ( ) + < ( + ) N A A e T T + + A A A e e i= r SA + S 3 EA E data < p L which accuts fr all fur elemets f ( S E ) uder the exteded mdel. The psterir prbability i (5) is based the likelihd (4) with the distributi f S established as described abve ad fixed thrughut the trial. sterir distributis uder this mdel ad the mdels discussed belw i the fllwig tw sectis were cmputed usig iterative defesive imprtace samplig [3] which requires e t lcate the mde f L(data ) prir() as a fucti f ad cmpute the gradiet at the mde at each iterati. We used the Nelder Mead methd [4] t fid the mde. All prgrammig was de i C which prvides speed reusability ad flexibility. This laguage als was chse t take advatage f the extesive i-huse library f C cmputer prgrams i the M.. Aders epartmet f Bistatistics ad Applied Mathematics which are { } i ik ik. (4) (5) Cliical Trials 2005; 2:

7 Evet times i early phase cliical trials 473 Table 3 Operatig characteristics f the Xelda Gemzar trial with T iterval cesred due t prgressi beig evaluated at eight-week itervals fr each patiet Mdellig iterval cesred T p Igrig iterval cesrig E E EA ET N. pats. Trial durati ET N. pats. Trial durati available frm the secd authr request. While may f the Bayesia cmputatis described here culd be carried ut i WiBUGS implemetig the simulatis described here usig this apprach wuld be highly cmplex. T simulate the trial we geerated T ad T A idepedetly fr several pairs f ( E EA ) values such that / E / EA / E with either E 4 the histrical mea value r E 7 the desired target. Fr each patiet T A was bserved ctiuusly ad T was bserved at eight-week itervals up t a maximum f 48 weeks (six evaluatis). The cut-ff p L was calibrated t btai ET 0.0 i the desirable case ( E EA ) ( ) fr which E 7. This yielded p L The stppig rule (5) was applied ctiuusly. The simulatis are summarized i the prti f Table 3 labeled Mdelig Iterval Cesred T. T quatify what is gaied by mdelig T ad T A as separate evets ad accutig fr the fact that T is iterval cesred we evaluated the OCs f the first desig based the simpler frmulati with stppig rule (2) give previusly whe i fact each patiet s prgressi times are iterval cesred. That is we simulated the bserved evet prcess fr (T T A ) with T iterval cesred but used the rule (2) uder the simple E-IG mdel fr T mi (T T A ). The results are summarized i the prti f Table 3 labeled Igrig Iterval Cesrig. These simulatis shw that igrig the fact that T is iterval cesred greatly reduces the desig s ET values. Thus the OCs f a desig that igres iterval cesrig may be very misleadig ad accutig fr the fact that prgressi is ly bserved peridically prvides a much safer desig. Accutig fr the effect f disease prgressi survival A piecewise likelihd Thus far we have assumed that the three evets ccur idepedetly. I kidey cacer ad may ther slid tumrs hwever the hazard f death icreases with disease prgressi. This additial cmplicati which is very imprtat cliically ad als may impact the way that a give mitrig rule behaves may be mdeled i a umber f ways. Here temprarily suppressig S ad E fr simplicity we will use a piecewise distributi uder which the pdf f T chages at T frm f (x) t F (T )f 2 (x T ) where f (x) ad f 2 (x) are pdfs defied fr x 0. The jit distributi f T ad T is give geerally by Uder this piecewise mdel still accutig fr the iterval cesrig f T the likelihd takes e f fur pssible frms summarized i Table 4. T express thigs mre cmpactly we cmbie the first tw rws f Table 4 which crrespd t the tw cases where. Uder the piecewise mdel (6) the prbability that prgressi is discvered at k ad is fllwed by either death r cesrig at T is k k ( T ) = f( y) F ( y) f ( T y) F ( T y) 2 2 (6) dy. (7) Similarly we cmbie the last tw rws f Table 4 which crrespd t the tw cases where where 0. The prbability that the last disease evaluati at k shws prgressi ad this is fllwed by death r cesrig at T is T ( k ) = ( ) F( ) 2( ) F2( f ( x y) = f ( x y) fp( y) = f( x) I( x< y) + F( y) f2( x y) I( x y) f ( y) x y> 0. { } k T f y y f T y T k p F + F ( T ) f ( T ) ( T ). The secd summad i (8) is eeded t iclude the evet T T that the patiet has t prgressed by the last fllw up time. ete T A mi(t A T ) ad A I(T A T A ). Assumig that T A ad (T T ) are idepedet the geeral y) dy (8) Cliical Trials 2005; 2:

8 474 F Thall LH Wte ad NM Tair Table 4 ssible utcmes ad likelihd ctributis fr the times t prgressi T ad death T uder the mdel with T iterval cesred ad the hazard f death chagig frm frm f /F befre T t f 2 /F 2 after T Outcme Likelihd ctributi k T k T T k k f ( y) ( y) f ( T y) dy F 2 0 k T k T T k k f ( y) F( y) F ( T y) dy 2 0 k T ad k T T T k f ( y) F( y) f ( T y) dy + F ( T ) f ( T ) 2 p 0 0 k T ad k T T f ( y) F( y) F ( T y) dy + F ( T ) F( T ) T k 2 likelihd fr all f the pssible bservatis f the three types f evets w may be expressed as L( T T k A A f f A f )= Uder the piecewise expetial mdel where the hazard f death chages frm t 2 at T detig 2 the geeral likelihd (6) takes the specific frm the margial pdf f T is ad the prbabilities (7) ad (8) take the frms ad A T k T k fa TA A T ( ) ( ) ( ) F ( A) A. f ( x y) = e I( x< y) + 2 e I( x y 2 ) 2 e e ( y x ) 2 f( x 2 )= e + + T 2 k T k e e e ( )= 2 2 k ( T )= e e 2 e k x y x y + e (9) (0) () (2) (3) Cmbiig these expressis with the fact that f A (T A ) AF A (T A ) A A A e T A A expetial case the likelihd (9) may be writte as N 2 A A T ( + ) L( data ) = e T T ( T ) ik i i i (4) where N A i ia ad T A i {T ia ia T i ( ia )}. T + A A A x ( + ) { x } 2 ( ) ( ) ( ) ik Ti i i= k i The hazard f death uder the piecewise mdel is h ( 2)( + ) e + 2e ( x 2 ) = x 2x e ( + ) ( ) + e. (5) This expressi reduces t uder the simple expetial mdel where 2 ad it cverges t as 2. The FS time mi{t T } ~ Exp( ) which is the same distributi as uder the mdel where T ad T are idepedet with T ~ f. Ituitively this is the case because the hazard f death after prgressi has effect mi{t T }. Csequetly the media FS time equals ( ). Sice T A is idepedet f (T T ) it fllws that the verall failure time T~Exp( A ) s i terms f the medias the early stppig rule is ( ) r S + S + S A + 3 (6) I particular the pst-prgressi death rate 2 plays rle i (6). Hwever if e wishes t mitr the death rate uder the piecewise mdel althugh media(t ) cat be cmputed i clsed frm e may frmulate a early stppig rule i terms f either the mea survival time E(T ) { 2 ( 2 )}/{ 2 ( ) 2 } r the hazard fucti h (x*) evaluated at sme fixed time x x* sice bth quatities ivlve all three parameters ( 2 ) characterizig f (x y) ad f (x). Establishig prirs 2 ( + ) x ( ) < x < E + E + E A data pl. I rder t establish prirs S ( S S2 SA S ) uder the piecewise mdel (4) we prceeded i tw stages. Recall that the prirs jr jr ad jr determie each ther. We first established 2 Cliical Trials 2005; 2:

9 Evet times i early phase cliical trials 475 prirs ( S SA S ) uder the simpler mdel that assumes T T ad T A are mutually idepedet ad we the exteded this prir t accut fr the effect f prgressi the hazard f death uder the piecewise mdel. We prceeded i this way because frm a cliicia s viewpit the piecewise mdel is rather cmplex. The simpler mdel thus serves as a cceptual bridge t check that the prirs S S2 ad S yield a prir S that makes sese. Fr cveiece agai temprarily suppress S ad E. The pssible utcmes described previusly fr ad A w pertai t ad. Allwig the pssibility that a patiet s fllw-up may be ctiued beyd T A we defie T A mi{t A T } ad the idicatr A I(T A T ) that a SAE is bserved. Accutig fr three separate evets ad iterval cesrig f T still assumig idepedece the geeral likelihd is (7) Uder the expetial mdel where each T r r ~ Exp( r ) with r ~ IG(a r b r ) the abve likelihd takes the specific frm L( data A ) (8) Re-itrducig S ad E the six hyperparameters (a S b S a S b S a SA b SA ) characterizig the three idepedet prirs ( S SA S ) may be elicited i may ways. See fr example Chaler et al. [5] r Kadae ad Wlfs [6]. A straightfrward apprach is t elicit the mea ad a 95% credible iterval fr each Sr which tgether determie (a Sr b Sr ). Hwever e must prceed with cauti whe elicitig prirs S S ad S A s that whe cmbied they yield a reasable prir the verall failure time media S. Sice the hazards f idepedet expetials are additive j j j ja fr j E S ad takig meas ad variaces gives the tw equatis ad ( ) L data f f f N = A = f ( T ) ( T i i ) fa( Ti A) i= ( ) i F ik N T T T i A e e i= A A A aj aj aj a = + + b b b b j aj b F F A T i A i i ia j i. ik j aj aj a = + + b b b ja ja ja j j j ja { ik e } k. ( ) i ia (9) (20) Thus give (a S b S a S b S a SA b SA ) it is imprtat t check that the values (a S b S ) resultig frm (9) ad (20) give a prir the verall failure time parameter that makes sese. Algebraically e must determie eight hyperparameters subject t the tw cstraits (9) ad (20) s there are really six pieces f ifrmati. I practice istead f determiig the six hyperparameters (a S b S a S b S a SA b SA ) the right-had sides f (9) ad (20) ad the hpig that the resultig (a S b S ) gives a reasable prir S e may elicit the six pieces f ifrmati while takig advatage f the physicia s familiarity with the verall failure rate. T d this e may first elicit the mea ad 95% CI f S t determie (a S b S ) ad the elicit fur f the six evet-specific hyperparameters. Substitutig these values it (9) ad (20) e may the check that the resultig prir f the remaiig cmpet evet time is reasable. This prcess may be iterated if eeded t calibrate sme f the hyperparameters. We tk this apprach which prduced prir meas ad 95% CI s E( S) 4 (3 5.2) fr the verall failure time media ad E( S ) 7 (2 0) ad E( S ) 2 (9 5) fr the medias f T ad T. The resultig hyperparameters were (a S b S a S b S a SA b SA ) ( ) ad (a S b S ) ( ) as give earlier. This issue is still preset whe usig a expetial evet time distributi such as a Weibull lgrmal r gamma sice i geeral the hazard f verall failure is determied by the hazards f the cmpet evets. T exted this prir t accmmdate the piecewise mdel we ext elicited prirs S ad S 2 subject t the cstrait that the resultig prir S has the abve mea ad 95% CI. This required a iterative prcess f repeatedly specifyig prirs S ad S 2 ad evaluatig the resultig prir f S util this had mea 2 ad 95% CI (9 5). This gave prir mea ad 95% CI f 4 (2 6) fr S ad 5.75 ( ) fr S 2 which imply that (a S b S a S 2 b S 2 ) ( ). As befre we assumed that the meas f the hazards fr E were the same as fr S but we iflated the variaces by multiplyig by 0 s that E( Er ) E( Sr ) ad var( Er ) 0 var( Sr ) fr r ad A. Fr the pst-prgressi death rate we used the smaller multiplier var( E 2 ) 2.5 var( S2 ) t stabilize the cmputatis. This gives the var( E2 ) 0.4 which is very clse t var( E ) 2.8. These values yielded the hyperparameters (a E b E a E b E a E 2 b E 2 a EA b EA ) ( ). T simulate the desig uder the piecewise mdel each sceari is determied by the fur parameters E ( E E 2 E E A ). We chse Cliical Trials 2005; 2:

10 476 F Thall LH Wte ad NM Tair Table 5 Operatig characteristics f the Xelda Gemzar trial based a mdel accutig fr the effect f prgressi the hazard f death. The desig is idetical that summarized i Table 3 but here the hazard f death chages at T uder the uderlyig piecewise hazard mdel. I the ull case ( E E2 E ) ( ) with verall E 4.0. I the alterative case ( E E2 E ) ( ) with verall E 7.0. I bth cases EA 48 iecewise hazard mdel Igrig effect f E ET N. pats. Trial dur. ET N. pats. Trial dur umerical values f E t crrespd t the simpler cases studied previusly with verall media failure time E either 4.0 r 7.0. As befre we calibrated the cut-ff i the stppig rule t btai ET 0.0 i the desirable case which gave p L T assess the effect f accutig fr the chagig hazard f death i each case we als simulated the trial usig the rule (5) based the previus mdel that assumes the hazard f death is t affected by prgressi. The simulatis summarized i Table 5 shw that the desig has very desirable prperties ad that igrig the fact that prgressi icreases the subsequet hazard f death iflates the ET with the ET icreasig 50% frm 0.0 t 0.5 whe E 7.0. A radmized phase II trial Each f the early stppig rules (2) (5) ad (6) is based a E-versus-S cmparis f evet time parameters. A itrisic prblem with cmparig data frm a sigle-arm trial f E t a histrical stadard S usig either frequetist r Bayesia methds is that ay treatmet effect is cfuded by betwee-study effects [7]. This prblem which ca be severe whe trial effects are large relative t treatmet effects arises either whe applyig early stppig rules r whe usig the fial data frm the trial f E t estimate the E-versus-S treatmet differece. These ccers may mtivate a radmized phase II trial f E versus S. The machiery used i each f the previus sectis t cduct a siglearm trial f E may be used with sme simple mdificatis t cstruct a radmized trial. T illustrate this i the geeral case csidered i the previus secti we assume prirs S ad E that are bth idetical t the ifrmative prir E specified i the Establishig prirs secti radmize the 84 patiets fairly betwee E ad S use the early stppig rule (6) fr futility ad als use the additial rule that the trial will be stpped early with E declared prmisig if ( ) < ( + ) r S + S + S A E E + EA > 099. data (2) The three pssible utcmes are that the trial is stpped early due t futility the trial is stpped early with E declared prmisig r the trial rus t cmpleti withut either decisi. I the third case the ivestigatrs may r may t decide t prceed with a phase III trial f E versus S. The simulati results are summarized i Table 6. As might be expected sice w a ifrmative prir is assumed S ad there are average 42 patiets per arm i the ull case the ET fr futility is smaller ad the trial durati is lger tha the cmparable values fr the sigle-arm trial i Table 5 assumig a ifrmative prir S. Hwever a great advatage f radmizig i phase II is that the phase II data may be icrprated it subsequet phase III cmpariss prvided that the patiet etry criteria are the same [8 9]. Table 6 Operatig characteristics f a radmized trial f Xelda Gemzar (X G) versus 5-FU Gemzar (5-FU G) accutig fr iterval cesred prgressi time ad the effect f prgressi the hazard f death. The ull parameter vectr is 0 ( E E2 E EA ) ( ). The alterative parameter vectr is ( ). These give verall media failure times ad 7.0 Early stppig prbabilities N. patiets 5-FU G X G Futility Select X G 5-FU G X G Trial durati Cliical Trials 2005; 2:

11 Evet times i early phase cliical trials 477 Rbustess Thus far we have assumed expetial r piecewise expetial distributis i rder t deal with the cmplicatis addressed i the previus three sectis. If the evet rates are t cstat hwever a mre cmplex distributi may be required. I this secti we examie the rbustess f the E-IG mdel based methd ad CMA ad als illustrate hw a mre cmplex evet time mdel may be implemeted. T d this we first cstruct a ew desig fr mitrig the verall failure rate as befre but w assumig that T fllws a geeralized gamma (GG) distributi. Frmally we assume that T has pdf { } t f( t ) = exp ( t / ) ( ) (22) where ad are all psitive-valued parameters fllwig idepedet lgrmal prirs. Thus six hyperparameters are required t determie the prirs. Settig yields a gamma distributi ad settig yields a Weibull distributi. We shall refer t this as the geeralized gammalgrmal (GG-LN) mdel. T establish lgrmal prirs uder S we used the same elicited mea 4 ad 95% CI ( ) fr S as befre ad als the elicited values 0.25 fr the mea ad 95% CI ( ) f F(2) ad the elicited mea 0.30 ad 95% CI ( ) fr F(6). We slved fr the six lgrmal hyperparameters usig the pealized least squares methd f Thall ad Ck [20]. This yielded lg( S ) distributed rmal with mea.340 ad variace deted S ~ LN( ) ad S ~ LN( ) S ~ LN( ). We assumed lgrmal prirs E E ad E havig the same meas but much larger variaces. Specifically we multiplied each prir variace uder S by the smallest value s that the histrical var S (T) 2. was iflated at least 0-fld which yielded the multiplicati factr 2.6 sice the resultig var E (T) Thus we assumed E ~ LN( ) LN( ) ad s. We did t use a arbitrarily large multipier sice this yields prirs fr which T is likely t take urealistically large values i tur prducig a desig with pr prperties. We used a early stppig rule f the same frm as (2) with p L calibrated t give ET 0.0 whe 7.0 fr T ~ GG with variace equal t that uder the crrespdig expetial distributi. This yielded p L sterirs were cmputed usig the imprtace samplig methd described earlier. T study the rbustess f the E-IG ad GG-LN mdel based rules ad CMA we simulated data frm a Weibull distributi havig 4.0 r 7.0 ad shape parameter r.2 ad als frm a lgrmal distributi havig the give ad variace equal t that f the crrespdig expetial distributi. The results are summarized i Table 7. Fr Weibull data with shape parameter 0.8 all three methds have iflated ET values i the rage whe 7.0. This case is difficult because the hazard is iitially high but mte decreasig s early i the trial a methd must recgize that 7.0 Table 7 Rbustess. Operatig characteristics f the expetial-iverse gamma (E-IG) mdel-based desig the geeralized gamma-lgrmal (GG-LN) mdel-based desig ad CMA fr the Xelda Gemzar trial whe failure times fllw a Weibull distributi with shape parameter r a lgrmal distributi esig ET N. pats. Trial durati ET N. pats. Trial durati T ~ Weibull 0.8 E-IG GG-LN CMA T ~ Weibull.0 E-IG GG-LN CMA T ~ Weibull.2 E-IG GG-LN CMA T ~ Lgrmal E-IG GG-LN CMA Cliical Trials 2005; 2:

12 478 F Thall LH Wte ad NM Tair despite the relatively large umber f early failures that idicate a uacceptably high evet rate. The case.0 is the expetial studied i Table. Here the GG-LN mdel has a slightly iflated ET 0.3 whe 7.0. Whe.2 which prduces mre later evets all three methds have ET values smaller tha the mial 0.0 whe 7.0. Fr lgrmal data this effect is mre pruced. I all cases whe 4.0 CMA is less safe with a substatially smaller ET larger sample size ad lger trial durati cmpared t the ther tw methds. espite the much greater flexibility f the GG-LN mdel the rigial E-IG mdel based methd perfrms very similarly ad is remarkably rbust i mst f the cases studied. Sice the Xelda trial data are available it is f iterest t assess the distributi f T. Of the 84 patiets as f 26 July 2005 there were 8 treatmet failures (46 disease prgressis ad 35 SAEs) with sample media 5.7 weeks virtually idetical t the histrical media with 5-FU G. Gdess-f-fit aalyses uder each f the evet time mdels discussed abve shwed that the lgrmal gave a excellet fit. etig the Kapla Meier estimate by Ŝ KM (t) ad the rmal pdf by uder the lgrmal { Ŝ KM (T i )} shuld be apprximately liear i lg(t i ). The plt shwed gd liearity with R Fr a Bayesia aalysis f these data assumig that T ~ LN( 2 ) with idepedet LN(00) prirs fr ad lg( 2 ) the psterir mea ad 95% CI were 5.9 ( ) weeks fr the media e ad 23.9 ( ) weeks fr the mea e 2/2. iscussi The examples discussed here were chse t illustrate hw e may deal with particular cmplicatis that cmmly ccur whe mitrig evet times i cliical trials. There are a several imprtat issues that we have t addressed. A very imprtat prblem is patiet hetergeeity. While i priciple this may be dealt with by icludig prgstic cvariates i the mdel ad mitrig prcedure it raises the practical issues f dealig with pssible treatmet-cvariate iteractis [2] ad specifyig prirs. This may be difficult fr cvariate parameters ad raises the additial issue f usig empirical versus elicited prirs. Fially it may be desirable t use multiple stppig rules e.g. by specifyig a separate rule fr the SAE rate. Ackwledgemets Refereces. retice R. Surrgate edpits i cliical trials: defiiti ad peratial criteria. Statistics i Medicie 989; 8: Flemig TR retice RL epe MS ad Glidde. Surrgate ad auxiliary edpits i cliical trials with ptetial applicatis i cacer ad AIS research. Statistics i Medicie 994; 3: Buyse M ad Mleberghs G. Criteria fr validati f surrgate edpits i radmized experimets. Bimetrics 998; 54: Begg CB ad Leug H. O the use f surrgate ed pits i radmized trials. J Ryal Statistical Sciety Ser A 2000; 63: Cwles MK. Bayesia estimati f the prprti f treatmet effect captured by a surrgate marker. Statistics i Medicie 2002; 2: Fllma A ad Albert S. Bayesia mitrig f evet rates with cesred data. Bimetrics 999; 55: Rser GL. Bayesia mitrig f cliical trials with failure-time edpits. Bimetrics 2005; 6: Cheug K ad Thall F. Mitrig the rates f cmpsite evets with cesred data i phase II cliical trials. Bimetrics 2002; 58: Thall F Wathe JK Bekele BN Champli RE Baker LO ad Bejami RS. Hierarchical Bayesia appraches t phase II trials i diseases with multiple subtypes. Statistics i Medicie 2003; 22: Spiegelhalter J Abrams KR ad Myles J. Bayesia appraches t cliical trials ad health care evaluati. New rk: Wiley Thall F ad Sim R. ractical Bayesia guidelies fr phase IIB cliical trials. Bimetrics 994; 50: Thall F ad Sug H-G. Sme extesis ad applicatis f a Bayesia strategy fr mitrig multiple utcmes i cliical trials. Statistics i Medicie 998; 7: Owe A ad Zhu. Safe ad effective imprtace samplig. Jural f the America Statistical Assciati 999; 95: Nelder JA ad Mead R. A simplex methd fr fucti miimizati. Cmputer Jural 965; 7: Chaler KM Church T Luis TA ad Matts J. Graphical elicitati f a prir distributi fr a cliical trial. The Statisticia 993; 42: Kadae JB ad Wlfs LJ. rirs fr the desig ad aalysis f cliical trials. I Berry ad Stagl eds. Bayesia bistatistics New rk: ekker 996: Estey EH ad Thall F. New desigs fr phase II cliical trials. Bld 2003; 02: Iue LT Thall F ad Berry A. Seamlessly expadig a radmized phase II trial t phase III. Bimetrics : Liu Q ad ledger G. hase 2 ad 3 cmbiati desigs t accelerate drug develpmet. Jural f the America Statistical Assciati 2005; 00: Thall F ad Ck J. se-fidig based efficacy-txicity trade-ffs. Bimetrics 2004; 60: Thall F Sug H-G ad Estey EH. Selectig therapeutic strategies based efficacy ad death i multi-curse cliical trials. Jural f the America Statistical Assciati 2002; 97: eter Thall s research was partially supprted by NCI grat RO CA Cliical Trials 2005; 2: