Advances in Implementing Non-Parametric Dose-Response Models in Adaptive Designs

Transcription

1 Advances n Implemenng Non-Paramerc Dose-Response Models n Adapve Desgns Professor Andy Greve Deparmen of Publc Healh Scences Kng s College London Thanks o : Peer Müller and Don Berry (MD Anderson), Mke K Smh (Pfzer) Chrs Wer (U. Glasgow), Davd Spegelhaler (U Cambrdge)

2 Movaon - ASTIN Sudy Revsed Two Implemenaon Issues Oulne Fng smoohng models sae-space models, mxed models and splnes Reducng compuaonal nensy of deermnng he Opmal Dose for he nex paen Conclusons

3 Movaon for ASTIN Sudy Approached by a clncal colleague (Dr Mchael Krams) on a sroke program (neuroproecon) Beleved ha many sroke programs had faled because he dose had no esablshed before phase III poor desgns Can we develop a beer desgn for choosng he dose? More effcen, more ehcal? 3

4 Improvemens o Sandard Dose-Response Desgn Increase number of doses placebo + large number - 5 Learn abou dose-response and adap Preven allocang paens o neffecve doses (ETHICAL) Model dose response nor parwse comparson Fuly analyss / early decson makng 4

5 Response Issues n Dose Selecon Increase Number of Doses & Adap Dose 5

6 ASTIN Sudy - Desgn Process Greve and Krams, Cln. Trals 005; : Randomzer Randomze o placebo or opmal dose New Paen Dose o val ranslaon Daa Inerface...Ongong... Updae paen daa Fnd opmal dose for learnng abou ED95 Dose Allocaor Connue Model predcs fnal oucomes Esmae doseresponse curve Decson rule Termnaor Predcve Model Surrogae/ Early Response Sop Bayesan Analyss 6

7 Issues How do we do he updang? How do we predc? How do we choose a dose? How do we sop? How do we model response? 7

8 Mean Change from Baselne SSS The Dose-Response Model: j = f( z j, ) Requremens Mean Response Dose Parameers To model f (z, ) we needed :. a flexble model, allowng nonmonoone curves.. analycal poseror updang (smulaon requred for ermnaor and allocaor) 3. effcen (analyc) compuaon of expeced ules Dose (mg/kg) 8

9 nd Order Polynomal Normal Dynamc Lnear Model Locally around z = Z j a sragh lne wh level j and slope j Parameers of he sragh lnes j j ( z Z j ) change beween doses by addng j a (small) evoluon nose. j j j j j j j j j j j Evoluon Varance = Smooher Z j Z j Z j 9

10 Fng NDLM n ASTIN Sudy Orgnal code wren by Peer Müller (Duke Unv and MD Anderson) Wren n C++ (000s of lnes) Re-engneered o FDA valdaon sandards by a compung company: Tessella A black box makng algorhmc changes dffcul. 0

11 #NDLM usng ASTIN fnal daa model{ # acual responses nobservaon equaon for ( n :I){ y[] ~ dnorm(mu[], sgmanv) # samplng model () mu[] <- hea[d[]] + bea*base[] # jus baselne covaraes () } # 'Dose-response curve, Normal evoluon for (k n :K){ hea[k] ~ dnorm(mu.hea[k], prec.hea); # level dsrbuon (3) mu.hea[k]<-hea[k-] + dela[k-] # nd order NDLM dela[k] ~ dnorm(dela[k-], prec.dela) # random walk onslope } hea[] ~ dnorm(mu.hea0,prec.hea0) # pror onplacebo level (vague) dela[] ~ dnorm(mu.dela0,prec.dela0) # pror onplacebo slope (vague) # Pror dsrbuons sgmanv ~ dgamma(0.00, 0.00) # vague pror onsamplng precson sgma <- / sqr(sgmanv) # samplng sd prec.hea ~ dgamma(0.00, 0.00) # vague pror onlevel precson sgma.hea <- / sqr(prec.hea) prec.dela ~ dgamma(0.00, 0.00) # vague pror onslope precson sgma.dela <- / sqr(prec.dela) bea ~ dunf(-00,00) #unform pror on regresson coeffcen Fng NDLM Usng WnBUGS Code by Davd Spegelhaler # read and conver daa - unused daa needs dummys for ( n : I) { base[] <- BASE[] - mean(base[]) # sandardse covaraes y[] <- CHANGE[] d[]<-dose[] # placebo=, ec up o d = 6 } }

12 Response Comparson of Implemenaons Fnal ASTIN Dose-Response Peer Müller Dose WnBUGS

13 General Lnear Sae Space Model Sandard form : Observaon equaon y h, ~ N( 0, ) (,..,T) and srucural equaons F u, u (u,...,up) ~ MVN( 0, u ) Transon Marx (p x p ) Desgn vecor (px) Sae vecor a me (p x p) dag(,..., delee hese gudes from slde maser before prnng uor gvng o he uclen up ) 3

14 4 Srucural Expanson Hans-Peer Pepho and Joseph O Oguu Smple Sae-Space Models n a Mxed Model Framework, Amer Sa, 6, 4-3 j j u u F F... u ) u F (F u F 0 Subsuon n he observaon equaon gves ) ( z, h z ), ( F h z u z x u u F h F h y p j j T j j General Mxed Model : y=x +Z +

15 SAS Code o Local Lnear Trend Model wh Baselne (ASTIN Model) proc mxed lognoe mehod=reml; model SSS=Dose base_sss/oup=pred ddfm=kr alphap=0.; random z_-z_6 z_-z_6/sub=nercep ype=oep(); In heory hs code could be used o f he model used for ASTIN. Proc Mxed now performs Bayesan Analyses usng he Pror Saemen Quesonable prors Jeffreys, fla 5

16 6 nd Order Random Walk Model If he fed model s no smooh enough mpose he resrcon whch s equvalen o 0 ), ~ N(, ), ~ N(, ), ~ N(, y 0 0 0

17 7 nd Order Log-lkelhood The log-lkelhood of n s Leng hen Objecve funcon of a splne n n y n n y

18 Splnes and Sae Space Model I follows ha: Cubc splnes are a specal case of he general sae space model Splnes can be fed by sae space model echnques n parcular as a mxed model The smooher can be esmaed as he rao of he varances of he observaon and srucural equaons Lnk: Mke Kenward s work on splnes & mxed models. Also relevan: 8

19 ASTIN Approach o Deermnng Dose (Ignorng he longudnal Aspec ) For each dose z smulae fuure observaons y zm (m=,,m) Add y zm o curren daa and repea MCMC analyss o ge p(g( D,y zm ) (g( ) s he arge parameer) Esmae Uly: U zm =-V(g( D,yz m ) Esmaed expeced uly U z by akng he smulaon average (over m) ALLOCATOR: U(z) Choose Z o maxmse uly THIS IS COMPUTATIONALLY INTENSIVE PARTICULARLY WHEN SIMULATING CLINICAL TRIALS DOSE 9

20 Smulaon Problem Sngle Scenaro Sudy Sudy k Sudy S Pa Pa Pa j Pa n MCMC Updae D/R model (T poseror smulaons) For each dose (d=,..,d) predc M paen response y zm (m=,..,m) For each y zm updae D/R model (Q poseror smulaons) THIS IS COMPUTATIONALLY INTENSIVE 0

21 Termnaon Rule Decson heorec Poseror Probables Pror Dsrbuon Unnformave (fla) pror Informave pror Facors of he Smulaon Expermen Varance of Pror Consan Less varable a low end of he dose range Level of Smoohng Hgh smoohng Low Smoohng Allocaon Crera Var(ED95) x Var(f(z*)) Var(f(z*)) Var(ED95) Deermnan of Covar ((ED95) and Var(f(z*)) Randomsaon Mehod Probably of allocaon proporonal o expeced uly Allocae o maxmum uly Probably unform over doses s 0.9f(z 0 ) < E[f(z) Y] <.0f(z 0 ) Probably unform over doses s 0.9f(z 0 ) < E[f(z) Y] 4 x 4 expermen /4 replcae Alasng Srucure

22 Change from baselne Smulaed Dose-Response Curves Hegh : or 4 ps delee hese gudes from slde maser before Doseprnng or gvng o he clen

23 Compuaonal Effcency 3

24 Generang random varables Suppose ha you can easly generae a random sample from densy g( ) or s already generaed BUT we wan o generae a sample from h( ) where h( ) f( )/ f( ) d Can we form a sample from h( ) gven only a sample from g( ) and he funconal form of f( )? The answer s yes - f here s M > 0 such ha f( )/g( ) M, 4

25 Accepance/Rejecon (Von Neumann) f(x) Y M. Smulae ~g( ). Smulae Y~U(0,M) 3. Accep f Y < f(x), else Rejec 5

26 The Weghed Boosrap If M s no readly avalable s sll possble o approxmaely resample from h( ) f( )/ f( ) follows. d as Gven (=,..., n) a sample from g( ), calculae w = f( )/g( ) and hen Draw * from he dscree dsrbuon over... n placng mass q on q w / n w Then * s approxmaely dsrbued accordng o h wh he approxmaon "mprovng as n ncreases. 6

27 Pror o Poseror Transformaon How does Bayes's Theorem generae a poseror sample from a pror sample? For fxed x, defne f x ( ) =l( ; x)p( ). ˆ;x If ˆ maxmzes l( ; x), le M = ( ) Then wh g( ) = p( ), apply he rejecon mehod o oban samples from he densy correspondng o he sandardzed f x he poseror densy p( lx). 7

28 Pror o Poseror Transformaon So Bayes Theorem, as a mechansm for generang a poseror sample from a pror sample, akes he form: For each n he pror sample accep no he poseror sample wh probably f x ( Mp( ) ) ( ( ;x) ˆ;x) oherwse rejec. 8

29 Pror o Poseror Transformaon The lkelhood acs as a resamplng probably; hose n he pror sample havng hgh lkelhood are more lkely o be reaned n he poseror sample. Snce p( x) = l( ; x)p( ), we can also sraghforwardly resample from he pror usng he weghed boosrap wh q n ( ;x)/ ( ;x) 9

30 Pared Comparson wh Tes Exenson of Bradley-Terry Model : Davdson JASA(970) Background more han reamen / paen preference for a reamen reamen preferences can be descrbed n erms of an underlyng connuum value of an ndvdual reamen s >0, = Probably Model (Bradley-Terry) Probably reamen preferred o j (>j) s P( j) delee hese gudes from slde maser before j prnng or gvng o he clen 30

31 Pared Comparson wh Tes Exenson of Bradley-Terry Model : Davdson JASA(970) Davdson(970) generalzed model o accoun for no preference (=j) Probably Model (Davdson) P( j) P( j ) P( j) j j j j j j j j s an ndex of dscrmnaon 3

32 Pared Comparson wh Tes Exenson of Bradley-Terry Model : Davdson JASA(970) Probably Model - Treamens P(A B) ( ) P(B A) ( ) P(A B) ( ( ) ) Lkelhood n paens : r prefer A s prefer B ( n r s)/ ( n r s)/ n r s ( ) ( ) n Preference Number of Paens A>B 9 B>A 3 delee hese gudes A=B from slde maser before 6 prnng or gvng o he clen 3

33 Poseror from Pror Resamplng For purposes of llusraon assume he jon pror dsrbuon for and, s unform over he recangle 0<

34 Poseror from Pror Resamplng Poseror generaed by weghed boosrap Skewness of he margnal poserors one posve, one negave Parameers posvely correlaed

35 Wer e al J Boph Sas 007;7: Ineresed n E(g( D,y m )) E[g( ) D,y m ] g( g( )p( )p(y D,y zm zm )p( )d D)d p(y zm )p( D)d From a MCMC sample (=,,T) can be approxmaed by g( )p(y zm ) p(y zm ) g( )w Where he weghs w p(y zm ) p(y zm ) are proporonal o he lkelhood of he predced observaon Wha abou M and T? 35

36 Wer e al J Boph Sas 007;7: Smulaons Invesgaed: M=,5,0,500 T=000, 000, 5000 or 0000 For M = wo approaches a sngle fuure observaon a each dose, or seng he observaon exacly a s expeced value usng he curren value n he MCMC run. Smulaons sugges ha he laer approach wh T=0000 provdes relable denfcaon of he opmal dose for learnng 36

37 Smh e al Pharmaceu. Sas. 006; 5: Fng he NDLM Usng WnBUGS #NDLM usng ASTIN fnal daa model{ # acual responses n observaon equaon for ( n :I){ y[] ~ dnorm(mu[],sgmanv) # samplng model () mu[] <- hea[d[]] + bea*base[] # jus baselne covaraes () } # 'Dose-response curve, Normal evoluon for (k n :K){ hea[k] ~ dnorm(mu.hea[k],prec.hea); # level dsrbuon (3) mu.hea[k]<-hea[k-]+ dela[k-] # nd order NDLM dela[k] ~ dnorm(dela[k-], prec.dela) # random walk on slope } hea[] ~ dnorm(mu.hea0,prec.hea0) # pror on placebo level (vague) dela[] ~ dnorm(mu.dela0,prec.dela0) # pror on placebo slope (vague) # Pror dsrbuons sgmanv ~ dgamma(0.00, 0.00) # vague pror on samplng precson sgma <- / sqr(sgmanv) # samplng sd prec.hea ~ dgamma(0.00, 0.00) # vague pror on level precson sgma.hea <- / sqr(prec.hea) prec.dela ~ dgamma(0.00, 0.00) # vague pror on slope precson sgma.dela <- / sqr(prec.dela) bea ~ dunf(-00,00) #unform pror on regresson coeffcen # read and conver daa - unused daa needs dummys for ( n : I) { base[] <- BASE[] - mean(base[]) # sandardse covaraes y[] <- CHANGE[] d[]<-dose[] # placebo=, ec up o d = 6 } } The WnBUGS code for fng he NDLM s smple BUT no necessarly effcen as he number of paens grows he processng of more daa slows WnBUGS ASTIN ended wh over 900 paens The process can be speeded by rewrng he model n erms of suffcen sascs Ousde WnBUGS suffcen sascs are deermned (usng SAS, S+, R ec) and passed o WnBUGS 37

38 Conclusons Desgn-focused adapve approaches wll be of connung neres Adapve desgns requre consderable smulaons Roune mplemenaon requres hough on how Whch models are approprae How models can be fed How doses can be chosen 38