A: Details of MCMC for Fitting DDP-GP Model

Transcription

1 Supplement: Bayesan Nonparametrc Estmaton for Dynamc Treatment Regmes wth Sequental Transton Tmes A: Detals of MCMC for Fttng DDP-GP Model Summary of Model p(y k x k, F k ) = F k (y k x k ) F k DDP-GP { {µ k h}, C k ; α k, {β k h}, σ k} (1) F k (y x k ) = wh k N(y; θh(x k k ), σ k ). (2) h=0 {θ k h(x k )} GP (µ k h(x k ), C k (x k )). h = 1, 2,... µ k h(x k ) = x k β k h. k = 1,..., n trans. We complete the model constructon by assumng βh k N(βk 0, Σ k 0), (σ k ) 2..d. Ga(λ 1, λ 2 ) and α k..d. Ga(λ 3, λ 4 ). Posteror Computaton To evaluate the posteror n a DDP-GP model, we frst margnalze (1) analytcally wth respect to the random probablty measures F k ( x k ). The form s not obvous from the earler defnton. Temporarly suppress the superscrpted transton ndex k. Consder generatng a sample (Y 1, x 1 ),, (Y n, x n ) by frst samplng from a covarate dstrbuton, p(x), and then from a condtonal transton tme dstrbuton, F ( x). We rewrte (2) as a herarchcal model wth a new latent ndcator varable γ for the normal mxture summand ndex h, (Y γ = h, x ) N(θ h (x ), σ 2 ) and p(γ = h) = w h, (3) for = 1,, n. Let θ ( ) = θ γ ( ) denote a realzaton of the stochastc process selected by γ. Next, we re-ndex the θ h ( ) such that n =1 I(γ = h) 1 for h = 1,..., H. That s, we let h = 1,..., H ndex the realzatons θ h ( ) that are selected by some of the γ s, so 1

2 that {θ 1,, θ H } are the unque values of the n realzatons { θ, = 1,..., n}. If clusters of patents are defned as S h = { : θ = θ h }, then the γ s are nterpreted as cluster membershp ndcators. Posteror smulaton makes use of these ndcators and the vectors θ h = (θ h (x 1 ),..., θ h (x n )). After margnalzaton wth respect to F x, we are left wth the margnal model for {γ, θ h (x ); = 1,..., n, h = 1,..., H}. For each transton k, we update parameters usng fnte DP algorthm as follows. Denote #{ : γ k = h} = n k h. Update σ k H (σ k ) 2 Inverse Gamma(λ 1 + nk 2, λ h=1 γ 2 + k =h(yk θh k(x )) 2 ) (4) 2 Update θ k h p(θ k h ) p(θ k h) :γ k =h p(y k θ k h(x k )) exp{ 1 2 (θk h X k β k h) (C k ) 1 (θ h X k β k h)} exp{ :γ k =h(yk θ k h (x )) 2 2(σ k ) 2 } N(((C k ) 1 + U U (σ k ) 2 I) 1 (U y k h (σ k ) 2 + (Ck ) 1 X k β k h), ((C k ) 1 + U U (σ k ) 2 I n k n k) 1 ), where y k h = {yk, γ k = h}, I n k n k s an nk n k dentty matrx, X k s an n k M k matrx wth the covarates x k of the -th patent n row. U s a n k h nk matrx: f patent s the j-th element of γ k = r, then U j = 1. All other elements are 0. Update β k h p(β k h ) p(β k h) exp{ 1 2 (θk h X k β k h) (C k ) 1 (θ k h X k β k h)} N(Σ k h[((x k ) (C k ) 1 θ k h + Σ k 0β k 0], Σ k h), where Σ k h = ((Xk ) (C k ) 1 X k + (Σ k 0) 1 ) 1. 2

3 Update w k h v k h Beta(1 + n k h, α k + j>h n k j ), where n k h = n k =1 I(γk = h) s the number of observatons such that γ k = h. Then wh k = vk h j>h (1 vk j ). Update γ k If y k s not censored, P r(γ k = h ) w k h p(y k θ k h(x ))p(θ k h(x k ) θ k h(x k ))d(θ k h(x k )), where x k = {x k j : γ k j = h, j }. If y k s censored, Let p k h(t) = p(y k θ k h(x ))p(θ k h(x k ) θ k h(x k ))d(θ k h(x k )). Then P r(γ k = h ) V k w k hp k h(t)dt, Update α k where V k s the observed tme for patent n transton k. Usng data augmentaton, we frst sample an m from beta dstrbuton beta(α k +1, n k ). Then we sample the new α k value from α k πga(λ 3 + H, λ 4 log(m)) + (1 π)ga(λ 3 + H 1, λ 4 log(m)), where π = 1 π λ 3+H 1 n k (λ 4 log(m)). 3

4 B: Determnng Pror Hyperparameters As prors for βh k n (1) we assume βk h N(βk 0, Σ k 0) for each transton k, h = 1, 2,.... For σ k we assume (σ k ) 2..d. Ga(λ 1, λ 2 ). And, fnally, α k..d. Ga(λ 3, λ 4 ). To apply the DDP-GP model, one must frst determne numercal values for the fxed hyperparameters {β k 0, Σ k 0, k = 1, 2,...} and λ = (λ 1, λ 2, λ 3, λ 4 ). Ths s a crtcal step. These numercal hyperparameter values must facltate posteror computaton, and they should not ntroduce napproprate nformaton nto the pror that would nvaldate posteror nferences. Wth ths n mnd, the hyperparameters (β k 0, Σ k 0) for the k th transton tme covarate effect dstrbuton may be obtaned va emprcal Bayes by dong a prelmnary fts of a lognormal dstrbuton Y k = log(t k ) N(x k β k 0, σ k 0) for each transton k. Smlarly, we assume a dagonal matrx for Σ k 0 wth the dagonal values also obtaned from the prelmnary ft of the lognormal dstrbuton. Once an emprcal estmate of σ k s obtaned, one can tune (λ 1, λ 2 ) so that the pror mean of σ k matches the emprcal estmate and the varance equals 1 or a sutably large value to ensure a vague pror. Fnally, nformaton about α k typcally s not avalable n practce. We use λ 3 = λ 4 = 1. Ths approach works n practce because the parameter β k 0 specfes the pror mean for the mean functon of the GP pror, whch n turn formalzes the regresson of T k on the covarates x k, ncludng treatment selecton. The mputed treatment effects hnge on the predctve dstrbuton under that regresson. Excessve pror shrnkage could smooth away the treatment effect that s the man focus. The use of an emprcal Bayes type pror n the present settng s smlar to emprcal Bayes prors n herarchcal models. Ths type of emprcal Bayes approach for hyperparameter selecton s commonly used when a full pror elctaton s ether not possble or s mpractcal. Inference s not senstve to values of the hyperparameters λ that determne the prors of σ k and α k for two reasons. Frst, the standard devaton σ k s the scale of the kernel that s used to smooth the dscrete random probablty measure generated by the DDP pror. It s mportant for reportng a smooth ft, that s for dsplay, but t s not crtcal for the mputed fts n our regresson settng. Assumng some regularty of the posteror mean functon, smoothng adds only mnor correctons. Second, the total mass parameter α k determnes the number of unque clusters formed n the underlyng Polya urn. However, because most clusters are small changng the pror of α k does not sgnfcantly change the posteror predctve values that 4

5 are the bass for the proposed nference. The conjugacy of the mpled multvarate normal on {θh k(xk ), = 0,..., n} and the normal kernel n (2) greatly smplfy computatons, snce any Markov chan Monte Carlo (MCMC) scheme for DP mxture models can be used. MacEachern and Müller (1998) and Neal (2000) descrbed specfc algorthms to mplement posteror MCMC smulaton n DPM models. Ishwaran and James (2001) developed alternatve computatonal algorthms based on fnte DPs, whch truncated (2) after a fnte number of terms. C: Survval Tme Regresson Smulaton Ths smulaton was desgned to study the DDP-GP regresson model by comparng nference for a survval functon wth the smulaton truth. In ths study, we dd not evaluate a regme effect, but rather focused on nference for the survval curve. For each subject, we generated T = survval tme, the covarates x 1 = tumor sze (0=small, 1=large) and x 2 = body weght, and x 3 = a bomarker (0=absent, 1=present). We assumed that small and large tumor szes each had probablty.50. Body weghts were computed by samplng from a unform dstrbuton, Unf(80, 150), wth the covarate x 2 defned by shftng and scalng to obtan mean 0 and varance 1. The bomarker was assocated wth tumor sze, as follows. Patents n the large tumor sze group were bomarker negatve wth probablty 0.7 and bomarker postve wth probablty 0.3. Patents wth small tumor sze were bomarker negatve wth probablty 0.3 and bomarker postve wth probablty 0.7. Let Y LN(m, s) denote a lognormal random varable Y = log T for T N(m, s). By a slght abuse of notaton, we also use LN(m, s) to denote the lognormal p.d.f. Let x = (1, x,1, x,2, x,3 ) denote the covarates for patent, here we nclude 1 n the covarate to ndcate the ntercept. We smulated each sample Y 1,, Y n of n observatons from a mxture of lognormal dstrbutons, Y x 0.4 LN(x β 1, σ 2 ) LN(x β 2, σ 2 ), where the true covarate parameters of the mxture components were β 1 = (1, 2, 2, 1) and β 2 = (2, 1, 3, 3), wth σ 2 = 0.4. For comparson, we also ft an AFT regresson model, assumng Y = log(t ) = x β + σɛ, = 1,..., n wth ɛ followng an extreme value dstrbuton, so that T follows a Webull dstrbuton. 5

6 In ths smulaton, we consdered four scenaros, wth n = 50, 100, or 200 observatons wthout censorng or n = 200 wth 23% censorng. For each scenaro, N = 1, 000 trals were smulated. For each smulated data set we ft a DDP-GP survval regresson model F (Y x ). For smulaton j, let S(t x) = p(t n+1 t x n+1,j = x, data) denote the posteror expected survval functon for a future patent wth covarate x. Usng the emprcal dstrbuton 1 n n =1 δ x to margnalze w.r.t. x j n+1,j and averagng across smulatons, we get S(t) = 1 N N j=1 1 n n S(t x j ). =1 Fgure S1 compares S( ) under the DDP-GP model wth the smulaton truth S 0 (t) = 1 N N j=1 1 n n S 0 (t x j ), =1 and maxmum lkelhood estmates (MLE) under Webull AFT, Lognormal AFT, and Exponental AFT models. In each scenaro, the true curve s gven as a sold black sold lne, the MLE of the survval functons under the AFT regresson model assumng Webull dstrbuton, Lognormal dstrbuton and Exponental dstrbuton as green, blue, magenta sold lnes respectvely, and the posteror mean survval functon under the DDP-GP model as a sold red lne wth pont-wse 90% credble bands as two dotted red lnes. In all four scenaros, the DDP-GP model based estmate relably recovered the shape of the true survval functon and avoded the excessve bas seen wth the Webull, lognormal and exponental MLE. As expected, the three scenaros wthout censorng show that ncreasng sample sze gves more accurate estmaton. Wth 23% censorng, the DDP-GP estmate becomes less accurate, but t stll s much closer to the smulaton truth than the AFT regresson models wth Webull, lognormal and exponental dstrbutons. D: Computng Mean Survval Tme The rsk sets of the seven transton tme n the leukema tral are defned as follows. Let G 0 = {1,..., n} denote the ntal rsk set at the start of nducton chemotherapy, and G (0,r) = { : s 1 = r} for r = D, C, R, so G 0 = G (0,D) G (0,C) G (0,R). Smlarly, G (C,P ) = 6

7 Npat=50, wthout censorng Npat=100, wthout censorng Survval Truth DDP GP Webull Lognormal Exponental Survval Truth DDP GP Webull Lognormal Exponental Tme Tme Npat=200, wthout censorng Npat=200, wth 23% censorng Survval Truth DDP GP Webull Lognormal Exponental Survval Truth DDP GP Webull Lognormal Exponental Tme Tme Fgure S1: Smulaton 1. True mean survval functons (black color) and estmated mean survval functons under the DDP-GP model (red color) for sample szes n = 50, 100, 200 and n = 200 wth 23% censorng for 1,000 smulatons. For comparsons, we also show the MLE under an AFT regresson wth Webull dstrbuton (green color), Lognormal dstrbuton (blue color) and Exponental dstrbuton (magenta). In all cases, the pont-wse 90% credble bands are also dsplayed as the regon between two dotted red lnes. { : s 1 = C, s 2 = P } s the later rsk set for T (P,D). To record rght censorng, let U denote the tme from the start of nducton to last 7

8 followup for patent. We assume that U s condtonally ndependent of the transton tmes gven pror transton tmes and other covarates. Censorng of event tmes occurs by competng rsk and/or loss to follow up. For a patent n the rsk set for event tme T k, let δ k δ (0,D) = 1 f a patent s not censored and 0 f patent s rght censored. For example, = 1 for G 0 f T (0,D) = mn(u, T (0,k), k = D, C, R). Smlarly, δ (R,D) + T (C,P ) G (0,R) f T (0,R) + T (R,D) < U and δ (P,D) = 1 for G (C,P ) f T (0,C) = 1 for + T (P,D) < U. Let V x, denote the observed tme for patent n rsk set G x, as follows. For G 0 let V 1, = mn(t (0,D), T (0,R), T (0,C), U ) denote the observed tme for the stage 1 event or censorng. For G (0,C) let V C = mn(t (C,D), T (C,P ), U T (0,C) ) denote the observed event tme for the competng rsks D and P and loss to followup. Smlarly, for G (0,R), let V R = mn(t (R,D), U T (R,D) ), and for G (C,P ) let V (C,P ), = mn(t (P,D), U T (0,C) T (C,P ) ). The jont lkelhood functon s the product L = L 1 L 2 L 3 L 4. The frst factor L 1 corresponds to response to nducton therapy, L 1 = G 0 k {D,R,C} f (0,k) (V 1, x (0,k) ) δ(0,k) S (0,k) (V 1, x (0,k) ) 1 δ(0,k). (5) where S k = 1 F k. The second factor L 2 corresponds to patents G (0,R) who experence resstance to nducton and receve salvage Z 2,1, L 2 = f (R,D) (V R x (R,D) G (0,R) ) δ(r,d) S (R,D) (V R x (R,D) ) 1 δ(r,d). (6) The thrd factor L 3 s the lkelhood contrbuton from patents achevng CR, L 3 = G (0,C) k=(c,d),(c,p ) f k (V C x k ) δk S k (V C x k ) 1 δk. (7) The fourth factor L 4 s the contrbuton from patents who experence tumor progresson after CR L 4 = f (P,D) (V CP, x (P,D) G (C,P ) ) δ(p,d) S (P,D) (V CP, x (P,D) ) 1 δ(p,d). (8) 8

9 + The mean survval tme of a patent treated wth regme Z = (Z 1, Z 2,1, Z 2,2 ) s [ ] η(z) = p(s 1 = D x 0, Z 1 )η (0,D) (x 0, Z 1 ) d p(x 0 ) { [ + p(s 1 = R x 0, Z 1 ) η R (x 0, Z 1 ) + ]} η (R,D) (x 0, Z 1, Z 2,1, T (0,R) )dµ(t (0,R) ) d p(x 0 ) [ [ p(s 1 = C x 0, Z 1 ) η C (x 0, Z 1 ) + p(s 2 = D s 1 = C, x 0, Z 1, T (0,C) )η (C,D) (x 0, Z 1, T C ) + p(s 2 = P s 1 = C, x 0, Z 1, T (0,C) )[η (C,P ) (x 0, Z 1, T (0,C) ) ] + η (P,D) (x 0, Z 1, Z 2,2 T (0,C), T (C,P ) )dµ(t (C,P ) )]dµ(t (0,C) ) d p(x 0 ). (9) We compute the IPTW estmates for overall mean survval wth regme Z as IP T W (Z) = n w (Z)T / =1 n w (Z), (10) =1 where w (Z) = I(Z = Z [ )δ I(s 1 = D) + I(s 1 = R)I (Z ˆK(U 2,1 )/ ˆPr(Z 2,1 s 1 = R, Z 1, x 0, T (0,R) ) ) +I(s 1 = C, s 2 = D) ] +I(s 1 = C, s 2 = P )I (Z 2,2 )/ ˆPr(Z 2,2 s 1 = C, s 2 = P, Z 1, x 0, T (0,C), T (C,P ) ). (11) In (11), ˆK s the Kaplan-Meer estmator of the censorng survval dstrbuton K(u) = P (U t) at tme t. I (Z) s s an ndctor of treatment Z and 0 otherwse, and ˆPr(Z 2,1 s 1 = C, Z 1, x 0, T (0,R) ) s the probablty of recevng salvage treatment Z 2,1 estmated usng logstc regresson, and smlarly for ˆPr(Z 2,2 s 1 = C, s 2 = P, Z 1, x 0, T (0,C), T (C,P ) ). The above estmator has been shown to be consstent under sutable assumptons (Wahed and Thall, 2013; Scharfsten et al., 1999). 9

10 E: Survval Regresson for T (C,P ) and T (P,D) Here we summarze results for the survval regresson for T (C,P ). Among the n = 210 patents, 102 (48.6%) acheved C, wth C rates of 37%, 48%, 53% and 56% n the FAI, FAI plus ATRA, FAI plus GCSF and FAI plus GCSF plus ATRA arms, respectvely. Of the 102 patents who acheved CR, 93 experenced dsease progresson before death or beng lost to follow-up. Among these 93 relapsed patents, 53 receved salvage treatment wth HDAC. For a hypothetcal future patent at age 61 and poor cytogenetc abnormalty, Fgure S2 summarzes survval regresson functons for each of the four nducton therapes, wth sold lnes representng T (0,C) = 20 and dashed lnes representng T (0,C) = 30. The four dashed lnes are below the four correspondng sold lnes, ndcatng that T (0,C) was assocated wth T (C,P ). Ths observaton concdes wth the well-known phenomenon n chemotherapy for AML or MDS that, regardless of nducton therapy, the longer t takes to acheve C, the shorter the perod that the patent remans n C. Smlarly, we summarze results for the survval regresson for T (P,D). For a patent wth poor cytogenetc abnormalty, Fgure S3 shows the posteror predcted survval functons under dfferent combnatons of nducton therapy and age. Panels (a) and (c) show the survval functons of a patent assgned salvage treatment HDAC wth age 46 or 76, whle panels (b) and (d) plot the correspondng survval functons for the patent assgned non HDAC as salvage. Four dfferent colors represent the four nducton therapes. Fgure S3 shows that resdual tme to D after dsease progresson followng C was assocated wth both age and salvage therapy. Older patents are more lkely to have shorter resdual lfe once ther dsease progressed, and patents gven HDAC as salvage de more quckly than patents gven non HDAC salvage. References Ishwaran, H. and James, L. F. (2001). Gbbs samplng methods for stck-breakng prors. Journal of the Amercan Statstcal Assocaton, 96(453). MacEachern, S. N. and Müller, P. (1998). Estmatng mxture of drchlet process models. Journal of Computatonal and Graphcal Statstcs, 7(2):

11 The effect of T C on T CP Survval FAI FAI+ATRA FAI+GCSF FAI+ATRA+GCSF Tme Fgure S2: The effect of T (0,C) on T (C,P ) at age 61 wth poor cytogenetc abnormalty. Black, red, green and blue curves represent nducton treatments FAI, FAI+ATRA, FAI+GCSF and FAI+ATRA+GCSF, respectvely. Sold lnes and dash lnes represent T (0,C) = 20 and T (0,C) = 30, respectvely. The longer t takes to acheve C, the shorter the perod of tme that the patent remaned n C. Neal, R. M. (2000). Markov chan samplng methods for drchlet process mxture models. Journal of computatonal and graphcal statstcs, 9(2): Scharfsten, D. O., Rotntzky, A., and Robns, J. M. (1999). Adjustng for nongnorable drop-out usng semparametrc nonresponse models. Journal of the Amercan Statstcal Assocaton, 94(448): Wahed, A. S. and Thall, P. F. (2013). Evaluatng jont effects of nducton salvage treatment regmes on overall survval n acute leukaema. Journal of the Royal Statstcal Socety: Seres C (Appled Statstcs), 62(1):

12 Salvage = HDAC, Age=46 Salvage = non HDAC, Age=46 Survval FAI FAI+ATRA FAI+GCSF FAI+ATRA+GCSF Survval FAI FAI+ATRA FAI+GCSF FAI+ATRA+GCSF Tme Tme (a) (b) Salvage = HDAC, Age=76 Salvage = non HDAC, Age=76 Survval FAI FAI+ATRA FAI+GCSF FAI+ATRA+GCSF Survval FAI FAI+ATRA FAI+GCSF FAI+ATRA+GCSF Tme Tme (c) (d) Fgure S3: AML-MDS tral data n transton (P, D): Panels (a) and (c) show the posteror estmated survval functons of patent at age 46 and 76 wth poor cytogenetc abnormalty assgned to salvage treatment HDAC for four nducton therapes respectvely. Panels (b) and (d) show the posteror estmated survval functons of patent at age 46 and 76 wth poor cytogenetc abnormalty assgned to salvage treatment non HDAC for four nducton therapes respectvely. Black, red, green and blue curves represent nducton treatments FAI, FAI+ATRA, FAI+GCSF and FAI+ATRA+GCSF, respectvely. 12