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Two-Stage Randomization for for Competingrisks and Survival outcomes

Klaus Holst & Thomas Scheike

2025-10-21

Two-Stage Randomization for for Competing risks and Survivaloutcomes

Under two-stage randomization we can estimate the average treatmenteffect\(E(Y(i,\bar k))\) of treatmentregime\((i,\bar k)\).

The estimator can be agumented in different ways: using the tworandomizations and the dynamic censoring augmentatation.

Estimating\(\mu_{i,\bar k} = P(Y(i,\bark,\epsilon=v) <= t)\), restricted mean\(E( \min(Y(i,\bar k),\tau))\) or years lost\(E( I(\epsilon=v) \cdot(\tau - \min(Y(i,\bar k),\tau)))\) using IPCW weightedestimating equations : \

The solved estimating eqution is\[\begin{align*}\sum_i \frac{I(min(T_i,t) < G_i)}{G_c(min(T_i ,t))} I(T \leq t,\epsilon=1 ) - AUG_0 - AUG_1 + AUG_C - p(i,j)) = 0\end{align*}\] using the covariates from augmentR0 to augmentwith\[\begin{align*}AUG_0 = \frac{A_0(i) - \pi_0(i)}{ \pi_0(i)} X_0 \gamma_0\end{align*}\] and using the covariates from augmentR1 to augmentwith\[\begin{align*}AUG_1 = \frac{A_0(i)}{\pi_0(i)} \frac{A_1(j) - \pi_1(j)}{ \pi_1(j)} X_1\gamma_1\end{align*}\] and censoring augmenting with\[\begin{align*} AUG_C = \int_0^t \gamma_c(s)^T (e(s) - \bar e(s)) \frac{1}{G_c(s) }dM_c(s)\end{align*}\] where\(\gamma_c(s)\) is chosen to minimize thevariance given the dynamic covariates specified by augmentC.

Standard errors are estimated using the influence function of allestimators and tests of differences can therefore be computedsubsequently.

Data must be given on start,stop,status survival format with

library(mets)set.seed(100)n<-200ddf<- mets:::gsim(n,covs=1,null=0,cens=1,ce=1,betac=c(0.3,1))true<-apply(ddf$TTt<2,2,mean)true#> [1] 0.740 0.715 0.340 0.350datat<- ddf$datat## set-random response on data, only relevant after status==2response<-rbinom(n,1,0.5)datat$response<-as.factor(response[datat$id]*datat$Count2)datat$A000<-as.factor(1)datat$A111<-as.factor(1)bb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.f~A0.f,augmentR1=~X11+X12+TR,augmentR0=~X01+X02,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f))bb#> Simple estimator :#>                              coef#> A0.f=1, response*A1.f=1 0.5723748 0.19721139#> A0.f=1, response*A1.f=2 0.7354573 0.08755875#> A0.f=2, response*A1.f=1 0.2834829 0.10028201#> A0.f=2, response*A1.f=2 0.4790373 0.10007200#>#> First Randomization Augmentation :#>                              coef#> A0.f=1, response*A1.f=1 0.5900338 0.21419118#> A0.f=1, response*A1.f=2 0.7428143 0.08889887#> A0.f=2, response*A1.f=1 0.2721992 0.10288280#> A0.f=2, response*A1.f=2 0.4671886 0.10273188#>#> Second Randomization Augmentation :#>                              coef#> A0.f=1, response*A1.f=1 0.5564110 0.24533756#> A0.f=1, response*A1.f=2 0.7185240 0.09841671#> A0.f=2, response*A1.f=1 0.2778309 0.09021056#> A0.f=2, response*A1.f=2 0.4685339 0.09839184#>#> 1st and 2nd Randomization Augmentation :#>                              coef#> A0.f=1, response*A1.f=1 0.5981472 0.26485855#> A0.f=1, response*A1.f=2 0.7302633 0.10170090#> A0.f=2, response*A1.f=1 0.2699247 0.09093482#> A0.f=2, response*A1.f=2 0.4620365 0.09859532estimate(coef=bb$riskG$riskG01[,1],vcov=crossprod(bb$riskG.iid$riskG01))#>                         Estimate Std.Err    2.5%  97.5%   P-value#> A0.f=1, response*A1.f=1   0.5981 0.26486 0.07903 1.1173 2.392e-02#> A0.f=1, response*A1.f=2   0.7303 0.10170 0.53093 0.9296 6.946e-13#> A0.f=2, response*A1.f=1   0.2699 0.09093 0.09170 0.4482 2.994e-03#> A0.f=2, response*A1.f=2   0.4620 0.09860 0.26879 0.6553 2.783e-06estimate(coef=bb$riskG$riskG01[,1],vcov=crossprod(bb$riskG.iid$riskG01),f=function(p)c(p[1]/p[2],p[3]/p[4]))#>                         Estimate Std.Err   2.5% 97.5%  P-value#> A0.f=1, response*A1.f=1   0.8191  0.3600 0.1135 1.525 0.022884#> A0.f=2, response*A1.f=1   0.5842  0.2249 0.1433 1.025 0.009399estimate(coef=bb$riskG$riskG01[,1],vcov=crossprod(bb$riskG.iid$riskG01),f=function(p)c(p[1]-p[2],p[3]-p[4]))#>                         Estimate Std.Err    2.5%   97.5% P-value#> A0.f=1, response*A1.f=1  -0.1321   0.266 -0.6534 0.38920  0.6194#> A0.f=2, response*A1.f=1  -0.1921   0.129 -0.4450 0.06076  0.1365
## 2 levels for each response , fixed weightsdatat$response.f<-as.factor(datat$response)bb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.f~A0.f*response.f,augmentR0=~X01+X02,augmentR1=~X11+X12,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f),estpr=c(0,0),pi0=0.5,pi1=0.5)bb#> Simple estimator :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.5195752 0.18171540#> A0.f=1, response.f*A1.f=2,1 0.5264182 0.18212146#> A0.f=1, response.f*A1.f=1,2 0.6343225 0.09083112#> A0.f=1, response.f*A1.f=2,2 0.6411655 0.08810024#> A0.f=2, response.f*A1.f=1,1 0.3023142 0.11081204#> A0.f=2, response.f*A1.f=2,1 0.3199096 0.09719853#> A0.f=2, response.f*A1.f=1,2 0.5294417 0.13399244#> A0.f=2, response.f*A1.f=2,2 0.5470372 0.13251846#>#> First Randomization Augmentation :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.6359737 0.21658343#> A0.f=1, response.f*A1.f=2,1 0.6425345 0.21728891#> A0.f=1, response.f*A1.f=1,2 0.7129831 0.06901176#> A0.f=1, response.f*A1.f=2,2 0.7195438 0.06513763#> A0.f=2, response.f*A1.f=1,1 0.2561355 0.12249210#> A0.f=2, response.f*A1.f=2,1 0.2790365 0.10488801#> A0.f=2, response.f*A1.f=1,2 0.4557345 0.14361723#> A0.f=2, response.f*A1.f=2,2 0.4786356 0.14226926#>#> Second Randomization Augmentation :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.4046498 0.19742912#> A0.f=1, response.f*A1.f=2,1 0.5687969 0.23376677#> A0.f=1, response.f*A1.f=1,2 0.5910450 0.09523643#> A0.f=1, response.f*A1.f=2,2 0.6498579 0.08872953#> A0.f=2, response.f*A1.f=1,1 0.3039991 0.10638060#> A0.f=2, response.f*A1.f=2,1 0.3311109 0.09138384#> A0.f=2, response.f*A1.f=1,2 0.5569428 0.11332978#> A0.f=2, response.f*A1.f=2,2 0.5164913 0.12796725#>#> 1st and 2nd Randomization Augmentation :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.5212352 0.21098387#> A0.f=1, response.f*A1.f=2,1 0.6994484 0.26830813#> A0.f=1, response.f*A1.f=1,2 0.6692919 0.07617047#> A0.f=1, response.f*A1.f=2,2 0.7328127 0.06615399#> A0.f=2, response.f*A1.f=1,1 0.2586853 0.11531506#> A0.f=2, response.f*A1.f=2,1 0.2935352 0.09702298#> A0.f=2, response.f*A1.f=1,2 0.4809089 0.11457277#> A0.f=2, response.f*A1.f=2,2 0.4602799 0.13349781## 2 levels for each response ,  estimated treat probabilitiesbb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.f~A0.f*response.f,augmentR0=~X01+X02,augmentR1=~X11+X12,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f),estpr=c(1,1))bb#> Simple estimator :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.5733301 0.25088070#> A0.f=1, response.f*A1.f=2,1 0.6022300 0.25101611#> A0.f=1, response.f*A1.f=1,2 0.6893786 0.07700325#> A0.f=1, response.f*A1.f=2,2 0.7182785 0.08178121#> A0.f=2, response.f*A1.f=1,1 0.2851197 0.09997901#> A0.f=2, response.f*A1.f=2,1 0.2841367 0.07893846#> A0.f=2, response.f*A1.f=1,2 0.4821020 0.10343782#> A0.f=2, response.f*A1.f=2,2 0.4811190 0.10002228#>#> First Randomization Augmentation :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.5934059 0.27131225#> A0.f=1, response.f*A1.f=2,1 0.6224367 0.27204244#> A0.f=1, response.f*A1.f=1,2 0.6962560 0.07772829#> A0.f=1, response.f*A1.f=2,2 0.7252868 0.08319005#> A0.f=2, response.f*A1.f=1,1 0.2738115 0.10244557#> A0.f=2, response.f*A1.f=2,1 0.2767634 0.08081569#> A0.f=2, response.f*A1.f=1,2 0.4661741 0.10481333#> A0.f=2, response.f*A1.f=2,2 0.4691260 0.10238718#>#> Second Randomization Augmentation :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.4372672 0.23706174#> A0.f=1, response.f*A1.f=2,1 0.5854429 0.26329507#> A0.f=1, response.f*A1.f=1,2 0.6685369 0.07730893#> A0.f=1, response.f*A1.f=2,2 0.7320359 0.08145477#> A0.f=2, response.f*A1.f=1,1 0.2745442 0.10426421#> A0.f=2, response.f*A1.f=2,1 0.2988823 0.07578047#> A0.f=2, response.f*A1.f=1,2 0.5019477 0.09984288#> A0.f=2, response.f*A1.f=2,2 0.4745778 0.10218550#>#> 1st and 2nd Randomization Augmentation :#>                                  coef#> A0.f=1, response.f*A1.f=1,1 0.4702327 0.25293879#> A0.f=1, response.f*A1.f=2,1 0.6127080 0.28587900#> A0.f=1, response.f*A1.f=1,2 0.6759149 0.07701506#> A0.f=1, response.f*A1.f=2,2 0.7433289 0.08370622#> A0.f=2, response.f*A1.f=1,1 0.2634297 0.10565890#> A0.f=2, response.f*A1.f=2,1 0.2948997 0.07656016#> A0.f=2, response.f*A1.f=1,2 0.4865367 0.09860797#> A0.f=2, response.f*A1.f=2,2 0.4699980 0.10193373## 2 and 3 levels for each response , fixed weightsdatat$A1.23.f<-as.numeric(datat$A1.f)dtable(datat,~A1.23.f+response)#>#>         response   0   1#> A1.23.f#> 1                120  23#> 2                120  25datat<-dtransform(datat,A1.23.f=2+rbinom(nrow(datat),1,0.5),            Count2==1& A1.23.f==2& response==0)dtable(datat,~A1.23.f+response)#>#>         response   0   1#> A1.23.f#> 1                120  23#> 2                111  25#> 3                  9   0datat$A1.23.f<-as.factor(datat$A1.23.f)dtable(datat,~A1.23.f+response|Count2==1)#>#>         response  0  1#> A1.23.f#> 1                21 23#> 2                10 25#> 3                 9  0###bb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.23.f~A0.f*response.f,augmentR0=~X01+X02,augmentR1=~X11+X12,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f),estpr=c(1,0),pi1=c(0.3,0.5))bb#> Simple estimator :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.5928568 0.20464868#> A0.f=1, response.f*A1.23.f=2,1 0.6055291 0.20241441#> A0.f=1, response.f*A1.23.f=3,1 0.5539792 0.19865003#> A0.f=1, response.f*A1.23.f=1,2 0.7203538 0.09728960#> A0.f=1, response.f*A1.23.f=2,2 0.7330260 0.08618558#> A0.f=1, response.f*A1.23.f=3,2 0.6814762 0.07905734#> A0.f=2, response.f*A1.23.f=1,1 0.3487105 0.14756981#> A0.f=2, response.f*A1.23.f=2,1 0.2724117 0.09182351#> A0.f=2, response.f*A1.23.f=3,1 0.2669705 0.10319223#> A0.f=2, response.f*A1.23.f=1,2 0.5551901 0.15399346#> A0.f=2, response.f*A1.23.f=2,2 0.4788913 0.12263546#> A0.f=2, response.f*A1.23.f=3,2 0.4734501 0.12853150#>#> First Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.6091794 0.22507560#> A0.f=1, response.f*A1.23.f=2,1 0.6246762 0.22364858#> A0.f=1, response.f*A1.23.f=3,1 0.5756603 0.21982886#> A0.f=1, response.f*A1.23.f=1,2 0.7242102 0.10066515#> A0.f=1, response.f*A1.23.f=2,2 0.7397070 0.08813234#> A0.f=1, response.f*A1.23.f=3,2 0.6906911 0.07986256#> A0.f=2, response.f*A1.23.f=1,1 0.3348012 0.15239160#> A0.f=2, response.f*A1.23.f=2,1 0.2718558 0.09060009#> A0.f=2, response.f*A1.23.f=3,1 0.2532931 0.10673462#> A0.f=2, response.f*A1.23.f=1,2 0.5362379 0.15754640#> A0.f=2, response.f*A1.23.f=2,2 0.4732925 0.12378699#> A0.f=2, response.f*A1.23.f=3,2 0.4547298 0.13192021#>#> Second Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.4707095 0.21804098#> A0.f=1, response.f*A1.23.f=2,1 0.6287053 0.22907037#> A0.f=1, response.f*A1.23.f=3,1 0.6778953 0.30311581#> A0.f=1, response.f*A1.23.f=1,2 0.6499002 0.12488660#> A0.f=1, response.f*A1.23.f=2,2 0.7656237 0.07140976#> A0.f=1, response.f*A1.23.f=3,2 0.6875010 0.08312068#> A0.f=2, response.f*A1.23.f=1,1 0.2656521 0.17396985#> A0.f=2, response.f*A1.23.f=2,1 0.2706333 0.08715854#> A0.f=2, response.f*A1.23.f=3,1 0.3235400 0.07994612#> A0.f=2, response.f*A1.23.f=1,2 0.4759907 0.16071698#> A0.f=2, response.f*A1.23.f=2,2 0.4671860 0.11714753#> A0.f=2, response.f*A1.23.f=3,2 0.5050170 0.11067832#>#> 1st and 2nd Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.5068915 0.22640669#> A0.f=1, response.f*A1.23.f=2,1 0.6601925 0.25328437#> A0.f=1, response.f*A1.23.f=3,1 0.7128045 0.32561624#> A0.f=1, response.f*A1.23.f=1,2 0.6681178 0.12092323#> A0.f=1, response.f*A1.23.f=2,2 0.7709766 0.07310528#> A0.f=1, response.f*A1.23.f=3,2 0.7088981 0.08171051#> A0.f=2, response.f*A1.23.f=1,1 0.2472033 0.17493179#> A0.f=2, response.f*A1.23.f=2,1 0.2703258 0.08650158#> A0.f=2, response.f*A1.23.f=3,1 0.3183141 0.08043082#> A0.f=2, response.f*A1.23.f=1,2 0.4540746 0.15704473#> A0.f=2, response.f*A1.23.f=2,2 0.4667557 0.11617870#> A0.f=2, response.f*A1.23.f=3,2 0.4960792 0.11127903## 2 and 3 levels for each response , estimatedbb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.23.f~A0.f*response.f,augmentR0=~X01+X02,augmentR1=~X11+X12,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f),estpr=c(1,1))bb#> Simple estimator :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.5733301 0.25088123#> A0.f=1, response.f*A1.23.f=2,1 0.6486249 0.25863637#> A0.f=1, response.f*A1.23.f=3,1 0.5558352 0.25058134#> A0.f=1, response.f*A1.23.f=1,2 0.6893788 0.07700333#> A0.f=1, response.f*A1.23.f=2,2 0.7646736 0.11893993#> A0.f=1, response.f*A1.23.f=3,2 0.6718839 0.07549715#> A0.f=2, response.f*A1.23.f=1,1 0.2851198 0.09997906#> A0.f=2, response.f*A1.23.f=2,1 0.2795959 0.10086233#> A0.f=2, response.f*A1.23.f=3,1 0.2893264 0.12751179#> A0.f=2, response.f*A1.23.f=1,2 0.4821024 0.10343788#> A0.f=2, response.f*A1.23.f=2,2 0.4765785 0.12129977#> A0.f=2, response.f*A1.23.f=3,2 0.4863090 0.13928327#>#> First Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.5934060 0.27131280#> A0.f=1, response.f*A1.23.f=2,1 0.6665511 0.27965619#> A0.f=1, response.f*A1.23.f=3,1 0.5783224 0.27132992#> A0.f=1, response.f*A1.23.f=1,2 0.6962562 0.07772837#> A0.f=1, response.f*A1.23.f=2,2 0.7694013 0.12078013#> A0.f=1, response.f*A1.23.f=3,2 0.6811727 0.07593118#> A0.f=2, response.f*A1.23.f=1,1 0.2738116 0.10244561#> A0.f=2, response.f*A1.23.f=2,1 0.2791785 0.10064910#> A0.f=2, response.f*A1.23.f=3,1 0.2740035 0.13235513#> A0.f=2, response.f*A1.23.f=1,2 0.4661745 0.10481340#> A0.f=2, response.f*A1.23.f=2,2 0.4715413 0.12263585#> A0.f=2, response.f*A1.23.f=3,2 0.4663664 0.14396909#>#> Second Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.4372664 0.23706211#> A0.f=1, response.f*A1.23.f=2,1 0.5867572 0.24940274#> A0.f=1, response.f*A1.23.f=3,1 0.6818858 0.34853309#> A0.f=1, response.f*A1.23.f=1,2 0.6685368 0.07730905#> A0.f=1, response.f*A1.23.f=2,2 0.7678408 0.07768404#> A0.f=1, response.f*A1.23.f=3,2 0.6794316 0.07967452#> A0.f=2, response.f*A1.23.f=1,1 0.2745441 0.10426433#> A0.f=2, response.f*A1.23.f=2,1 0.2695053 0.09394894#> A0.f=2, response.f*A1.23.f=3,1 0.3345432 0.09931154#> A0.f=2, response.f*A1.23.f=1,2 0.5019478 0.09984303#> A0.f=2, response.f*A1.23.f=2,2 0.4603587 0.11962056#> A0.f=2, response.f*A1.23.f=3,2 0.5047271 0.12508846#>#> 1st and 2nd Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.23.f=1,1 0.4702319 0.25293915#> A0.f=1, response.f*A1.23.f=2,1 0.6117617 0.26980849#> A0.f=1, response.f*A1.23.f=3,1 0.7132649 0.36786766#> A0.f=1, response.f*A1.23.f=1,2 0.6759149 0.07701519#> A0.f=1, response.f*A1.23.f=2,2 0.7733904 0.07957475#> A0.f=1, response.f*A1.23.f=3,2 0.7066309 0.07677686#> A0.f=2, response.f*A1.23.f=1,1 0.2634296 0.10565903#> A0.f=2, response.f*A1.23.f=2,1 0.2691166 0.09330578#> A0.f=2, response.f*A1.23.f=3,1 0.3296791 0.10056744#> A0.f=2, response.f*A1.23.f=1,2 0.4865368 0.09860813#> A0.f=2, response.f*A1.23.f=2,2 0.4605674 0.11777635#> A0.f=2, response.f*A1.23.f=3,2 0.4971161 0.12478451## 2 and 1 level for each responsedatat$A1.21.f<-as.numeric(datat$A1.f)dtable(datat,~A1.21.f+response|Count2==1)#>#>         response  0  1#> A1.21.f#> 1                21 23#> 2                19 25datat<-dtransform(datat,A1.21.f=1,Count2==1& response==1)dtable(datat,~A1.21.f+response|Count2==1)#>#>         response  0  1#> A1.21.f#> 1                21 48#> 2                19  0datat$A1.21.f<-as.factor(datat$A1.21.f)dtable(datat,~A1.21.f+response|Count2==1)#>#>         response  0  1#> A1.21.f#> 1                21 48#> 2                19  0bb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.21.f~A0.f*response.f,augmentR0=~X01+X02,augmentR1=~X11+X12,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f),estpr=c(1,1))bb#> Simple estimator :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6352226 0.11813045#> A0.f=1, response.f*A1.21.f=2,1 0.6641226 0.12002926#> A0.f=2, response.f*A1.21.f=1,1 0.3865954 0.09118205#> A0.f=2, response.f*A1.21.f=2,1 0.3856124 0.07700508#>#> First Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6482593 0.12794220#> A0.f=1, response.f*A1.21.f=2,1 0.6772901 0.13063539#> A0.f=2, response.f*A1.21.f=1,1 0.3729074 0.09195877#> A0.f=2, response.f*A1.21.f=2,1 0.3758593 0.07808807#>#> Second Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6435175 0.11802262#> A0.f=1, response.f*A1.21.f=2,1 0.6882148 0.11332360#> A0.f=2, response.f*A1.21.f=1,1 0.3954927 0.09148962#> A0.f=2, response.f*A1.21.f=2,1 0.3782504 0.08450470#>#> 1st and 2nd Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6685224 0.13033444#> A0.f=1, response.f*A1.21.f=2,1 0.7138566 0.12655030#> A0.f=2, response.f*A1.21.f=1,1 0.3839378 0.08986106#> A0.f=2, response.f*A1.21.f=2,1 0.3758166 0.08434474## known weightsbb<-binregTSR(Event(entry,time,status)~+1+cluster(id),datat,time=2,cause=c(1),response.code=2,treat.model0=A0.f~+1,treat.model1=A1.21.f~A0.f*response.f,augmentR0=~X01+X02,augmentR1=~X11+X12,augmentC=~X01+X02+A11t+A12t+X11+X12+TR,cens.model=~strata(A0.f),estpr=c(1,0),pi1=c(0.5,1))bb#> Simple estimator :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6410543 0.12005600#> A0.f=1, response.f*A1.21.f=2,1 0.6486576 0.11864403#> A0.f=2, response.f*A1.21.f=1,1 0.3780709 0.09171015#> A0.f=2, response.f*A1.21.f=2,1 0.3940667 0.08377301#>#> First Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6532872 0.12997608#> A0.f=1, response.f*A1.21.f=2,1 0.6625853 0.12899925#> A0.f=2, response.f*A1.21.f=1,1 0.3647406 0.09331039#> A0.f=2, response.f*A1.21.f=2,1 0.3842369 0.08422015#>#> Second Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6435110 0.12122751#> A0.f=1, response.f*A1.21.f=2,1 0.6751763 0.11432293#> A0.f=2, response.f*A1.21.f=1,1 0.3957743 0.08413935#> A0.f=2, response.f*A1.21.f=2,1 0.3772145 0.09110553#>#> 1st and 2nd Randomization Augmentation :#>                                     coef#> A0.f=1, response.f*A1.21.f=1,1 0.6709750 0.13165583#> A0.f=1, response.f*A1.21.f=2,1 0.7043078 0.12732297#> A0.f=2, response.f*A1.21.f=1,1 0.3861201 0.08279377#> A0.f=2, response.f*A1.21.f=2,1 0.3757694 0.09080739

Two-Stage Randomization CALGB-9823

We here illustrate some analysis of one SMART conducted by Cancer andLeukemia Group B Protocol 8923 (CALGB 8923), Stone and others (2001).388 patients were randomized to an initial treatment of GM-CSF (A1 ) orstandard chemotherapy (A2 ). Patients with complete remission andinformed consent to second stage were then re-randomized to onlycytarabine (B1 ) or cytarabine plus mitoxantrone (B2 ).

We first compute the weighted risk-set estimator based on estimatedweights\[\begin{align*}\Lambda_{A1,B1}(t) & = \sum_i \int_0^t \frac{w_i(s)}{Y^w(s)} dN_i(s)\end{align*}\] where\(w_i(s) =I(A0_i=A1) + (t>T_R) I(A1_i=B1)/\pi_1(X_i)\), that is 1 whenyou start on treatment\(A1\) and thenfor those that changes to\(B1\) attime\(T_R\) then is scaled up with theproportion doing this. This is equivalent to the IPTW (inverseprobability of treatment weighted estimator). We estimate the treatmentregimes\(A1, B1\) and\(A2, B1\) by letting\(A10\) indicate those that are consistentwith ending on\(B1\).\(A10\) then starts being\(1\) and becomes\(0\) if the subject is treated with\(B2\), but stays\(1\) if the subject is treated with\(B1\). We can then look at the two stratawhere\(A0=0,A10=1\) and\(A0=1,A10=1\). Similary, for those that endbeing consistent with\(B2\). Thusdefining\(A11\) to start being\(1\), then stays\(1\) if\(B2\) is taken, and becomes\(0\) if the second randomization is\(B1\).

We here use the propensity score model\(P(A1=B1|A0)\) that uses the observedfrequencies on arm\(B1\) among thosestarting out on either\(A1\) or\(A2\).

data(calgb8923)calgt<- calgb8923tm=At.f~factor(Count2)+age+sex+wbctm=At.f~factor(Count2)tm=At.f~factor(Count2)*A0.fhead(calgt)#>   id V X Z   TR R     U delta  stop age   wbc sex race      time status start#> 1  1 0 0 0 0.00 0 13.33     1 13.33  64 128.0   1    1 13.338219      1  0.00#> 2  2 1 1 0 0.00 0 17.80     1 17.80  71   4.3   2    1 17.802995      1  0.00#> 3  3 1 0 0 0.00 0  1.27     1  1.27  71  43.6   2    1  1.271527      1  0.00#> 4  4 1 0 1 0.00 0 24.77     1 24.77  63  72.3   2    1  0.730000      2  0.00#> 5  4 1 0 1 0.73 1 24.77     1 24.77  63  72.3   2    1 24.772515      1  0.73#> 6  5 0 1 0 0.00 0 10.37     1 10.37  65   1.4   1    1 10.374479      1  0.00#>   A0.f A0 A1 A11 A12 A1.f A10 At.f lbnr__id Count1 Count2 consent trt2 trt1#> 1    0  0  0   1   0    0   0    0        1      0      0      -1   -1    1#> 2    1  1  0   1   0    0   0    1        1      0      0      -1   -1    2#> 3    0  0  0   1   0    0   0    0        1      0      0      -1   -1    1#> 4    0  0  0   1   0    0   0    0        1      0      0      -1   -1    1#> 5    0  0  1   1   1    1   1    1        2      0      1       1    1    1#> 6    1  1  0   1   0    0   0    1        1      0      0      -1   -1    2ll0<-phreg_IPTW(Event(start,time,status==1)~strata(A0,A10)+cluster(id),calgt,treat.model=tm)pll0<-predict(ll0,expand.grid(A0=0:1,A10=0,id=1))ll1<-phreg_IPTW(Event(start,time,status==1)~strata(A0,A11)+cluster(id),calgt,treat.model=tm)pll1<-predict(ll1,expand.grid(A0=0:1,A11=1,id=1))plot(pll0,se=1,lwd=2,col=1:2,lty=1,xlab="time (months)",xlim=c(0,30))plot(pll1,add=TRUE,col=3:4,se=1,lwd=2,lty=1,xlim=c(0,30))abline(h=0.25)legend("topright",c("A1B1","A2B1","A1B2","A2B2"),col=c(1,2,3,4),lty=1)

summary(pll1,times=12)#> $pred#> [1] 0.4556676 0.5029214#>#> $se.pred#> [1] 0.04212703 0.04165848#>#> $lower#> [1] 0.3801487 0.4275556#>#> $upper#> [1] 0.5461888 0.5915721#>#> $times#> [1] 12#>#> attr(,"class")#> [1] "summarypredictrecreg"summary(pll0,times=12)#> $pred#> [1] 0.3950461 0.4279272#>#> $se.pred#> [1] 0.04263714 0.04342426#>#> $lower#> [1] 0.3197260 0.3507467#>#> $upper#> [1] 0.4881098 0.5220911#>#> $times#> [1] 12#>#> attr(,"class")#> [1] "summarypredictrecreg"

The propensity score mode can be extended to use covariates to getincreased efficiency. Note also that the propensity scores for\(A0\) will cancel out in the differentstrata.

SessionInfo

sessionInfo()#> R version 4.5.1 (2025-06-13)#> Platform: aarch64-apple-darwin25.0.0#> Running under: macOS Tahoe 26.0.1#>#> Matrix products: default#> BLAS:   /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRblas.dylib#> LAPACK: /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRlapack.dylib;  LAPACK version 3.12.1#>#> locale:#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8#>#> time zone: Europe/Copenhagen#> tzcode source: internal#>#> attached base packages:#> [1] stats     graphics  grDevices utils     datasets  methods   base#>#> other attached packages:#> [1] timereg_2.0.7  survival_3.8-3 mets_1.3.8#>#> loaded via a namespace (and not attached):#>  [1] cli_3.6.5           knitr_1.50          rlang_1.1.6#>  [4] xfun_0.53           jsonlite_2.0.0      future.apply_1.20.0#>  [7] listenv_0.9.1       lava_1.8.1          htmltools_0.5.8.1#> [10] sass_0.4.10         rmarkdown_2.30      grid_4.5.1#> [13] evaluate_1.0.5      jquerylib_0.1.4     fastmap_1.2.0#> [16] numDeriv_2016.8-1.1 yaml_2.3.10         mvtnorm_1.3-3#> [19] lifecycle_1.0.4     compiler_4.5.1      codetools_0.2-20#> [22] ucminf_1.2.2        Rcpp_1.1.0          future_1.67.0#> [25] lattice_0.22-7      digest_0.6.37       R6_2.6.1#> [28] parallelly_1.45.1   parallel_4.5.1      splines_4.5.1#> [31] Matrix_1.7-4        bslib_0.9.0         tools_4.5.1#> [34] globals_0.18.0      cachem_1.1.0
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