# Two-Stage Randomization for Cox Type rate models

#### Klaus Holst & Thomas Scheike

#### 2025-01-11

# Two-Stage Randomization for counting process outcomes

Specify rate models of \(N_1(t)\).

- survival data
- competing risks, cause specific hazards.
- recurrent events data

Under simple randomization we can estimate the rate Cox model

- \(\lambda_0(t) \exp(A_0 \beta_0)\)

Under two-stage randomization we can estimate the rate Cox model

- \(\lambda_0(t) \exp(A_0 \beta_0 + A_1(t) \beta_1 )\)

Starting point is that Cox’s partial likelihood score can be used for estimating parameters \[\begin{align*} U(\beta) & = \int (A(t) - e(t)) dN_1(t) \end{align*}\] where \(A(t)\) is the combined treatments over time.

- the solution will converge to \(\beta^*\) the Struthers-Kalbfleisch solution of the score and will have robust standard errors Lin-Wei.

The estimator can be agumented in different ways using additional covariates at the time of randomization and a censoring augmentation. The solved estimating eqution is \[\begin{align*} \sum_i U_i - AUG_0 - AUG_1 + AUG_C = 0 \end{align*}\] using the covariates from augmentR0 to augment with \[\begin{align*} AUG_0 = ( A_0 - \pi_0(X_0) ) X_0 \gamma_0 \end{align*}\] where possibly \(P(A_0=1|X_0)=\pi_0(X_0)\) but does not depend on covariates under randomization, and furhter using the covariates from augmentR1, to augment with R indiciating that the randomization takes place or not, \[\begin{align*} AUG_1 = R ( A_1 - \pi_1(X_1)) X_1 \gamma_1 \end{align*}\] and the dynamic censoring augmenting
\[\begin{align*} AUG_C = \int_0^t \gamma_c(s)^T (e(s) - \bar e(s)) \frac{1}{G_c(s) } dM_c(s) \end{align*}\] where \(\gamma_c(s)\) is chosen to minimize the variance given the dynamic covariates specified by augmentC.

The propensity score models are always estimated unless it is requested to use some fixed number \(\pi_0=1/2\) for example, but always better to be adaptive and estimate \(\pi_0\). Also \(\gamma_0\) and \(\gamma_1\) are estimated to reduce variance of \(U_i\).

- The treatment’s must be given as factors.
- Treatment for 2nd randomization may depend on response.
  - Treatment probabilities are estimated by default and uncertainty from this adjusted for.
  - treat.model must then typically allow for interaction with treatment number and covariates
- Randomization augmentation for 1’st and 2’nd randomization possible.
  - typesR=c(“R0”,“R1”,“R01”)
- Censoring model possibly stratified on observed covariates (at time 0).
  - default model is to stratify after randomization R0
  - cens.model can be specified
- Censoring augmentation done dynamically over time with time-dependent covariates.
  - typesC=c(“C”,“dynC”), C fixed coefficients and dynC dynamic
  - done for each strata in censoring model

Standard errors are estimated using the influence function of all estimators and tests of differences can therefore be computed subsequently.

- variance adjustment for censoring augmentation computed subtracting variance gain
- influence functions given for case with R0 only

The times of randomization is specified by

- treat.var is “1” when a randomization is given
  - default is to assume that all time-points corresponds to a treatment, the survival case without time-dependent covariates
  - recurrent events situation must be specified as first record of each subject, see below example.

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

- one code of status indicating events of interest
- one code for the censorings

The phreg_rct can be used for counting process style data, and thus covers situations with

- recurrent events
- survival data
- cause-specific hazards (competing risks)

and will in all cases compute augmentations

- dynamic censoring augmentation
  - dynC
- RCT augmentation
  - R0, R1 and R01

# Simple Randomization: Lu-Tsiatis marginal Cox model

    library(mets)
    set.seed(100)

    ## Lu, Tsiatis simulation
    data <- mets:::simLT(0.7,100)
    dfactor(data) <- Z.f~Z

    out <- phreg_rct(Surv(time,status)~Z.f,data=data,augmentR0=~X,augmentC=~factor(Z):X)
    summary(out)
    #>                 Estimate   Std.Err       2.5%     97.5%   P-value
    #> Marginal-Z.f1 0.29263400 0.2739159 -0.2442313 0.8294993 0.2853693
    #> R0_C:Z.f1     0.07166242 0.2234066 -0.3662065 0.5095313 0.7483838
    #> R0_dynC:Z.f1  0.08321604 0.2221710 -0.3522312 0.5186633 0.7079889
    #> attr(,"class")
    #> [1] "summary.phreg_rct"
    ###out <- phreg_rct(Surv(time,status)~Z.f,data=data,augmentR0=~X,augmentC=~X)
    ###out <- phreg_rct(Surv(time,status)~Z.f,data=data,augmentR0=~X,augmentC=~factor(Z):X,cens.model=~+1)

Results consitent with speff of library(speff2trial)

    ###library(speff2trial)
    library(mets)
    data(ACTG175)
    ###
    data <- ACTG175[ACTG175$arms==0 | ACTG175$arms==1, ]
    data <- na.omit(data[,c("days","cens","arms","strat","cd40","cd80","age")])
    data$days <- data$days+runif(nrow(data))*0.01
    dfactor(data) <- arms.f~arms
    notrun <- 1

    if (notrun==0) {
    fit1 <- speffSurv(Surv(days,cens)~cd40+cd80+age,data=data,trt.id="arms",fixed=TRUE)
    summary(fit1)
    }
    #
    # Treatment effect
    #             Log HR       SE   LowerCI   UpperCI           p
    # Prop Haz  -0.70375  0.12352  -0.94584  -0.46165  1.2162e-08
    # Speff     -0.72430  0.12051  -0.96050  -0.48810  1.8533e-09

    out <- phreg_rct(Surv(days,cens)~arms.f,data=data,augmentR0=~cd40+cd80+age,augmentC=~cd40+cd80+age)
    summary(out)
    #>                    Estimate   Std.Err       2.5%      97.5%      P-value
    #> Marginal-arms.f1 -0.7036460 0.1224406 -0.9436251 -0.4636669 9.092786e-09
    #> R0_C:arms.f1     -0.7265342 0.1197607 -0.9612610 -0.4918075 1.306891e-09
    #> R0_dynC:arms.f1  -0.7204699 0.1196158 -0.9549125 -0.4860272 1.710025e-09
    #> attr(,"class")
    #> [1] "summary.phreg_rct"

The study is actually block-randomized according (?) so the standard should be computed with an adjustment that is equivalent to augmenting with this block as factor

    dtable(data,~strat+arms)
    #>
    #>       arms   0   1
    #> strat
    #> 1          223 213
    #> 2           96 106
    #> 3          213 203
    dfactor(data) <- strat.f~strat
    out <- phreg_rct(Surv(days,cens)~arms.f,data=data,augmentR0=~strat.f)
    summary(out)
    #>                    Estimate   Std.Err       2.5%      97.5%      P-value
    #> Marginal-arms.f1 -0.7036460 0.1224406 -0.9436251 -0.4636669 9.092786e-09
    #> R0_none:arms.f1  -0.7009844 0.1217138 -0.9395390 -0.4624298 8.447051e-09
    #> attr(,"class")
    #> [1] "summary.phreg_rct"

# Recurrent events: Simple Randomization

Recurrents events simulation with death and censoring.

    n <- 1000
    beta <- 0.15;
    data(base1cumhaz)
    data(base4cumhaz)
    data(drcumhaz)
    dr <- scalecumhaz(drcumhaz,1)
    base1 <- scalecumhaz(base1cumhaz,1)
    base4 <- scalecumhaz(base4cumhaz,0.5)
    cens <- rbind(c(0,0),c(2000,0.5),c(5110,3))
    ce <- 3; betao1 <- 0

    varz <- 1; dep=4; X <- z <- rgamma(n,1/varz)*varz
    Z0 <- NULL
    px <- 0.5
    if (betao1!=0) px <- lava::expit(betao1*X)
    A0 <- rbinom(n,1,px)
    r1 <- exp(A0*beta[1])
    rd <- exp( A0 * 0.15)
    rc <- exp( A0 * 0 )
    ###
    rr <-    mets:::simLUCox(n,base1,death.cumhaz=dr,r1=r1,Z0=X,dependence=dep,var.z=varz,cens=ce/5000)
    rr$A0 <- A0[rr$id]
    rr$z1 <- attr(rr,"z")[rr$id]
    rr$lz1 <- log(rr$z1)
    rr$X <- rr$lz1
    rr$lX <- rr$z1
    rr$statusD <- rr$status
    rr <- dtransform(rr,statusD=2,death==1)
    rr <- count.history(rr)
    rr$Z <- rr$A0
    data <- rr
    data$Z.f <- as.factor(data$Z)
    data$treattime <- 0
    data <- dtransform(data,treattime=1,lbnr__id==1)
    dlist(data,start+stop+statusD+A0+z1+treattime+Count1~id|id %in% c(4,5))
    #> id: 4
    #>      start   stop    statusD A0 z1    treattime Count1
    #> 4      0.000   9.565 1       0  0.471 1         0
    #> 1003   9.565 372.057 1       0  0.471 0         1
    #> 1468 372.057 389.831 0       0  0.471 0         2
    #> ------------------------------------------------------------
    #> id: 5
    #>   start stop  statusD A0 z1    treattime Count1
    #> 5 0     213.9 2       1  2.338 1         0

Now we fit the model

    fit2 <- phreg_rct(Event(start,stop,statusD)~Z.f+cluster(id),data=data,
         treat.var="treattime",typesR=c("non","R0"),typesC=c("non","C","dynC"),
         augmentR0=~z1,augmentC=~z1+Count1)
    summary(fit2)
    #>                Estimate    Std.Err       2.5%     97.5%      P-value
    #> Marginal-Z.f1 0.2870649 0.09632565 0.09827011 0.4758597 0.0028810700
    #> non_C:Z.f1    0.2826049 0.09631924 0.09382262 0.4713871 0.0033457707
    #> non_dynC:Z.f1 0.1926888 0.08864883 0.01894025 0.3664373 0.0297337758
    #> R0_non:Z.f1   0.3110880 0.08049844 0.15331399 0.4688621 0.0001113067
    #> R0_C:Z.f1     0.3066141 0.08049078 0.14885504 0.4643731 0.0001393570
    #> R0_dynC:Z.f1  0.2164684 0.07113356 0.07704922 0.3558877 0.0023413376
    #> attr(,"class")
    #> [1] "summary.phreg_rct"

- Censoring model was stratified on Z.f
- treatment probabilities were estimated using the data

# Twostage Randomization: Recurrent events

    n <- 500
    beta=c(0.3,0.3);betatr=0.3;betac=0;betao=0;betao1=0;ce=3;fixed=1;sim=1;dep=4;varz=1;ztr=0; ce <- 3
    ## take possible frailty
    Z0 <- rgamma(n,1/varz)*varz
    px0 <- 0.5; if (betao!=0) px0 <- expit(betao*Z0)
    A0 <- rbinom(n,1,px0)
    r1 <- exp(A0*beta[1])
    #
    px1 <- 0.5; if (betao1!=0) px1 <- expit(betao1*Z0)
    A1 <- rbinom(n,1,px1)
    r2 <- exp(A1*beta[2])
    rtr <- exp(A0*betatr[1])
    rr <-  mets:::simLUCox(n,base1,death.cumhaz=dr,cumhaz2=base1,rtr=rtr,betatr=0.3,A0=A0,Z0=Z0,
            r1=r1,r2=r2,dependence=dep,var.z=varz,cens=ce/5000,ztr=ztr)
    rr$z1 <- attr(rr,"z")[rr$id]
    rr$A1 <- A1[rr$id]
    rr$A0 <- A0[rr$id]
    rr$lz1 <- log(rr$z1)
    rr <- count.history(rr)
    rr$A1t <- 0
    rr <- dtransform(rr,A1t=A1,Count2==1)
    rr$At.f <- rr$A0
    rr$A0.f <- factor(rr$A0)
    rr$A1.f <- factor(rr$A1)
    rr <- dtransform(rr, At.f = A1, Count2 == 1)
    rr$At.f <- factor(rr$At.f)
    dfactor(rr)  <-  A0.f~A0
    rr$treattime <- 0
    rr <- dtransform(rr,treattime=1,lbnr__id==1)
    rr$lagCount2 <- dlag(rr$Count2)
    rr <- dtransform(rr,treattime=1,Count2==1 & (Count2!=lagCount2))
    dlist(rr,start+stop+statusD+A0+A1+A1t+At.f+Count2+z1+treattime+Count1~id|id %in% c(5,10))
    #> id: 5
    #>   start stop  statusD A0 A1 A1t At.f Count2 z1     treattime Count1
    #> 5 0     132.3 3       1  1  0   1    0      0.2316 1         0
    #> ------------------------------------------------------------
    #> id: 10
    #>     start stop    statusD A0 A1 A1t At.f Count2 z1      treattime Count1
    #> 10   0.00   33.12 2       1  0  0   1    0      0.06891 1         0
    #> 509 33.12 1363.53 0       1  0  0   0    1      0.06891 1         0

Now fitting the model and computing different augmentations (true values 0.3 and 0.3)

    sse <- phreg_rct(Event(start,time,statusD)~A0.f+A1t+cluster(id),data=rr,
         typesR=c("non","R0","R1","R01"),typesC=c("non","C","dynC"),treat.var="treattime",
         treat.model=At.f~factor(Count2),
         augmentR0=~z1,augmentR1=~z1,augmentC=~z1+Count1+A1t)
    summary(sse)
    #>                 Estimate   Std.Err        2.5%     97.5%      P-value
    #> Marginal-A0.f1 0.3179631 0.1418023  0.04003574 0.5958904 0.0249420566
    #> Marginal-A1t   0.3290147 0.1472363  0.04043683 0.6175925 0.0254434247
    #> non_C:A0.f1    0.3002782 0.1391490  0.02755115 0.5730053 0.0309308283
    #> non_C:A1t      0.4151190 0.1405664  0.13961382 0.6906241 0.0031451130
    #> non_dynC:A0.f1 0.3104992 0.1314476  0.05286660 0.5681318 0.0181692114
    #> non_dynC:A1t   0.4374223 0.1300991  0.18243265 0.6924119 0.0007731772
    #> R0_non:A0.f1   0.4142867 0.1176773  0.18364338 0.6449300 0.0004306830
    #> R0_non:A1t     0.3382185 0.1470378  0.05002979 0.6264072 0.0214360247
    #> R0_C:A0.f1     0.3962505 0.1144662  0.17190089 0.6206002 0.0005367253
    #> R0_C:A1t       0.4242165 0.1403584  0.14911904 0.6993140 0.0025079567
    #> R0_dynC:A0.f1  0.4066624 0.1049692  0.20092652 0.6123983 0.0001070147
    #> R0_dynC:A1t    0.4464941 0.1298744  0.19194497 0.7010432 0.0005862621
    #> R1_non:A0.f1   0.3269008 0.1416813  0.04921043 0.6045911 0.0210383383
    #> R1_non:A1t     0.2104421 0.1254571 -0.03544923 0.4563335 0.0934636201
    #> R1_C:A0.f1     0.3092275 0.1390258  0.03674199 0.5817130 0.0261319083
    #> R1_C:A1t       0.2976811 0.1175580  0.06727171 0.5280905 0.0113347106
    #> R1_dynC:A0.f1  0.3194771 0.1313171  0.06210024 0.5768540 0.0149798139
    #> R1_dynC:A1t    0.3202318 0.1048176  0.11479297 0.5256706 0.0022496121
    #> R01_non:A0.f1  0.4092668 0.1176428  0.17869115 0.6398424 0.0005034879
    #> R01_non:A1t    0.2280764 0.1243165 -0.01557936 0.4717322 0.0665584775
    #> R01_C:A0.f1    0.3912859 0.1144307  0.16700576 0.6155659 0.0006275647
    #> R01_C:A1t      0.3150865 0.1163399  0.08706445 0.5431086 0.0067623432
    #> R01_dynC:A0.f1 0.4016977 0.1049305  0.19603760 0.6073577 0.0001290708
    #> R01_dynC:A1t   0.3375831 0.1034497  0.13482542 0.5403408 0.0011013908
    #> attr(,"class")
    #> [1] "summary.phreg_rct"

- treat.model has A0 and A1 coded as At.f and we here allow a model that depends on the randomization to be as adaptive as possible.
- for the observational case one can here also adjust for covarites.
- Censoring model was stratified on A0.f

# Causal assumptions for Twostage Randomization: Recurrent events

We take interest in \(N_1\) but also have death \(N_d\).

Now we need that given \(X_0\)

- \(A_0 \perp N_1^0(), N_1^1(), N_d^0(), N_d^1() | X_0\)
- positivity \(1 > \pi_0(X_0) > 0\)

and given \(\bar X_1\) the history accumulated at time \(T_R\) of 2nd randomization

- \(A_1 \perp N_1^0(), N_1^1(), N_d^0(), N_d^1() | \bar X_1\)
- positivity \(1 > \pi_1(X_1) > 0\)

and

- consistency

to link the counterfactual quantities to observed data.

We must use IPTW weighted Cox score and augment as before

In addition we need that the censoring is independent given for example \(A_0\)

- Independent censoring

To use the phreg_rct in this situation

- RCT=FALSE
- propensity score models must be specified
- marginal estimate is based on IPTW cox model phreg_IPTW
  - when the same model is used for the propensity scores and the augmentation models the augmentation models are not needed due to the adaptive nature from fitting the propensity score models.

    fit2 <- phreg_rct(Event(start,stop,statusD)~Z.f+cluster(id),data=data,
         treat.var="treattime",typesR=c("non","R0"),typesC=c("non","C","dynC"),
         RCT=FALSE,treat.model=Z.f~z1,augmentR0=~z1,augmentC=~z1+Count1)
    summary(fit2)
    #>                Estimate    Std.Err       2.5%     97.5%      P-value
    #> Marginal-Z.f1 0.3111195 0.08058494 0.15317593 0.4690631 0.0001130326
    #> non_C:Z.f1    0.3067348 0.08057748 0.14880581 0.4646637 0.0001408301
    #> non_dynC:Z.f1 0.2169383 0.07119837 0.07739205 0.3564845 0.0023117164
    #> R0_non:Z.f1   0.3111195 0.08058494 0.15317593 0.4690631 0.0001130326
    #> R0_C:Z.f1     0.3067348 0.08057748 0.14880581 0.4646637 0.0001408301
    #> R0_dynC:Z.f1  0.2169383 0.07119837 0.07739205 0.3564845 0.0023117164
    #> attr(,"class")
    #> [1] "summary.phreg_rct"

and for twostage randomization

    sse <- phreg_rct(Event(start,time,statusD)~A0.f+A1t+cluster(id),data=rr,
         typesR=c("non","R0","R1","R01"),typesC=c("non","C","dynC"),
         treat.var="treattime",
         RCT=FALSE, treat.model=At.f~z1*factor(Count2),
         augmentR0=~z1,augmentR1=~z1,augmentC=~z1+Count1+A1t)
    summary(sse)
    #>                 Estimate    Std.Err        2.5%     97.5%      P-value
    #> Marginal-A0.f1 0.3817476 0.13188068  0.12326622 0.6402290 0.0037958891
    #> Marginal-A1t   0.2259319 0.12344157 -0.01600910 0.4678730 0.0672089296
    #> non_C:A0.f1    0.3765255 0.12594711  0.12967369 0.6233773 0.0027938653
    #> non_C:A1t      0.3187675 0.11256529  0.09814361 0.5393914 0.0046280182
    #> non_dynC:A0.f1 0.3797091 0.11214290  0.15991306 0.5995051 0.0007093493
    #> non_dynC:A1t   0.3514300 0.09297578  0.16920079 0.5336591 0.0001569535
    #> R0_non:A0.f1   0.3817476 0.13188068  0.12326622 0.6402290 0.0037958891
    #> R0_non:A1t     0.2259319 0.12344157 -0.01600910 0.4678730 0.0672089296
    #> R0_C:A0.f1     0.3765255 0.12594711  0.12967369 0.6233773 0.0027938653
    #> R0_C:A1t       0.3187675 0.11256529  0.09814361 0.5393914 0.0046280182
    #> R0_dynC:A0.f1  0.3797091 0.11214290  0.15991306 0.5995051 0.0007093493
    #> R0_dynC:A1t    0.3514300 0.09297578  0.16920079 0.5336591 0.0001569535
    #> R1_non:A0.f1   0.3817476 0.13188068  0.12326622 0.6402290 0.0037958891
    #> R1_non:A1t     0.2259319 0.12344157 -0.01600910 0.4678730 0.0672089296
    #> R1_C:A0.f1     0.3765255 0.12594711  0.12967369 0.6233773 0.0027938653
    #> R1_C:A1t       0.3187675 0.11256529  0.09814361 0.5393914 0.0046280182
    #> R1_dynC:A0.f1  0.3797091 0.11214290  0.15991306 0.5995051 0.0007093493
    #> R1_dynC:A1t    0.3514300 0.09297578  0.16920080 0.5336591 0.0001569535
    #> R01_non:A0.f1  0.3817476 0.13185641  0.12331378 0.6401814 0.0037894525
    #> R01_non:A1t    0.2259319 0.12307282 -0.01528637 0.4671502 0.0663934395
    #> R01_C:A0.f1    0.3765255 0.12592170  0.12972349 0.6233275 0.0027883528
    #> R01_C:A1t      0.3187675 0.11216079  0.09893641 0.5385986 0.0044823275
    #> R01_dynC:A0.f1 0.3797091 0.11211436  0.15996900 0.5994492 0.0007071245
    #> R01_dynC:A1t   0.3514300 0.09248564  0.17016144 0.5326985 0.0001447938
    #> attr(,"class")
    #> [1] "summary.phreg_rct"

# SessionInfo

    sessionInfo()
    #> R version 4.4.2 (2024-10-31)
    #> Platform: aarch64-apple-darwin24.2.0
    #> Running under: macOS Sequoia 15.2
    #>
    #> Matrix products: default
    #> BLAS:   /Users/klaus/.asdf/installs/R/4.4.2/lib/R/lib/libRblas.dylib
    #> LAPACK: /Users/klaus/.asdf/installs/R/4.4.2/lib/R/lib/libRlapack.dylib;  LAPACK version 3.12.0
    #>
    #> 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] mets_1.3.5     timereg_2.0.6  survival_3.8-3
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] cli_3.6.3           knitr_1.49          rlang_1.1.4
    #>  [4] xfun_0.50           jsonlite_1.8.9      listenv_0.9.1
    #>  [7] future.apply_1.11.3 lava_1.8.1          htmltools_0.5.8.1
    #> [10] sass_0.4.9          rmarkdown_2.29      grid_4.4.2
    #> [13] evaluate_1.0.1      jquerylib_0.1.4     fastmap_1.2.0
    #> [16] mvtnorm_1.3-2       yaml_2.3.10         lifecycle_1.0.4
    #> [19] numDeriv_2016.8-1.1 compiler_4.4.2      codetools_0.2-20
    #> [22] ucminf_1.2.2        Rcpp_1.0.13-1       future_1.34.0
    #> [25] lattice_0.22-6      digest_0.6.37       R6_2.5.1
    #> [28] parallelly_1.41.0   parallel_4.4.2      splines_4.4.2
    #> [31] bslib_0.8.0         Matrix_1.7-1        tools_4.4.2
    #> [34] globals_0.16.3      cachem_1.1.0