# WIP: Cooking survival data, 5 minute recipes

#### 2025-10-21

# Overview

Simulation of survival data is important for both theoretical and practical work. In a practical setting we might wish to validate that standard errors are valid even in a rather small sample, or validate that a complicated procedure is doing as intended. Therefore it is useful to have simple tools for generating survival data that looks as much as possible like particular data. In a theoretical setting we often are interested in evaluating the finite sample properties of a new procedure in different settings that often are motivated by a specific practical problem. The aim is provide such tools.

Bender et al. in a nice paper discussed how to generate survival data based on the Cox model, and restricted attention to some of the many useful parametric survival models (weibull, exponential). We here use piecwise linear baseline functions that make it easy to simulate data that follows closely the baseline given by the data using semi or nonparametric models. THis makes it easy to capture important aspects of the data.

Different survival models can be cooked, and we here give recipes for hazard and cumulative incidence based simulations. More recipes are given in vignette about recurrent events.

- hazard based.
- cumulative incidence.
- recurrent events (see recurrent events vignette).

     library(mets)
     options(warn=-1)
     set.seed(10) # to control output in simulations

# Hazard based, Cox models

Given a survival time \(T\) with cumulative hazard \(\Lambda(t)=\int_0^t \lambda(s) ds\), it follows that with \(E \sim Exp(1)\) (exponential with rate 1), that \(\Lambda^{-1}(E)\) will have the same distribution as \(T\).

This provides the basis for simulations of survival times with a given hazard and is a consequence of this simple calculation \[ P(\Lambda^{-1}(E) > t) = P(E > \Lambda(t)) = \exp( - \Lambda(t)) = P(T > t). \]

Similarly if \(T\) given \(X\) have hazard on Cox form \[ \lambda_0(t) \exp( X^T \beta) \] where \(\beta\) is a \(p\)-dimensional regression coefficient and \(\lambda_0(t)\) a baseline hazard funcion, then it is useful to observe also that \(\Lambda^{-1}(E/HR)\) with \(HR=\exp(X^T \beta)\) has the same distribution as \(T\) given \(X\).

Therefore if the inverse of the cumulative hazard can be computed we can generate survival with a specified hazard function. One useful observation is note that for a piecewise linear continuous cumulative hazard on an interval \([0,\tau]\) \(\Lambda_l(t)\) it is easy to compute the inverse.

Further, we can approximate any cumulative hazard with a piecewise linear continous cumulative hazard and then simulate data according to this approximation. Recall that fitting the Cox model to data will give a piecewise constant cumulative hazard and the regression coefficients so with these at hand we can first approximate the piecewise constant “Breslow”-estimator with a linear upper (or lower bound) by simply connecting the values by straight lines.

# Delayed entry

If \(T\) given \(X\) have hazard on Cox form \[ \lambda_0(t) \exp( X^T \beta) \] and we wish to generate data according to this hazard for those that are alive at time \(s\), that is draw from the distribution of \(T\) given \(T>s\) (all given \(X\) ), then we note that
\[ \Lambda_0^{-1}( \Lambda_0(s) + E/HR)) \] with \(HR=\exp(X^T \beta))\) and with \(E \sim Exp(1)\) has the distributiion we are after.

This is again a consequence of a simple calculation \[ P_X(\Lambda^{-1}(\Lambda(s)+ E/HR) > t) = P_X(E > HR( \Lambda(t) - \Lambda(s)) ) = P_X(T>t | T>s) \]

The engine is to simulate data with a given linear cumulative hazard. First generating survival data based on the cumulative hazard cumhaz:j

     nsim <- 200
     chaz <-  c(0,1,1.5,2,2.1)
     breaks <- c(0,10,   20,  30,   40)
     cumhaz <- cbind(breaks,chaz)
     X <- rbinom(nsim,1,0.5)
     beta <- 0.2
     rrcox <- exp(X * beta)

     pctime <- rchaz(cumhaz,n=nsim)
     pctimecox <- rchaz(cumhaz,rrcox)

Now looking at a simple cox model

     library(mets)
     n <- 100
     data(bmt)
     bmt$bmi <- rnorm(408)
     dcut(bmt) <- gage~age
     data <- bmt
     cox1 <- phreg(Surv(time,cause==1)~tcell+platelet+age,data=bmt)

     dd <- sim.phreg(cox1,n,data=bmt)
     dtable(dd,~status)
    #>
    #> status
    #>  0  1
    #> 54 46
     scox1 <- phreg(Surv(time,status==1)~tcell+platelet+age,data=dd)
     cbind(coef(cox1),coef(scox1))
    #>                [,1]       [,2]
    #> tcell    -0.6517920 -0.8108950
    #> platelet -0.5207454 -0.5471871
    #> age       0.4083098  0.4390413
     par(mfrow=c(1,1))
     plot(scox1,col=2); plot(cox1,add=TRUE,col=1)

     ## changing the parametes
     cox10 <- cox1
     cox10$coef <- c(0,0.4,0.3)
     dd <- sim.phreg(cox10,n,data=bmt)
     dtable(dd,~status)
    #>
    #> status
    #>  0  1
    #> 41 59
     scox1 <- phreg(Surv(time,status==1)~tcell+platelet+age,data=dd)
     cbind(coef(cox10),coef(scox1))
    #>          [,1]      [,2]
    #> tcell     0.0 0.1752103
    #> platelet  0.4 0.4409485
    #> age       0.3 0.1086505
     par(mfrow=c(1,1))
     plot(scox1,col=2); plot(cox10,add=TRUE,col=1)

Multiple Cox models for cause specific hazards can be combined, and we start by drawing the covariates manually, below we just call the sim.phregs function that draws covariates from the data,

     data(bmt);
     cox1 <- phreg(Surv(time,cause==1)~tcell+platelet,data=bmt)
     cox2 <- phreg(Surv(time,cause==2)~tcell+platelet,data=bmt)

     X1 <- bmt[,c("tcell","platelet")]
     n <- nsim
     xid <- sample(1:nrow(X1),n,replace=TRUE)
     Z1 <- X1[xid,]
     Z2 <- X1[xid,]
     rr1 <- exp(as.matrix(Z1) %*% cox1$coef)
     rr2 <- exp(as.matrix(Z2) %*% cox2$coef)

     d <-  rcrisk(cox1$cum,cox2$cum,rr1,rr2)
     dd <- cbind(d,Z1)

     scox1 <- phreg(Surv(time,status==1)~tcell+platelet,data=dd)
     scox2 <- phreg(Surv(time,status==2)~tcell+platelet,data=dd)
     par(mfrow=c(1,2))
     plot(cox1); plot(scox1,add=TRUE,col=2)
     plot(cox2); plot(scox2,add=TRUE,col=2)

     cbind(cox1$coef,scox1$coef,cox2$coef,scox2$coef)
    #>                [,1]       [,2]       [,3]       [,4]
    #> tcell    -0.4232606 -0.6808494  0.3991068  0.2044248
    #> platelet -0.5654438 -0.3404486 -0.2461474 -0.1388716

Now fully nonparametric model with stratified baselines and specific call of sim.base function

     data(sTRACE)
     dtable(sTRACE,~chf+diabetes)
    #>
    #>     diabetes   0   1
    #> chf
    #> 0            223  16
    #> 1            230  31
     coxs <-   phreg(Surv(time,status==9)~strata(diabetes,chf),data=sTRACE)
     strata <- sample(0:3,nsim,replace=TRUE)
     simb <- sim.base(coxs$cumhaz,nsim,stratajump=coxs$strata.jumps,strata=strata)
     cc <-   phreg(Surv(time,status)~strata(strata),data=simb)
     plot(coxs,col=1); plot(cc,add=TRUE,col=2)

We now fit 3 cause-specific hazard models and generate competing risks data with hazards taken from the fitted Cox models. Here a complex situation with stratified baselines of some of the models.

     ## stratified with phreg
     cox0 <- phreg(Surv(time,cause==0)~tcell+platelet,data=bmt)
     cox1 <- phreg(Surv(time,cause==1)~tcell+platelet,data=bmt)
     cox2 <- phreg(Surv(time,cause==2)~strata(tcell)+platelet,data=bmt)
     coxs <- list(cox0,cox1,cox2)
    ### dd <- sim.cause.cox(coxs,nsim,data=bmt)
     dd <- sim.phregs(coxs,n,data=bmt)

     ## checking that  cause specific hazards are as given, make n larger

     scox0 <- phreg(Surv(time,status==1)~tcell+platelet,data=dd)
     scox1 <- phreg(Surv(time,status==2)~tcell+platelet,data=dd)
     scox2 <- phreg(Surv(time,status==3)~strata(tcell)+platelet,data=dd)
     cbind(cox0$coef,scox0$coef)
    #>               [,1]       [,2]
    #> tcell    0.1912407 0.04260751
    #> platelet 0.1563789 0.62751528
     cbind(cox1$coef,scox1$coef)
    #>                [,1]       [,2]
    #> tcell    -0.4232606 -0.5092389
    #> platelet -0.5654438 -0.4858872
     cbind(cox2$coef,scox2$coef)
    #>                [,1]      [,2]
    #> platelet -0.2271912 0.2167204
     par(mfrow=c(1,3))
     plot(cox0); plot(scox0,add=TRUE,col=2);
     plot(cox1); plot(scox1,add=TRUE,col=2);
     plot(cox2); plot(scox2,add=TRUE,col=2);

     ########################################
     ## second example
     ########################################

     cox1 <- phreg(Surv(time,cause==1)~strata(tcell)+platelet,data=bmt)
     cox2 <- phreg(Surv(time,cause==2)~tcell+strata(platelet),data=bmt)
     coxs <- list(cox1,cox2)
    ### dd <- sim.cause.cox(coxs,nsim,data=bmt)
     dd <- sim.phregs(coxs,n,data=bmt)
     scox1 <- phreg(Surv(time,status==1)~strata(tcell)+platelet,data=dd)
     scox2 <- phreg(Surv(time,status==2)~tcell+strata(platelet),data=dd)
     cbind(cox1$coef,scox1$coef)
    #>                [,1]       [,2]
    #> platelet -0.5658612 -0.6092701
     cbind(cox2$coef,scox2$coef)
    #>            [,1]      [,2]
    #> tcell 0.4153706 0.5130511
     par(mfrow=c(1,2))
     plot(cox1); plot(scox1,add=TRUE);
     plot(cox2); plot(scox2,add=TRUE);

- sim.phreg only for phreg, but can deal with strata
- sim.cox for other cox models, see last part of vignette, can only deal with covariates that can be identified from the names of its coefficients (so factors should be coded accordingly).

One more example

     library(mets)
     n <- 100
     data(bmt)
     bmt$bmi <- rnorm(408)
     dcut(bmt) <- gage~age
     data <- bmt
     cox1 <- phreg(Surv(time,cause==1)~strata(tcell,platelet),data=bmt)
     cox2 <- phreg(Surv(time,cause==2)~strata(gage,tcell),data=bmt)
     cox3 <- phreg(Surv(time,cause==0)~strata(platelet)+bmi,data=bmt)
     coxs <- list(cox1,cox2,cox3)

     dd <- sim.phregs(coxs,n,data=bmt,extend=0.002)
     dtable(dd,~status)
    #>
    #> status
    #>  0  1  2  3
    #>  9 46 17 28
     scox1 <- phreg(Surv(time,status==1)~strata(tcell,platelet),data=dd)
     scox2 <- phreg(Surv(time,status==2)~strata(gage,tcell),data=dd)
     scox3 <- phreg(Surv(time,status==3)~strata(platelet)+bmi,data=dd)
     cbind(coef(cox1),coef(scox1), coef(cox2),coef(scox2), coef(cox3),coef(scox3))
    #>          [,1]      [,2]
    #> bmi 0.1281967 0.2921947
     par(mfrow=c(1,3))
     plot(scox1,col=2); plot(cox1,add=TRUE,col=1)
     plot(scox2,col=2); plot(cox2,add=TRUE,col=1)
     plot(scox3,col=2); plot(cox3,add=TRUE,col=1)

# Multistate models: The Illness Death model

Using a hazard based simulation with delayed entry we can then simulate data from for example the general illness-death model. Here the cumulative hazards need to be specified.

First we set up some cumulative hazards, then we simulate some data and re-estimate the cumulative baselines

     data(CPH_HPN_CRBSI)
     dr <- CPH_HPN_CRBSI$terminal
     base1 <- CPH_HPN_CRBSI$crbsi
     base4 <- CPH_HPN_CRBSI$mechanical
     dr2 <- scalecumhaz(dr,1.5)
     cens <- rbind(c(0,0),c(2000,0.5),c(5110,3))

     iddata <- simMultistate(nsim,base1,base1,dr,dr2,cens=cens)
     dlist(iddata,.~id|id<3,n=0)
    #> id: 1
    #>       time status entry death from to start     stop
    #> 1 119.8711      3     0     1    1  3     0 119.8711
    #> ------------------------------------------------------------
    #> id: 2
    #>       time status entry death from to start     stop
    #> 2 682.9688      3     0     1    1  3     0 682.9688

     ### estimating rates from simulated data
     c0 <- phreg(Surv(start,stop,status==0)~+1,iddata)
     c3 <- phreg(Surv(start,stop,status==3)~+strata(from),iddata)
     c1 <- phreg(Surv(start,stop,status==1)~+1,subset(iddata,from==2))
     c2 <- phreg(Surv(start,stop,status==2)~+1,subset(iddata,from==1))
     ###
     par(mfrow=c(2,2))
     plot(c0)
     lines(cens,col=2)
     plot(c3,main="rates 1-> 3 , 2->3")
     lines(dr,col=1,lwd=2)
     lines(dr2,col=2,lwd=2)
     ###
     plot(c1,main="rate 1->2")
     lines(base1,lwd=2)
     ###
     plot(c2,main="rate 2->1")
     lines(base1,lwd=2)

# Cumulative incidence

In this section we discuss how to simulate competing risks data that have a specfied cumulative incidence function. We consider for simplicity a competing risks model with two causes and denote the cumulative incidence curves as \(F_1(t,X) = P(T < t, \epsilon=1|X)\) and \(F_2(t,X) = P(T < t, \epsilon=2|X)\). Here given some covariate \(X\).

To generate data with the required cumulative incidence functions a simple approach is to first figure out if the subject dies and then from what cause, then finally draw the survival time according to the conditional distribution.

For simplicity we consider survival times in a fixed interval \([0,\tau]\), and first flip a coin with and probabilities \(1-F_1(\tau,X)-F_2(\tau,X)\) to decide if the subject is a survivor or dies. Then if subject dies we then flip a coin with probabilities \(F_1(\tau,X)/(F_1(\tau,X)+F_2(\tau,X))\) and \(F_2(\tau,X)/(F_1(\tau,X)+F_2(\tau,X))\) to decide if it is a cause \(!\), \(\epsilon=1\), or a cause 2, \(\epsilon=2\). Finally we draw the survival time using the cumulative incidence distribution. The timing of a cause \(j\) event is thus \(T = (\tilde F_1^{-1}(U,X)\) with \(\tilde F_1(s,X) = F_1(s,X)/F_1(\tau,X)\) and \(U\) is a uniform.

Then indeed \(P(T \leq t, \epsilon=j|X) = F_j(t,X)\) for \(j=1,2\).

We again note and use that if \(\tilde F_j(s)\) and \(F_j(s)\) are piecewise linear continuous functions then the inverse is easy to compute.

# Cumulative incidence I

We here simulate two causes of death with two binary covarites of logistic type \[\begin{align*} F_1(t,X) &= \frac{ \Lambda_1(t,\rho_1) exp(X^T \beta)}{1+\Lambda_1(t,\rho_1) exp(X^T \beta)} \end{align*}\] and \(F_2\) here enforcing the sum condition \(F_1+F_2 \leq 1\) \[\begin{align*} F_2(t,X) & = \frac{ \Lambda_2(t,\rho_2) exp(X^T \beta)}{1+\Lambda_2(t,\rho_2) exp(X^T \beta)} [ 1- F_1(\tau,X) ] \end{align*}\] or not \[\begin{align*} F_2(t,X) & = \frac{ \Lambda_2(t,\rho_2) exp(X^T \beta)}{1+\Lambda_2(t,\rho_2) exp(X^T \beta)} \end{align*}\]

The baselines are given as \(\Lambda_j(t) = \rho_1 (1- exp(-t/r_j))\) where \(\rho_j\) and \(r_j\) are postive constants, and here \(\tau=6\).

To simulate the survival time we use a piecwise linear approximation of the cumulative incidence functions and will thus depends on some grid for linear approximation. Our linear approximation can be made arbitrarily close to any specific smooth cumulative incidence function.

    library(mets)
    nsim <- 100
    rho1 <- 0.4; rho2 <- 2
    beta <- c(0.3,-0.3,-0.3,0.3)

    dats <- simul.cifs(nsim,rho1,rho2,beta,rc=0.5,depcens=0,type="logistic")

    # Fitting regression model with CIF logistic-link
    cif1 <- cifreg(Event(time,status)~Z1+Z2,dats)
    summary(cif1)
    #>
    #>    n events
    #>  100     13
    #>
    #>  100 clusters
    #> coeffients:
    #>    Estimate     S.E.  dU^-1/2 P-value
    #> Z1  0.19032  0.30822  0.27962  0.5369
    #> Z2 -0.82349  0.64255  0.60309  0.2000
    #>
    #> exp(coeffients):
    #>    Estimate    2.5%  97.5%
    #> Z1  1.20964 0.66114 2.2132
    #> Z2  0.43890 0.12457 1.5463

    dats <- simul.cifs(n,rho1,rho2,beta,rc=0.5,depcens=0,type="cloglog")
    ciff <- cifregFG(Event(time,status)~Z1+Z2,dats)
    summary(ciff)
    #>
    #>    n events
    #>  100     26
    #>
    #>  100 clusters
    #> coeffients:
    #>    Estimate     S.E.  dU^-1/2 P-value
    #> Z1  0.33760  0.19992  0.20825  0.0913
    #> Z2 -0.62671  0.38164  0.40632  0.1006
    #>
    #> exp(coeffients):
    #>    Estimate    2.5%  97.5%
    #> Z1  1.40158 0.94722 2.0739
    #> Z2  0.53435 0.25291 1.1290

We can also use the parameters based on fitted models

     data(bmt)
     ################################################################
     #  simulating several causes with specific cumulatives
     ################################################################
     cif1 <-  cifreg(Event(time,cause)~tcell+age,data=bmt,cause=1)
     cif2 <-  cifreg(Event(time,cause)~tcell+age,data=bmt,cause=2)

     ## dd <- sim.cifs(list(cif1,cif2),nsim,data=bmt)
     dds <- sim.cifsRestrict(list(cif1,cif2),nsim,data=bmt)

     scif1 <-  cifreg(Event(time,cause)~tcell+age,data=dds,cause=1)
     scif2 <-  cifreg(Event(time,cause)~tcell+age,data=dds,cause=2)

     cbind(cif1$coef,scif1$coef)
    #>             [,1]       [,2]
    #> tcell -0.7966937 -2.5383710
    #> age    0.4164386  0.5450321
     cbind(cif2$coef,scif2$coef)
    #>              [,1]      [,2]
    #> tcell  0.66688269 1.7602329
    #> age   -0.03248603 0.4363223
     par(mfrow=c(1,2))
     plot(cif1); plot(scif1,add=TRUE,col=2)
     plot(cif2); plot(scif2,add=TRUE,col=2)

# CIF Delayed entry

Now assume that given covariates \(F_1(t;X) = P(T < t, \epsilon=1|X)\) and \(F_2(t;X) = P(T < t, \epsilon=2|X)\) are two cumulative incidence functions that satistifes the needed constraints. We wish to generate data that follows these two piecewise linear cumulative indidence functions with delayed entry at time \(s\). We should thus generate data that follows the cumulative incidence functions \[ \tilde F_1(t,s;X)= \frac{F_1(t;X) - F_1(s;;X)}{ 1 - F_1(s;X) - F_2(s;X)} \] and \[ \tilde F_2(t,s;X)= \frac{F_2(t;X) - F_2(s;;X)}{ 1 - F_1(s;X) - F_2(s;X)} \] this can be done according to the recipe in the previous section.
To be specific (ignoring the \(X\) in the formula) \[ F_1^{-1}( F_1(s) + U \cdot (1 - F_1(s;X) - F_2(s;X)) ) \] where \(U\) is a uniform, will have distribution given by \(\tilde F_1(t,s)\).

# Recurrent events

See also recurrent events vignette

     data(CPH_HPN_CRBSI)
     dr <- CPH_HPN_CRBSI$terminal
     base1 <- CPH_HPN_CRBSI$crbsi
     base4 <- CPH_HPN_CRBSI$mechanical

     n <- 100
     rr <- simRecurrent(n,base1,death.cumhaz=dr)
     ###
     par(mfrow=c(1,3))
     showfitsim(causes=1,rr,dr,base1,base1,which=1:2)

     rr <- simRecurrentII(n,base1,base4,death.cumhaz=dr)
     dtable(rr,~death+status)
    #>
    #>       status   0   1   2
    #> death
    #> 0             10 280  40
    #> 1             90   0   0
     showfitsim(causes=2,rr,dr,base1,base4,which=1:2)

     cumhaz <- list(base1,base1,base4)
     drl <- list(dr,base4)
     rr <- simRecurrentList(n,cumhaz,death.cumhaz=drl)
     dtable(rr,~death+status)
    #>
    #>       status   0   1   2   3
    #> death
    #> 0              2 194 192  30
    #> 1             71   0   0   0
    #> 2             27   0   0   0
     showfitsimList(rr,cumhaz,drl)

- sim.recurrent can simulate based on cox hazard for events and death based on phreg
  - similar to sim.phreg

     data(hfactioncpx12)
     hf <- hfactioncpx12
     hf$x <- as.numeric(hf$treatment)
     n <- 100

     ##  to fit non-parametric models with just a baseline
     xr <- phreg(Surv(entry,time,status==1)~cluster(id),data=hf)
     dr <- phreg(Surv(entry,time,status==2)~cluster(id),data=hf)

     simcoxcox <- sim.recurrent(xr,dr,n=n,data=hf)

     recGL <- recreg(Event(entry,time,status)~+cluster(id),hf,death.code=2)
     simglcox <- sim.recurrent(recGL,dr,n=n,data=hf)

# Simulations based on coxph

     cox <-  survival::coxph(Surv(time,status==9)~vf+chf+wmi,data=sTRACE)
     sim1 <- sim.cox(cox,nsim,data=sTRACE)
     cc <- survival::coxph(Surv(time,status)~vf+chf+wmi,data=sim1)
     cbind(cox$coef,cc$coef)
    #>           [,1]       [,2]
    #> vf   0.2970218 -0.2184569
    #> chf  0.8018334  0.5825885
    #> wmi -0.8920005 -1.8603130
     cor(sim1[,c("vf","chf","wmi")])
    #>             vf        chf         wmi
    #> vf  1.00000000  0.1435916  0.06617519
    #> chf 0.14359163  1.0000000 -0.50148928
    #> wmi 0.06617519 -0.5014893  1.00000000
     cor(sTRACE[,c("vf","chf","wmi")])
    #>              vf        chf         wmi
    #> vf   1.00000000  0.1346711 -0.08966805
    #> chf  0.13467109  1.0000000 -0.37464791
    #> wmi -0.08966805 -0.3746479  1.00000000

     cox <-  phreg(Surv(time, status==9)~vf+chf+wmi,data=sTRACE)
     sim3 <- sim.cox(cox,nsim,data=sTRACE)
     cc <-  phreg(Surv(time, status)~vf+chf+wmi,data=sim3)
     cbind(cox$coef,cc$coef)
    #>           [,1]       [,2]
    #> vf   0.2970218  0.1326161
    #> chf  0.8018334  1.0879842
    #> wmi -0.8920005 -0.5919069
     plot(cox,se=TRUE); plot(cc,add=TRUE,col=2)

     coxs <-  phreg(Surv(time,status==9)~strata(chf,vf)+wmi,data=sTRACE)
     sim3 <- sim.phreg(coxs,nsim,data=sTRACE)
     cc <-   phreg(Surv(time, status)~strata(chf,vf)+wmi,data=sim3)
     cbind(coxs$coef,cc$coef)
    #>           [,1]       [,2]
    #> wmi -0.8683355 -0.9816674
     plot(coxs,col=1); plot(cc,add=TRUE,col=2)

More Cox games with cause specific hazards

     data(bmt)
     # coxph
     cox1 <- survival::coxph(Surv(time,cause==1)~tcell+platelet,data=bmt)
     cox2 <- survival::coxph(Surv(time,cause==2)~tcell+platelet,data=bmt)
     coxs <- list(cox1,cox2)
     dd <- sim.cause.cox(coxs,nsim,data=bmt)
     scox1 <- survival::coxph(Surv(time,status==1)~tcell+platelet,data=dd)
     scox2 <- survival::coxph(Surv(time,status==2)~tcell+platelet,data=dd)
     cbind(cox1$coef,scox1$coef)
    #>                [,1]       [,2]
    #> tcell    -0.4231551 -0.9391252
    #> platelet -0.5646181 -0.1879697
     cbind(cox2$coef,scox2$coef)
    #>                [,1]       [,2]
    #> tcell     0.3991911  0.0480784
    #> platelet -0.2456203 -0.3839736

# 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