In kkholst/mets: Analysis of Multivariate Event Times

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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 more 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 recent paper also 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).

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 \cite{} 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.

 nsim <- 100
 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 we generate data that resemble Cox models for the bmt 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)

Now model with no covariates and specific call of sim.base function

 data(sTRACE)
 dtable(sTRACE,~chf+diabetes)
 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)

More Cox games

 cox <-  coxph(Surv(time,status==9)~vf+chf+wmi,data=sTRACE)
 sim1 <- sim.cox(cox,nsim,data=sTRACE)
 cc <- coxph(Surv(time,status)~vf+chf+wmi,data=sim1)
 cbind(cox$coef,cc$coef)
 cor(sim1[,c("vf","chf","wmi")])
 cor(sTRACE[,c("vf","chf","wmi")])

 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)
 plot(cox,se=TRUE); plot(cc,add=TRUE,col=2)

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

More Cox games with cause specific hazards

 data(bmt)
 # coxph          
 cox1 <- coxph(Surv(time,cause==1)~tcell+platelet,data=bmt)
 cox2 <- coxph(Surv(time,cause==2)~tcell+platelet,data=bmt)
 coxs <- list(cox1,cox2)
 dd <- sim.cause.cox(coxs,nsim,data=bmt)
 scox1 <- coxph(Surv(time,status==1)~tcell+platelet,data=dd)
 scox2 <- coxph(Surv(time,status==2)~tcell+platelet,data=dd)
 cbind(cox1$coef,scox1$coef)
 cbind(cox2$coef,scox2$coef)

Startified Cox models using phreg

 ## 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)
 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)
 cbind(cox1$coef,scox1$coef)
 cbind(cox2$coef,scox2$coef)
 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); 

 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)
 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)
 cbind(cox2$coef,scox2$coef)
 par(mfrow=c(1,2))
 plot(cox1); plot(scox1,add=TRUE); 
 plot(cox2); plot(scox2,add=TRUE);

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(base1cumhaz)
 data(base4cumhaz)
 data(drcumhaz)
 dr <- drcumhaz
 dr2 <- drcumhaz
 dr2[,2] <- 1.5*drcumhaz[,2]
 base1 <- base1cumhaz
 base4 <- base4cumhaz
 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)

 ### 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))
 bplot(c0)
 lines(cens,col=2) 
 bplot(c3,main="rates 1-> 3 , 2->3")
 lines(dr,col=1,lwd=2)
 lines(dr2,col=2,lwd=2)
 ###
 bplot(c1,main="rate 1->2")
 lines(base1,lwd=2)
 ###
 bplot(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 $F_1(t) = P(T < t, \epsilon=1)$ and $F_2(t) = P(T < t, \epsilon=2)$.

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 $1-F_1(\tau)-F_2(\tau)$ to decide if the subject is a survivor or dies. If the subject dies we then flip a coin with probabilities $F_1(\tau)/(F_1(\tau)+F_2(\tau))$ and $F_2(\tau)/(F_1(\tau)+F_2(\tau))$ to decide if $\epsilon=1$ or $\epsilon=2$, and finally draw a $T = (\tilde F_1^{-1}(U)$ with $\tilde F_1(s) = F_1(s)/F_1(\tau)$ and $U$ is a uniform.

We again note that if $\tilde F_1(s)$ and $F_1(s)$ are piecewise linear continuous functions then the inverses are easy to compute.

Cumulative incidence regression

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.

Possibly $F_1(t;X) = 1 - \exp( \Lambda_1(t) \exp( X^T \beta_1)$ $F_2(t;X) = 1 - \exp( \Lambda_2(t) \exp( X^T \beta_2)$ given estimators of $\Lambda_1$ and $\lambda_2$ and $\beta_1$ and $\beta_2$. We can obtain a piecewise linear continuous approximation, $F_1^L(t;X)$
by linearly connecting estimates $\hat F_1(t_j;X) = 1 - \exp( \hat \Lambda_1(t) \exp( X^T \hat \beta_1)$. Now with these at hand
$F_1^L(t;X)$ and $F_2^L(t;X)$ we can generate data with these cumulative incidence functions.

Here both the cumulative incidence are on the specified form if the restriction is not important. Using sim.cifs but sim.cifs enforces the restriction. Here $F_1$ will be on the specified form, and $F_2$ not.

 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)
 cbind(cif2$coef,scif2$coef)
 par(mfrow=c(1,2))   
 plot(cif1); plot(scif1,add=TRUE,col=2)
 plot(cif2); plot(scif2,add=TRUE,col=2)

Parametric form with baselines
- Fine-Gray form
- logistic link

We assumed that $F_1(t,X) = 1-\exp( \Lambda_1(t) \exp( X^T \beta_1))$ with $\Lambda_1(t) = \rho_1 \cdot (1-exp(-t))$ and $\beta_1 = (0,-0.1)$, and that the other cause was given by
$F_2(t,X) = 1-\exp( \Lambda_2(t) \exp( X^T \beta_2)) ( 1 - F_1(+\infty,X))$ with $\Lambda_2(t) = \rho_2 \cdot (1-exp(-t))$ and $\beta_2 = (-0.5,0.3)$, a parametrization that satisfies the constraint $F_1+F_2 \leq 1$.

 set.seed(100)
 rho1 <- 0.2; rho2 <- 10
 n <- nsim
 beta=c(0.0,-0.1,-0.5,0.3)
 dats <- simul.cifs(n,rho1,rho2,beta,rc=0.2)
 dtable(dats,~status)
 dsort(dats) <- ~time
 fg <- cifreg(Event(time,status)~Z1+Z2,data=dats,cause=1,propodds=NULL)
 summary(fg)

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 $\tilde F_1(t,s)$.