Nothing
WIP: Cooking survival data, 5 minute recipes
In mets: Analysis of Multivariate Event Times
knitr::opts_chunk$set(
collapse = TRUE,
##dev="png",
dpi=50,
fig.width=7.15, fig.height=5.5,
out.width="600px",
fig.retina=1,
comment = "#>"
)
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 \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. 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$bmi <- rnorm(408)
dcut(bmt) <- gage~age
data <- bmt
cox1 <- phreg(Surv(time,cause==1)~tcell+platelet+age,data=bmt)
dd <- sim.phreg(cox1,n,data=bmt)
dtable(dd,~status)
scox1 <- phreg(Surv(time,status==1)~tcell+platelet+age,data=dd)
cbind(coef(cox1),coef(scox1))
par(mfrow=c(1,1))
plot(scox1,col=2); plot(cox1,add=TRUE,col=1)
## changing the parametes
cox10 <- cox1
cox10$coef <- c(0,0.4,0.3)
dd <- sim.phreg(cox10,n,data=bmt)
dtable(dd,~status)
scox1 <- phreg(Surv(time,status==1)~tcell+platelet+age,data=dd)
cbind(coef(cox10),coef(scox1))
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)
Now fully nonparametric model with stratified baselines 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)
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)
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);
########################################
## 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)
cbind(cox2$coef,scox2$coef)
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$bmi <- rnorm(408)
dcut(bmt) <- gage~age
data <- bmt
cox1 <- phreg(Surv(time,cause==1)~strata(tcell,platelet),data=bmt)
cox2 <- phreg(Surv(time,cause==2)~strata(gage,tcell),data=bmt)
cox3 <- phreg(Surv(time,cause==0)~strata(platelet)+bmi,data=bmt)
coxs <- list(cox1,cox2,cox3)
dd <- sim.phregs(coxs,n,data=bmt,extend=0.002)
dtable(dd,~status)
scox1 <- phreg(Surv(time,status==1)~strata(tcell,platelet),data=dd)
scox2 <- phreg(Surv(time,status==2)~strata(gage,tcell),data=dd)
scox3 <- phreg(Surv(time,status==3)~strata(platelet)+bmi,data=dd)
cbind(coef(cox1),coef(scox1), coef(cox2),coef(scox2), coef(cox3),coef(scox3))
par(mfrow=c(1,3))
plot(scox1,col=2); plot(cox1,add=TRUE,col=1)
plot(scox2,col=2); plot(cox2,add=TRUE,col=1)
plot(scox3,col=2); plot(cox3,add=TRUE,col=1)
Multistate models: The Illness Death model
Using a hazard based simulation with delayed entry we can then simulate data
from for example the general illness-death model. Here the cumulative hazards
need to be specified.
First we set up some cumulative hazards, then we simulate some data
and re-estimate the cumulative baselines
data(CPH_HPN_CRBSI)
dr <- CPH_HPN_CRBSI$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)
### 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)
dats <- simul.cifs(n,rho1,rho2,beta,rc=0.5,depcens=0,type="cloglog")
ciff <- cifregFG(Event(time,status)~Z1+Z2,dats)
summary(ciff)
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)
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)
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$mechanical
n <- 100
rr <- simRecurrent(n,base1,death.cumhaz=dr)
###
par(mfrow=c(1,3))
showfitsim(causes=1,rr,dr,base1,base1,which=1:2)
rr <- simRecurrentII(n,base1,base4,death.cumhaz=dr)
dtable(rr,~death+status)
showfitsim(causes=2,rr,dr,base1,base4,which=1:2)
cumhaz <- list(base1,base1,base4)
drl <- list(dr,base4)
rr <- simRecurrentList(n,cumhaz,death.cumhaz=drl)
dtable(rr,~death+status)
showfitsimList(rr,cumhaz,drl)
- sim.recurrent can simulate based on cox hazard for events and death based on phreg
- similar to sim.phreg
data(hfactioncpx12)
hf <- hfactioncpx12
hf$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$coef)
cor(sim1[,c("vf","chf","wmi")])
cor(sTRACE[,c("vf","chf","wmi")])
cox <- phreg(Surv(time, status==9)~vf+chf+wmi,data=sTRACE)
sim3 <- sim.cox(cox,nsim,data=sTRACE)
cc <- phreg(Surv(time, status)~vf+chf+wmi,data=sim3)
cbind(cox$coef,cc$coef)
plot(cox,se=TRUE); plot(cc,add=TRUE,col=2)
coxs <- phreg(Surv(time,status==9)~strata(chf,vf)+wmi,data=sTRACE)
sim3 <- sim.phreg(coxs,nsim,data=sTRACE)
cc <- phreg(Surv(time, status)~strata(chf,vf)+wmi,data=sim3)
cbind(coxs$coef,cc$coef)
plot(coxs,col=1); plot(cc,add=TRUE,col=2)
More Cox games with cause specific hazards
data(bmt)
# coxph
cox1 <- survival::coxph(Surv(time,cause==1)~tcell+platelet,data=bmt)
cox2 <- survival::coxph(Surv(time,cause==2)~tcell+platelet,data=bmt)
coxs <- list(cox1,cox2)
dd <- sim.cause.cox(coxs,nsim,data=bmt)
scox1 <- survival::coxph(Surv(time,status==1)~tcell+platelet,data=dd)
scox2 <- survival::coxph(Surv(time,status==2)~tcell+platelet,data=dd)
cbind(cox1$coef,scox1$coef)
cbind(cox2$coef,scox2$coef)
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knitr::opts_chunk$set( collapse = TRUE, ##dev="png", dpi=50, fig.width=7.15, fig.height=5.5, out.width="600px", fig.retina=1, comment = "#>" )
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 \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. 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$bmi <- rnorm(408) dcut(bmt) <- gage~age data <- bmt cox1 <- phreg(Surv(time,cause==1)~tcell+platelet+age,data=bmt) dd <- sim.phreg(cox1,n,data=bmt) dtable(dd,~status) scox1 <- phreg(Surv(time,status==1)~tcell+platelet+age,data=dd) cbind(coef(cox1),coef(scox1)) par(mfrow=c(1,1)) plot(scox1,col=2); plot(cox1,add=TRUE,col=1) ## changing the parametes cox10 <- cox1 cox10$coef <- c(0,0.4,0.3) dd <- sim.phreg(cox10,n,data=bmt) dtable(dd,~status) scox1 <- phreg(Surv(time,status==1)~tcell+platelet+age,data=dd) cbind(coef(cox10),coef(scox1)) 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)
Now fully nonparametric model with stratified baselines 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)
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) 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); ######################################## ## 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) cbind(cox2$coef,scox2$coef) 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$bmi <- rnorm(408) dcut(bmt) <- gage~age data <- bmt cox1 <- phreg(Surv(time,cause==1)~strata(tcell,platelet),data=bmt) cox2 <- phreg(Surv(time,cause==2)~strata(gage,tcell),data=bmt) cox3 <- phreg(Surv(time,cause==0)~strata(platelet)+bmi,data=bmt) coxs <- list(cox1,cox2,cox3) dd <- sim.phregs(coxs,n,data=bmt,extend=0.002) dtable(dd,~status) scox1 <- phreg(Surv(time,status==1)~strata(tcell,platelet),data=dd) scox2 <- phreg(Surv(time,status==2)~strata(gage,tcell),data=dd) scox3 <- phreg(Surv(time,status==3)~strata(platelet)+bmi,data=dd) cbind(coef(cox1),coef(scox1), coef(cox2),coef(scox2), coef(cox3),coef(scox3)) par(mfrow=c(1,3)) plot(scox1,col=2); plot(cox1,add=TRUE,col=1) plot(scox2,col=2); plot(cox2,add=TRUE,col=1) plot(scox3,col=2); plot(cox3,add=TRUE,col=1)
Multistate models: The Illness Death model
Using a hazard based simulation with delayed entry we can then simulate data from for example the general illness-death model. Here the cumulative hazards need to be specified.
First we set up some cumulative hazards, then we simulate some data and re-estimate the cumulative baselines
data(CPH_HPN_CRBSI) dr <- CPH_HPN_CRBSI$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) ### 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) dats <- simul.cifs(n,rho1,rho2,beta,rc=0.5,depcens=0,type="cloglog") ciff <- cifregFG(Event(time,status)~Z1+Z2,dats) summary(ciff)
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) 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)
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$mechanical n <- 100 rr <- simRecurrent(n,base1,death.cumhaz=dr) ### par(mfrow=c(1,3)) showfitsim(causes=1,rr,dr,base1,base1,which=1:2) rr <- simRecurrentII(n,base1,base4,death.cumhaz=dr) dtable(rr,~death+status) showfitsim(causes=2,rr,dr,base1,base4,which=1:2) cumhaz <- list(base1,base1,base4) drl <- list(dr,base4) rr <- simRecurrentList(n,cumhaz,death.cumhaz=drl) dtable(rr,~death+status) showfitsimList(rr,cumhaz,drl)
- sim.recurrent can simulate based on cox hazard for events and death based on phreg
- similar to sim.phreg
data(hfactioncpx12) hf <- hfactioncpx12 hf$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$coef) cor(sim1[,c("vf","chf","wmi")]) cor(sTRACE[,c("vf","chf","wmi")]) cox <- phreg(Surv(time, status==9)~vf+chf+wmi,data=sTRACE) sim3 <- sim.cox(cox,nsim,data=sTRACE) cc <- phreg(Surv(time, status)~vf+chf+wmi,data=sim3) cbind(cox$coef,cc$coef) plot(cox,se=TRUE); plot(cc,add=TRUE,col=2) coxs <- phreg(Surv(time,status==9)~strata(chf,vf)+wmi,data=sTRACE) sim3 <- sim.phreg(coxs,nsim,data=sTRACE) cc <- phreg(Surv(time, status)~strata(chf,vf)+wmi,data=sim3) cbind(coxs$coef,cc$coef) plot(coxs,col=1); plot(cc,add=TRUE,col=2)
More Cox games with cause specific hazards
data(bmt) # coxph cox1 <- survival::coxph(Surv(time,cause==1)~tcell+platelet,data=bmt) cox2 <- survival::coxph(Surv(time,cause==2)~tcell+platelet,data=bmt) coxs <- list(cox1,cox2) dd <- sim.cause.cox(coxs,nsim,data=bmt) scox1 <- survival::coxph(Surv(time,status==1)~tcell+platelet,data=dd) scox2 <- survival::coxph(Surv(time,status==2)~tcell+platelet,data=dd) cbind(cox1$coef,scox1$coef) cbind(cox2$coef,scox2$coef)
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