---
title: "WIP: Cooking survival data, 5 minute recipes"
aouthor: Klaus Holst & Thomas Scheike
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    fig_caption: yes
    # fig_width: 7.15
    # fig_height: 5.5        
vignette: >
  %\VignetteIndexEntry{Cooking survival data, 5 minutes recipes}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
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 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).

```{r}
 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.

```{r}
 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

```{r}
 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

```{r}
 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 

```{r}
 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.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

```{r}
 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)
```
 
 Stratified Cox models using phreg

```{r}
 ## 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); 
```

 - sim.phreg only for phreg, but a bit more flexible, can deal with strata 
 - sim.cox can only deal with covariates that can be identified from
     the names of its coefficients (so factors should be coded accordingly) or use sim.phreg


```{r}
 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+platelet,data=bmt)
 cox3 <- phreg(Surv(time,cause==0)~strata(platelet)+bmi,data=bmt)
 coxs <- list(cox1,cox2,cox3)

 dd <- sim.phregs(coxs,1000,data=bmt,extend=0.002)
 scox1 <- phreg(Surv(time,status==1)~strata(tcell)+platelet,data=dd)
 scox2 <- phreg(Surv(time,status==2)~strata(gage)+tcell+platelet,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

```{r}
 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 I
================================

We here simulate two causes of death with two binary covariates

```{r}
cif1 <- cbind(c(0,10,20,100),c(0,0.1,0.15,0.2))
cif2 <- cbind(c(0,10,20,100),c(0,0.4,0.45,0.5))

n <- 100; lrr1=c(0.2,0.1); lrr2=c(0.2,0.1); cens=NULL
### A binary, L binary
A <- rbinom(n,1,0.5)
L <- rbinom(n,1,0.5)
###
rr1 <- exp(cbind(A,L) %*% lrr1)
rr2 <- exp(cbind(A,L) %*% lrr2)
## model is fine
mmm<-max(rr1)*max(cif1[,2])+max(rr2)*max(cif2[,2])
mcif1 <- max(cif1[,2])
mcif2 <- max(cif2[,2])
if (mmm>1) warning(" models not satisfying sum <=1\n")
### here log-link model 
T1 <- simsubdist(cif1,rr1,type="cif")
T2 <- simsubdist(cif2,rr2,type="cif")
###
dies <- rbinom(n,1,rr1*mcif1+rr2*mcif2)
sel1 <- rbinom(n,1,mcif2/(mcif1+mcif2))+1
epsilon  <- dies*(sel1)
T1$epsilon <- epsilon
###
T1$A <- A; T1$L <- L
## times given 
T1$time <- T1$timecause
T1$time2 <- T2$timecause
T1$status <- epsilon
T1 <- dtransform(T1,time=100,epsilon==0)
T1 <- dtransform(T1,status=0,epsilon==0)
###
T1 <- dtransform(T1,time=time2,epsilon==2)
T1 <- dtransform(T1,status=2,epsilon==2)

dtable(T1,~status)

par(mfrow=c(1,2))
lrr1=c(0.2,0.1);lrr2=c(0.2,0.1)
pcif1 <- cif(Event(time,status)~strata(A,L),T1,cause=1)
pcif2 <- cif(Event(time,status)~strata(A,L),T1,cause=2)
###
newd <- data.frame(expand.grid(A=0:1,L=0:1))
rr1 <- c(exp(as.matrix(newd) %*% lrr1))
rr2 <- c(exp(as.matrix(newd) %*% lrr2))
###
cifm1 <- cbind(cif1[,1],cif1[,2] %o% rr1)
cifm2 <- cbind(cif2[,1],cif2[,2] %o% rr2)
###
par(mfrow=c(1,2))
plot(pcif1,ylim=c(0,0.3)); 
matlines(cifm1[,1],cifm1[,-1],col=1,lwd=2)
###
plot(pcif2,ylim=c(0,0.7))
matlines(cifm2[,1],cifm2[,-1],col=1,lwd=2)
```



Cumulative incidence regression models
======================================

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. 

```{r}
 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 $F_1+F_2 \leq 1$. 


```{r}
 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 $\tilde F_1(t,s)$.

Recurrent events
=================

See also recurrent events vignette

```{r}
 data(base1cumhaz)
 data(base4cumhaz)
 data(drcumhaz)
 dr <- drcumhaz
 base1 <- base1cumhaz
 base4 <- base4cumhaz

 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)
 par(mfrow=c(2,2))
 showfitsim(causes=2,rr,dr,base1,base4,which=1:2)

 cumhaz <- list(base1,base1,base4)
 drl <- list(dr,base4)
 rr <- simRecurrentIII(n,cumhaz,death.cumhaz=drl)
 dtable(rr,~death+status)
 showfitsimIII(rr,cumhaz,drl) 
```

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




SessionInfo
============


```{r}
sessionInfo()
```
