Transforms cumulative hazard estimates to estimate survival analysis
parameters specified by differential equations. We demonstrate how the
method works on some commonly studied parameters.
Survival
The survival function \(S\) solves a
differential equation with initial value 1; \[\begin{equation} S_t = 1 - \int_0^t S_{s}
dA_s.\end{equation}\]
We generate a sample of exponentially distributed survival times with
independent censoring, and calculate cumulative hazard estimates
n1 <- 300
Samp1 <- rexp(n1,1)
cTimes <- runif(n1)*4
Samp1 <- pmin(Samp1,cTimes)
isNotCensored <- cTimes != Samp1
startTimes1 <- rep(0,n1)
aaMod1 <- aalen(Surv(startTimes1,Samp1,isNotCensored)~1)
We want to estimate the survival function. The cumulative hazard
estimates are inserted as a time-ordered matrix where each column is an
increment of the vector of cumulatie hazard estimates;
dA_est1 <- diff(c(0,aaMod1$cum[,2]))
hazMatrix <- matrix(dA_est1,nrow=1)
In this case, \(F\) takes the simple
form \(F(x) = -x\), and its Jacobian is
therefore \(J_{F}(x) = -1\). These must
be provided as matrix-valued functions;
F_survival <- function(X)matrix(-X,nrow=1,ncol=1)
F_survival_JacobianList <- list(function(X)matrix(-1,nrow=1,ncol=1))
We obtain plugin estimates of \(S\)
and its covariance using the call
param <- pluginEstimate(n1, hazMatrix, F_survival, F_survival_JacobianList,matrix(1,nrow=1,ncol=1),matrix(0,nrow=1,ncol=1))
Finally, we may plot the results with approximate 95 % confidence
intervals
times1 <- aaMod1$cum[,1]
plot(times1,param$X,type="s",xlim=c(0,4),xlab="t",ylab="",main="Survival")
lines(times1,param$X + 1.96 * sqrt(param$covariance[1,1,]),type="s",lty=2)
lines(times1,param$X - 1.96 * sqrt(param$covariance[1,1,]),type="s",lty=2)
lines(times1,exp(-times1),col=2)
legend("topright",c("estimated","true"),lty=1,col=c(1,2),bty="n")
Restricted mean survival
The restricted mean survival function \(R\) solves the system \[\begin{equation} \begin{pmatrix} S_t \\ R_t
\end{pmatrix} = \begin{pmatrix}1 \\ 0 \end{pmatrix} + \int_0^t
\begin{pmatrix} -S_{s} & 0 \\ 0 & S_s \end{pmatrix}d
\begin{pmatrix}A_s \\ s \end{pmatrix},\end{equation}\]
where \(S\) is the survival
function. Here, the colums of \(F\),
\(F_1\) and \(F_2\), are given by \[\begin{equation} F_1(x_1,x_2) = \begin{pmatrix}
-x_1 \\ 0 \end{pmatrix}, F_2(x_1,x_2) = \begin{pmatrix} 0 \\ x_1
\end{pmatrix}.\end{equation}\] The Jacobian matrices are
therefore \[\begin{equation} J_{F_1}(x_1,x_2)
= \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix},
J_{F_2}(x_1,x_2) = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.
\end{equation}\] We define these along with initial values
below
F_restrict <- function(X)matrix(c(-X[1],0,0,X[1]),nrow=2)
F_restrict_JacobianList <- list(function(X)matrix(c(-1,0,0,0),nrow=2),
function(X)matrix(c(0,0,1,0),nrow=2,byrow=T))
X0_restrict <- matrix(c(1,0),nrow=2)
V0_restrict <- matrix(0,nrow=2,ncol=2)
The restricted mean survival is a ‘regular’ (i.e. Lebesgue) integral,
and we must therefore provide the time increments. We choose the time
interval \([0,4]\) in \(10^4\) increments:
fineTimes <- seq(0,4,length.out = 1e4+1)
tms <- sort(unique(c(fineTimes,times1)))
hazMatrix <- matrix(0,nrow=2,ncol=length(tms))
hazMatrix[1,match(times1,tms)] <- dA_est1
hazMatrix[2,] <- diff(c(0,tms))
We obtain plugin estimates using the call (note the last argument
that in this example can be used to improve efficiency);
param <- pluginEstimate(n1, hazMatrix, F_restrict, F_restrict_JacobianList,X0_restrict,V0_restrict,isLebesgue = 2)
We plot the results
plot(tms,param$X[2,],type="s",xlim=c(0,4),ylim=c(0,1.5),xlab="t",ylab="",main="Restricted mean survival")
lines(tms,param$X[2,] + 1.96 * sqrt(param$covariance[2,2,]),type="s",lty=2)
lines(tms,param$X[2,] - 1.96 * sqrt(param$covariance[2,2,]),type="s",lty=2)
lines(tms,1 - exp(-tms),col=2)
legend("topleft",c("estimated","true"),lty=1,col=c(1,2),bty="n")
Relative survival
We generate another set of exponentially distributed survival times
and compare the two groups. A new matrix of cumulative hazard increments
must be created;
n2 <- 200
Samp2 <- rexp(n2,1.3)
censTimes2 <- runif(n2)*3
Samp2 <- pmin(Samp2,censTimes2)
isNotCensored2 <- censTimes2 != Samp2
startTimes2 <- rep(0,n2)
aaMod2 <- aalen(Surv(startTimes2,Samp2,isNotCensored2)~1)
dA_est2 <- diff(c(0,aaMod2$cum[,2]))
times2 <- aaMod2$cum[,1]
times <- sort(unique(c(times1,times2)))
hazMatrix <- matrix(0,nrow=2,ncol=length(times))
hazMatrix[1,match(times1,times)] <- dA_est1
hazMatrix[2,match(times2,times)] <- dA_est2
The relative survival function \(RS\) between groups 1 and 2 solve the
equation \[\begin{equation} RS_t = 1 +
\int_0^t \begin{pmatrix} -RS_s & RS_s\end{pmatrix} d\begin{pmatrix}
A_s^1 \\ A_s^2 \end{pmatrix}, \end{equation}\]
i.e. \(F(x) = (-x,x)\). The
Jacobians of the columns of \(F\) are
\(J_{F_1}(x) = -1\), and \(J_{F_2}(x) = 1\). We specify these
functions as follows:
F_relsurv <- function(X)matrix(c(-X,X),nrow=1)
F_relsurv_JacobianList <- list(function(X)matrix(-1,nrow=1),
function(X)matrix(1,nrow=1))
The estimates are then obtained by the call
param <- pluginEstimate(n1+n2, hazMatrix, F_relsurv, F_relsurv_JacobianList,matrix(1,nrow=1,ncol=1),matrix(0,nrow=1,ncol=1))
We may plot the results with approximate 95% confidence intervals
plot(times,param$X,type="s",xlim=c(0,4),ylim=c(-0.5,4),xlab="t",ylab="",main="Relative survival")
lines(times,param$X + 1.96 * sqrt(param$covariance[1,1,]),type="s",lty=2)
lines(times,param$X - 1.96 * sqrt(param$covariance[1,1,]),type="s",lty=2)
lines(times,exp(-times*(1/1.3 - 1)),col=2)
legend("topleft",c("estimated","true"),lty=1,col=c(1,2),bty="n")
Cumulative incidence
We may be in a situation with two competing risks. The cumulative
incidences \(C^1,C^2\) solve the system
\[\begin{equation} \begin{pmatrix} S_t \\
C_t^1 \\ C_t^2 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}
+ \int_0^t \begin{pmatrix} -S_s & -S_s \\ S_s & 0 \\ 0 & S_s
\end{pmatrix} d \begin{pmatrix} A^1_s \\ A^2_s \end{pmatrix},
\end{equation}\] where \(S\) is
the survival. The first columns of \(F\) are
\[\begin{equation}F_1(x_1,x_2,x_2) =
\begin{pmatrix} -x_1 \\ x_1 \\ 0 \end{pmatrix}, F_2(x_1,x_2,x_2) =
\begin{pmatrix} -x_1 \\ 0 \\ x_1 \end{pmatrix}
\end{equation}\]
The reader may verify that the Jacobian matrix of \(F_1\) and \(F_1\) are \[\begin{equation} J_{F_1}(x_1,x_2,x_3) =
\begin{pmatrix} -1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0
& 0 \end{pmatrix} , J_{F_2}(x_1,x_2,x_3) = \begin{pmatrix} -1 &
0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}
\end{equation}\]
We specify the function \(F\) and
the list of the Jacobian matrices below, along with the initial
values:
F_cuminc <- function(X)matrix(c(-X[1],-X[1],X[1],0,0,X[1]),nrow=3,byrow=T)
F_cuminc_JacobianList <- list(function(X)matrix(c(-1,0,0,1,0,0,0,0,0),nrow=3,byrow=T),
function(X)matrix(c(-1,0,0,0,0,0,1,0,0),nrow=3,byrow=T))
X0_cuminc <- matrix(c(1,0,0),nrow=3)
v0_cuminc <- matrix(0,nrow=3,ncol=3)
Now for obtaining hazard estimates. We generate survival times first,
before sampling two causes of death (and censoring). We then estimate
the cause-specific cumulative hazards, and transform the estimates:
# Generate the data
dfr <- data.frame(from=0,to=rexp(n1+n2,1),from.state = 1)
# Adding causes of death (to.state = 2 or 3) and censoring (to.state = 0)
dfr$to.state <- sample(c(0,2,3),n1+n2,replace=T,prob=c(0.2,0.2,0.6))
# Obtaining cause-specific cumulative hazard estimates and extracting the increments
aamod22 <- aalen(Surv(from,to,to.state %in% c(2)) ~1, data=dfr)
aamod33 <- aalen(Surv(from,to,to.state %in% c(3)) ~1, data=dfr)
tmss = sort(unique(c(aamod22$cum[,1],aamod33$cum[,1])))
hazMatrix = matrix(0,nrow=2,ncol=length(tmss))
mt1 = match(aamod22$cum[,1],tmss)
mt2 = match(aamod33$cum[,1],tmss)
hazMatrix[1,mt1] = diff(c(0, aamod22$cum[,2] ))
hazMatrix[2,mt2] = diff(c(0, aamod33$cum[,2] ))
# Transforming the estimates
param <- pluginEstimate(n1+n2,hazMatrix,F_cuminc,F_cuminc_JacobianList,X0_cuminc,v0_cuminc)
Finally, we plot the estimates of the cumulative incidences \(C^1\) and \(C^2\) with confidence intervals:
plot(tmss,param$X[2,],type="s",xlim=c(0,4),ylim=c(0,1),xlab="t",ylab="",main="Cumulative incidence")
lines(tmss,param$X[2,] + 1.96 * sqrt(param$covariance[2,2,]),type="s",lty=2)
lines(tmss,param$X[2,] - 1.96 * sqrt(param$covariance[2,2,]),type="s",lty=2)
lines(tmss,param$X[3,],type="s",col=3)
lines(tmss,param$X[3,] + 1.96 * sqrt(param$covariance[3,3,]),type="s",lty=2,col=3)
lines(tmss,param$X[3,] - 1.96 * sqrt(param$covariance[3,3,]),type="s",lty=2,col=3)
legend("topleft",c(expression(C^1),expression(C^2)),lty=1,col=c(1,3),bty="n")
Restricted mean survival – comparing two groups
By re-specifying the ODE system we can compare two groups. We
illustrate this in the restricted mean survival example. Consider the
system
\[\begin{equation} \begin{pmatrix} R_t^1
\\ R^2 \\ RD \\ S^1 \\ S^2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0
\\ 1 \\ 1 \end{pmatrix} + \int_0^t \begin{pmatrix} S^1_{s} & 0 &
0 \\ S^2_{s} & 0 & 0 \\ S^1_{s} - S^2_{s} & 0 & 0 \\ 0
& -S_s^1 & 0 \\ 0 & 0 & -S_s^2 \end{pmatrix}d
\begin{pmatrix}s \\ A_s^1 \\ A_s^2
\end{pmatrix},\end{equation}\]
where \(S^i\) is the survival
function in group \(i\).
We specify matrix-valued functions and initial values as before:
F_restrict_diff <- function(X)matrix(c(X[4],0,0,
X[5],0,0,
X[4]-X[5],0,0,
0,-X[4],0,
0,0,-X[5]),nrow=5,byrow=T)
F_restrict_diff_JacobianList <- list(function(X)matrix(c(0,0,0,1,0,
0,0,0,0,1,
0,0,0,1,-1,
rep(0,10)),nrow=5,byrow=T),
function(X)matrix(c(rep(0,18),-1,rep(0,6)),nrow=5,byrow=T),
function(X)matrix(c(rep(0,24),-1),nrow=5,byrow=T))
X0_restrict_diff <- matrix(c(0,0,0,1,1),nrow=5)
V0_restrict_diff <- matrix(0,nrow=5,ncol=5)
We compare RMS for males and females in the flchain data set
in the survival package. Specifying hazmatrix:
aa_male = aalen(Surv(futime,death==1)~1, data = flchain[flchain$sex=="M",])
aa_female = aalen(Surv(futime,death==1)~1, data = flchain[flchain$sex=="F",])
tms <- sort(unique(c(fineTimes,aa_male$cum[,1], aa_female$cum[,1])))
mt_male = match(aa_male$cum[,1],tms)
mt_female = match(aa_female$cum[,1],tms)
hazMatrix <- matrix(0,nrow=3,ncol=length(tms))
hazMatrix[1,] <- diff(c(0,tms))
hazMatrix[2,mt_male] <- diff(c(0,aa_male$cum[,2]))
hazMatrix[3,mt_female] <- diff(c(0,aa_female$cum[,2]))
plugEst <- pluginEstimate(1, hazMatrix, F_restrict_diff, F_restrict_diff_JacobianList,X0_restrict_diff,V0_restrict_diff,isLebesgue = 1)
param = list(plugEst=plugEst, tms=tms)
Plotting \(\hat{RD} = \hat R^1 - \hat
R^2\) with 95% confidence intervals:
ylims = c(1.1*min(param$plugEst$X[3,] - 1.96 * sqrt(param$plugEst$covariance[3,3,])),
1.1*max(param$plugEst$X[3,] + 1.96 * sqrt(param$plugEst$covariance[3,3,])))
plot(param$tms,param$plugEst$X[3,],type="s",ylim=ylims,ylab="",main="RMS difference",xlab="years")
lines(param$tms,param$plugEst$X[3,] + 1.96 * sqrt(param$plugEst$covariance[3,3,]),type="s",lty=2)
lines(param$tms,param$plugEst$X[3,] - 1.96 * sqrt(param$plugEst$covariance[3,3,]),type="s",lty=2)
abline(a=0,b=0,col=2)
Cumulative incidence – comparing two groups
Consider a similar example with differences of cumulative incidence
functions
\[\begin{equation} \begin{pmatrix} S^a_t
\\ S_t^b \\ C_t^{a,1} \\ C_t^{a,2} \\ C_t^{b,1} \\ C_t^{b,2} \\ CD_t^1
\\ CD_t^2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\
0 \\ 0 \end{pmatrix} + \int_0^t \begin{pmatrix} -S_s^a & -S_s^a
& 0 & 0 \\ 0 & 0 & -S_s^b & -S_s^b \\ S_s^a & 0
& 0 & 0 \\ 0 & S_s^a & 0 & 0 \\ 0 & 0 &
S_s^b & 0 \\ 0 & 0 & 0 & S_s^b \\ S_s^a & 0 &
-S_s^b & 0 \\ 0 & S_s^a & 0 & -S_s^b \end{pmatrix} d
\begin{pmatrix} A^{a,1}_s \\ A^{a,2}_s \\ A^{b,1}_s \\ A^{b,2}_s
\end{pmatrix}, \end{equation}\] where \(S^a\) is the survival in group \(a\) and \(S^b\) is the survival in group \(b\), and \(C^{a,j}\) is the cumulative incidence for
cause \(j\) in group \(a\) and similarly for group \(b\). \(CD^i =
C^{a,i} - C^{b,i}\) is the cumulative incidence difference for
cause \(i\). We specify the ODE
system.
F_cuminc_diff <- function(X)matrix( c(-X[1],-X[1],0,0,
0,0,-X[2],-X[2],
X[1],0,0,0,
0,X[1],0,0,
0,0,X[2],0,
0,0,0,X[2],
X[1],0,-X[2],0,
0,X[1],0,-X[2]) ,nrow=8,byrow=T)
F_cuminc_diff_JacobianList <- list(function(X)matrix(c(-1, rep(0,7),
rep(0,8),
1, rep(0,7),
rep(0,8),
rep(0,8),
rep(0,8),
1, rep(0,7),
rep(0,8)),nrow=8,byrow=T),
function(X)matrix(c(-1, rep(0,7),
rep(0,8),
rep(0,8),
1, rep(0,7),
rep(0,8),
rep(0,8),
rep(0,8),
1, rep(0,7)),nrow=8,byrow=T),
function(X)matrix(c(rep(0,8),
0,-1, rep(0,6),
rep(0,8),
rep(0,8),
0,1, rep(0,6),
rep(0,8),
0,-1, rep(0,6),
rep(0,8)),nrow=8,byrow=T),
function(X)matrix(c(rep(0,8),
0,-1, rep(0,6),
rep(0,8),
rep(0,8),
rep(0,8),
0,1, rep(0,6),
rep(0,8),
0,-1, rep(0,6)),nrow=8,byrow=T))
X0_cuminc_diff <- matrix(c(1,1,0,0,0,0,0,0),nrow=8)
v0_cuminc_diff <- matrix(0,nrow=8,ncol=8)
We inspect cumulative incidences of the competing events PCM, and
death without malignancy, in the mgus2 data set. We compare
males and females in this data set. Specifying hazmatrix:
mgus2$etime <- with(mgus2, ifelse(pstat==0, futime, ptime))
event <- with(mgus2, ifelse(pstat==0, 2*death, 1))
mgus2$event <- factor(event, 0:2, labels=c("censor", "pcm", "death"))
fr_M <- mgus2[mgus2$sex == "M",]
fr_M$time <- fr_M$etime/12
fr_F <- mgus2[mgus2$sex == "F",]
fr_F$time <- fr_F$etime/12
fit_PCM_M = aalen(Surv(time,event=="pcm")~1,data=fr_M)
fit_death_M = aalen(Surv(time,event=="death")~1,data=fr_M)
fit_PCM_F = aalen(Surv(time,event=="pcm")~1,data=fr_F)
fit_death_F = aalen(Surv(time,event=="death")~1,data=fr_F)
tms2 = sort(unique(c(fit_PCM_M$cum[,1],fit_PCM_F$cum[,1],
fit_death_M$cum[,1],fit_death_F$cum[,1])))
dA_PCM_M = dA_death_M = dA_PCM_F = dA_death_F = rep(0, length(tms2))
dA_PCM_M[match(fit_PCM_M$cum[,1], tms2)] = diff(c(0,fit_PCM_M$cum[,2]))
dA_PCM_F[match(fit_PCM_F$cum[,1], tms2)] = diff(c(0,fit_PCM_F$cum[,2]))
dA_death_M[match(fit_death_M$cum[,1], tms2)] = diff(c(0,fit_death_M$cum[,2]))
dA_death_F[match(fit_death_F$cum[,1], tms2)] = diff(c(0,fit_death_F$cum[,2]))
hazMatrix <- matrix(0,nrow=4,ncol=length(tms2))
hazMatrix[1,] <- dA_PCM_M
hazMatrix[2,] <- dA_death_M
hazMatrix[3,] <- dA_PCM_F
hazMatrix[4,] <- dA_death_F
plugEst <- pluginEstimate(1, hazMatrix, F_cuminc_diff, F_cuminc_diff_JacobianList,X0_cuminc_diff,v0_cuminc_diff)
param = list(plugEst=plugEst, tms=tms2)
Plotting \(\hat{CD}^1\) and \(\hat{CD}^2\) with 95% confidence
intervals:
# Colors
male_color <- "red"
female_color <- "blue"
diff_color <- "gray"
par(mfrow=c(1,2))
# Plot with colors and labels
plot(param$tms, param$plugEst$X[3,],type="s",ylim=c(-0.2,0.9),main="Cum.inc. PCM",
ylab="",xlab="Years", col=male_color)
lines(param$tms, param$plugEst$X[3,] + 1.96 * sqrt(param$plugEst$covariance[3,3,]),type="s",lty=2, col=male_color)
lines(param$tms, param$plugEst$X[3,] - 1.96 * sqrt(param$plugEst$covariance[3,3,]),type="s",lty=2, col=male_color)
lines(param$tms, param$plugEst$X[5,],type="s", col=female_color)
lines(param$tms, param$plugEst$X[5,] + 1.96 * sqrt(param$plugEst$covariance[5,5,]),type="s",lty=2, col=female_color)
lines(param$tms, param$plugEst$X[5,] - 1.96 * sqrt(param$plugEst$covariance[5,5,]),type="s",lty=2, col=female_color)
lines(param$tms, param$plugEst$X[7,],type="s", col=diff_color)
lines(param$tms, param$plugEst$X[7,] + 1.96 * sqrt(param$plugEst$covariance[7,7,]),type="s",lty=2, col=diff_color)
lines(param$tms, param$plugEst$X[7,] - 1.96 * sqrt(param$plugEst$covariance[7,7,]),type="s",lty=2, col=diff_color)
abline(a=0,b=0,col=2)
# Add legend
legend("topleft", legend = c("Males", "Females", "Difference"), col = c(male_color, female_color, diff_color), lty = 1,bty="n")
# Plot with colors and labels
plot(param$tms, param$plugEst$X[4,],type="s",ylim=c(-0.2,0.9),main="Cum.inc. death",
ylab="",xlab="Years", col=male_color)
lines(param$tms, param$plugEst$X[4,] + 1.96 * sqrt(param$plugEst$covariance[4,4,]),type="s",lty=2, col=male_color)
lines(param$tms, param$plugEst$X[4,] - 1.96 * sqrt(param$plugEst$covariance[4,4,]),type="s",lty=2, col=male_color)
lines(param$tms, param$plugEst$X[6,],type="s", col=female_color)
lines(param$tms, param$plugEst$X[6,] + 1.96 * sqrt(param$plugEst$covariance[6,6,]),type="s",lty=2, col=female_color)
lines(param$tms, param$plugEst$X[6,] - 1.96 * sqrt(param$plugEst$covariance[6,6,]),type="s",lty=2, col=female_color)
lines(param$tms, param$plugEst$X[8,],type="s", col=diff_color)
lines(param$tms, param$plugEst$X[8,] + 1.96 * sqrt(param$plugEst$covariance[8,8,]),type="s",lty=2, col=diff_color)
lines(param$tms, param$plugEst$X[8,] - 1.96 * sqrt(param$plugEst$covariance[8,8,]),type="s",lty=2, col=diff_color)
abline(a=0,b=0,col=2)
# Add legend
legend("topleft", legend = c("Males", "Females", "Difference"), col = c(male_color, female_color, diff_color), lty = 1,bty="n")
