pluginEstimate {transform.hazards}

SDE plugin estimator solver

Description

Calculates recursive estimator for given hazard estimates, integrand function and gradients.

Usage

pluginEstimate(n, hazMatrix, F_fun, JacobianList, X0, V0, isLebesgue = NULL)

Arguments

n

Total number of indiivduals

hazMatrix

Matrix consisting of hazards(rows) and their increments(columns) along the same time scale

F_fun

Integrand function F = (F_1,F_2,...) for the differential equation system

JacobianList

The Jacobian matrices of F_1, F_2, ... organized in a list

X0

Matrix containing initial values for the parameter

V0

Matrix containing initial values for the variance

isLebesgue

(Optional, to improve efficientcy) Provide the index of A (e.g. 2) that is a regular dt integral, i.e. not a cumulative hazard.

Value

list containing the parameter estimate X, and its covariance estimates V/n

Author(s)

Pål Christie Ryalen [e-mail]

References

Ryalen, P.C., Stensrud, M.J., Røysland, K.: Transforming cumulative hazards, arXiv, to appear in Biometrika 2018.

Examples

###############################################################
#########################  Survival  ##########################
###############################################################
library(timereg)

n <- 100
dfr <- data.frame(to = rexp(n,1),from=0,event=1)

fit <- survfit(Surv(from,to,event==1)~1,data=dfr)

times <- fit$time
dN <- fit$n.event
Y <- fit$n.risk
Y[1] <- n

dA <- matrix(dN/Y,nrow=1,ncol=length(dN))

# Function specification
F_fun_Survival <- function(x)-matrix(x,1,1)
JacobianListSurvival <- list(function(x)-matrix(1,1,1))
X0_Survival <- matrix(1,1,1)
V0_Survival <- matrix(0,1,1)

paramEst_survival <- pluginEstimate(
  n,dA,F_fun_Survival,JacobianListSurvival,X0_Survival,V0_Survival
)

KM <- cumprod(1 - dA)
Greenwood <- KM^2 * cumsum(dA^2)

plot(
  times,paramEst_survival$X,type="s",main="SDE plugin survival estimates",
  ylab="",xlab="time"
)
lines(times,paramEst_survival$X + 1.96*sqrt(paramEst_survival$covariance[1,1,]),type="s")
lines(times,paramEst_survival$X - 1.96*sqrt(paramEst_survival$covariance[1,1,]),type="s")
lines(seq(0,10,length.out=100),exp(-seq(0,10,length.out=100)),col=2)
legend("topright",c("SDE plugin estimates","Exact"),lty=1,col=c(1,2),bty="n")

plot(
  times,paramEst_survival$covariance,type="s",
  main="SDE plugin variance vs Greenwood variance",ylab="",xlab="time"
)
lines(times,Greenwood,type="s",col=4,lty=1)
legend("topright",c("SDE plugin estimates","Greenwood"),lty=1,col=c(1,4),bty="n")

#################################################################################
############ Competing risks and Cumulative incidence(two states) ###############
#################################################################################

n <- 200
x1 <- rexp(n,1)
x2 <- rexp(n,1.3)
to.states <- ifelse(x1<x2,0,1)

dfr <- data.frame(from=0,to=c(x1[to.states==0],x2[to.states==1]),to.state=to.states)
dfr <- dfr[order(dfr$to),]

nrisk <- c(n,n:1)
dA1 <- c(0,1*(dfr$to.state==0))/nrisk
dA2 <- c(0,1*(dfr$to.state==1))/nrisk

hazMatrix <- rbind(dA1,dA2)

F_fun_cuminc <- function(X)rbind(c(X[2],0),c(-X[2],-X[2]))
JacobianList_cuminc <- list( function(X)matrix(c(0,0,1,-1),nrow=2),
                             function(X)matrix(c(0,0,0,-1),nrow=2) )

X0_cuminc <- matrix(c(0,1),nrow=2,ncol=1)
V0_cuminc <- matrix(0,nrow=2,ncol=2)

paramEst_cuminc <- pluginEstimate(n,hazMatrix,F_fun_cuminc,JacobianList_cuminc,X0_cuminc,V0_cuminc)

times <- c(0,dfr$to)

plot(
  times,paramEst_cuminc$X[1,],type="s",ylab="",xlab="time",
  main="SDE plugin cumulative incidence estimate",ylim=c(0,0.7)
)
lines(times,paramEst_cuminc$X[1,] + 1.96*sqrt(paramEst_cuminc$covariance[1,1,]),type="s")
lines(times,paramEst_cuminc$X[1,] - 1.96*sqrt(paramEst_cuminc$covariance[1,1,]),type="s")
lines(seq(0,10,length.out = 1000),10/1000*cumsum(exp(-seq(0,10,length.out = 1000)*(1 + 1.3))),col=2)
legend("topright",c("SDE plugin estimates","Exact"),lty=1,col=c(1,2),bty="n")


##########################################################################################
####################  Relative survival(two different populations)  ######################
##########################################################################################

n <- 300
t1 <- sort(rexp(n,1))
t2 <- sort(rexp(n,1.3))
times <- sort(c(0,t1,t2))
nrisk <- 300:1
dA1 <- 1*(t1 %in% times)/nrisk
dA2 <- 1*(t2 %in% times)/nrisk

tmatch1 <- match(t1,times)
tmatch2 <- match(t2,times)

hazMatrix <- matrix(0,nrow=2,ncol=length(times))
hazMatrix[1,tmatch1] <- dA1
hazMatrix[2,tmatch2] <- dA2

F_fun_RelSurv <- function(X)matrix(c(-X,X),ncol=2)
JacobianList_RelSurv <- list(function(X)matrix(-1,nrow=1,ncol=1),
                             function(X)matrix(1,nrow=1,ncol=1))

X0_RelSurv <- matrix(1,nrow=1,ncol=1)
V0_RelSurv <- matrix(0,nrow=1,ncol=1)


paramEst_relsurv <- pluginEstimate(
  n,hazMatrix,F_fun_RelSurv,JacobianList_RelSurv,X0_RelSurv,V0_RelSurv
)


plot(
  times,paramEst_relsurv$X[1,],type="s",ylab="",xlab="time",
  main="SDE plugin relative survival estimate",ylim=c(-1,5.8),xlim=c(0,4)
)
lines(times,paramEst_relsurv$X[1,] + 1.96*sqrt(paramEst_relsurv$covariance[1,,]),type="s")
lines(times,paramEst_relsurv$X[1,] - 1.96*sqrt(paramEst_relsurv$covariance[1,,]),type="s")
lines(seq(0,10,length.out = 100),exp(seq(0,10,length.out = 100)*(1.3-1)),col=2)
legend("topleft",c("SDE plugin estimates","Exact"),lty=1,col=c(1,2),bty="n")