get_h_xll: Local linear future conditional hazard rate estimator

Description

Calculates the local linear future hazard rate function, conditional on a marker value x, across a set of time values t.

Usage

get_h_xll(data, marker_name, event_time_name, time_name, event_name, x, b)

Value

A vector of $\hat h_x(t)$ for a grid of possible time values $t$.

Arguments

data: A data frame of time dependent data points. Missing values are allowed.
marker_name: The column name of the marker values in the data frame data.
event_time_name: The column name of the event times in the data frame data.
time_name: The column name of the times the marker values were observed in the data frame data.
event_name: The column name of the events in the data frame data.
x: Numeric value of the last observed marker value.
b: Bandwidth parameter.

Details

The function get_h_xll implements the local linear future conditional hazard estimator $$\hat{h}_x(t) = \frac{\sum_{i=1}^n \int_0^T\hat{\alpha}_i(X_i(t+s))Z_i(t+s)Z_i(s)K_{b}(x-X_i(s))\mathrm {d}s}{\sum_{i=1}^n\int_0^TZ_i(t+s)Z_i(s)K_{b}(x-X_i(s))\mathrm {d}s},$$ across a grid of possible time values $t$, where $X$ is the marker, $Z$ is the exposure and $\alpha(z)$ is the marker-only hazard, see get_alpha for more details.

Examples

Run this code

library(survival)
library(JM)

# Compare Local constant and local linear estimator, use KM for reference
# Albumin marker, use landmarking
Landmark <- 2
pbcT1 <- pbc2[which(pbc2$year< Landmark  & pbc2$years> Landmark),]
b=0.9

arg1ll<-get_h_xll(pbcT1, 'albumin', event_time_name = 'years', time_name = 'year',
                  event_name = 'status2', 2, 0.9) 
arg1lc<-get_h_x(pbcT1, 'albumin', event_time_name = 'years', time_name = 'year',
                event_name = 'status2', 2, 0.9) 

#Caclulate the local contant and local linear survival functions
br_s  = seq(Landmark, 14,  length=99)
sfalb2ll<- make_sf(    (br_s[2]-br_s[1])/4 , arg1ll)
sfalb2lc<- make_sf(    (br_s[2]-br_s[1])/4 , arg1lc)

#For comparison, also calculate the Kaplan-Meier
kma2<- survfit(Surv(years , status2) ~ 1, data = pbcT1)

#Plot the survival functions:
plot(br_s, sfalb2ll,  type="l", col=1, lwd=2, ylab="Survival probability", xlab="Marker level")
lines(br_s, sfalb2lc,  lty=2, lwd=2, col=2)
lines(kma2$time, kma2$surv, type="s",  lty=2, lwd=2, col=3)

legend("topright", c(  "Local linear HQM", "Local constant HQM", "Kaplan-Meier"), 
        lty=c(1, 2, 2), col=1:3, lwd=2, cex=1.7)
        

if (FALSE) {       
#Compare JM, HQM and KM for Bilirubin       
b = 10 
Landmark <- 1
lmeFit <- lme(serBilir ~ year, random = ~ year | id, data = pbc2)
coxFit <- coxph(Surv(years, status2) ~ serBilir, data = pbc2.id, x = TRUE)

 
jointFit0 <- jointModel(lmeFit, coxFit, timeVar = "year", 
                                method = "piecewise-PH-aGH")
pbcT1 <- pbc2[which(pbc2$year< Landmark  & pbc2$years> Landmark),]
 

timesS1 <- seq(1,14,by=0.5)
predT1 <- survfitJM(jointFit0, newdata = pbcT1,survTimes = timesS1)
nm<-length(predT1$summaries)

mat.out1<-matrix(nrow=length(timesS1), ncol=nm)
for(r in 1:nm)
{
  SurvLand <- predT1$summaries[[r]][,"Mean"][1]
  mat.out1[,r] <- predT1$summaries[[r]][,"Mean"]/SurvLand
}
sfit1y<-rowMeans(mat.out1, na.rm=TRUE)


arg1<- get_h_x(pbcT1, 'serBilir', event_time_name = 'years',  
        time_name = 'year', event_name = 'status2', 1, 10) 
br_s1  = seq(Landmark, 14,  length=99)
sfbil1<- make_sf(  (br_s1[2]-br_s1[1])/5.4 , arg1)
kma1<- survfit(Surv(years , status2) ~ 1, data = pbcT1)

plot(br_s1, sfbil1, type="l", ylim=c(0,1), xlim=c(Landmark,14), 
                    ylab="Survival probability", xlab="years",lwd=2)
lines(timesS1, sfit1y, col=2,  lwd=2, lty=2)
lines(kma1$time, kma1$surv, type="s", lty=2, lwd=2, col=3 )
legend("bottomleft", c("HQM est.", "Joint Model est.", "Kaplan-Meier"), 
                                    lty=c(1,2,2), col=1:3, lwd=2, cex=1.7)
}