RMST
Regression for rmst outcome \(E(T \wedge t | X) = exp(X^T \beta)\) based on IPCW adjustment for censoring, thus solving the estimating equation \[\begin{align*} & X^T [ (T \wedge t) \frac{I(C > T \wedge t)}{G_C(T \wedge t,X)} - exp(X^T \beta) ] = 0 . \end{align*}\] This is done with the resmeanIPCW function. For fully saturated model with full censoring model this is equal to the integrals of the Kaplan-Meier estimators as illustrated below.
We can also compute the integral of the Kaplan-Meier or Cox based survival estimator to get the RMST (with the resmean.phreg function) \[ \int_0^t S(s|X) ds \].
For competing risks the years lost can be computed via cumulative
incidence functions (cif.yearslost) or as IPCW estimator since
\[ E( I(\epsilon=1) ( t - T \wedge t ) | X) =
\int_0^t F_1(s) ds. \] For fully saturated model with full
censoring model these estimators are equivalent as illustrated
below.
set.seed(100)
data(bmt); bmt$time <- bmt$time+runif(nrow(bmt))*0.001
# E( min(T;t) | X ) = exp( a+b X) with IPCW estimation
out <- resmeanIPCW(Event(time,cause!=0)~tcell+platelet+age,bmt,
time=50,cens.model=~strata(platelet),model="exp")
summary(out)
#>
#> n events
#> 408 245
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 3.014348 0.063466 2.889956 3.138740 0.0000
#> tcell 0.106223 0.142443 -0.172960 0.385406 0.4558
#> platelet 0.246994 0.098833 0.053285 0.440704 0.0125
#> age -0.185946 0.043533 -0.271270 -0.100622 0.0000
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 20.37580 17.99252 23.0748
#> tcell 1.11207 0.84117 1.4702
#> platelet 1.28017 1.05473 1.5538
#> age 0.83032 0.76241 0.9043
### same as Kaplan-Meier for full censoring model
bmt$int <- with(bmt,strata(tcell,platelet))
out <- resmeanIPCW(Event(time,cause!=0)~-1+int,bmt,time=30,
cens.model=~strata(platelet,tcell),model="lin")
estimate(out)
#> Estimate Std.Err 2.5% 97.5% P-value
#> inttcell=0, platelet=0 13.60 0.8316 11.97 15.23 3.826e-60
#> inttcell=0, platelet=1 18.90 1.2696 16.41 21.39 3.999e-50
#> inttcell=1, platelet=0 16.19 2.4061 11.48 20.91 1.705e-11
#> inttcell=1, platelet=1 17.77 2.4536 12.96 22.58 4.461e-13
out1 <- phreg(Surv(time,cause!=0)~strata(tcell,platelet),data=bmt)
rm1 <- resmean.phreg(out1,times=30)
summary(rm1)
#> strata times rmean se.rmean years.lost
#> tcell=0, platelet=0 0 30 13.60294 0.8315422 16.39706
#> tcell=0, platelet=1 1 30 18.90129 1.2693293 11.09871
#> tcell=1, platelet=0 2 30 16.19119 2.4006192 13.80881
#> tcell=1, platelet=1 3 30 17.76617 2.4421866 12.23383
## competing risks years-lost for cause 1
out <- resmeanIPCW(Event(time,cause)~-1+int,bmt,time=30,cause=1,
cens.model=~strata(platelet,tcell),model="lin")
estimate(out)
#> Estimate Std.Err 2.5% 97.5% P-value
#> inttcell=0, platelet=0 12.105 0.8508 10.438 13.773 6.162e-46
#> inttcell=0, platelet=1 6.884 1.1741 4.583 9.185 4.536e-09
#> inttcell=1, platelet=0 7.261 2.3533 2.648 11.873 2.033e-03
#> inttcell=1, platelet=1 5.780 2.0925 1.679 9.882 5.737e-03
## same as integrated cumulative incidence
rmc1 <- cif.yearslost(Event(time,cause)~strata(tcell,platelet),data=bmt,times=30,cause=1)
summary(rmc1)
#> strata times intF11 intF12 se.intF11 se.intF12
#> tcell=0, platelet=0 0 30 12.105127 4.291933 0.8508102 0.6161447
#> tcell=0, platelet=1 1 30 6.884185 4.214527 1.1741034 0.9056995
#> tcell=1, platelet=0 2 30 7.260772 6.548042 2.3532912 1.9703328
#> tcell=1, platelet=1 3 30 5.780360 6.453473 2.0924927 2.0815028
#> total.years.lost
#> tcell=0, platelet=0 16.39706
#> tcell=0, platelet=1 11.09871
#> tcell=1, platelet=0 13.80881
#> tcell=1, platelet=1 12.23383
## plotting the years lost for different horizon's and the two causes
par(mfrow=c(1,3))
plot(rm1,years.lost=TRUE,se=1)
## cause refers to column of cumhaz for the different causes
plot(rmc1,cause=1,se=1)
plot(rmc1,cause=2,se=1)