Average treatment effect (ATE) for Restricted mean survival and years lost of Competing risks
Klaus Holst & Thomas Scheike
2024-02-20
Source:vignettes/rmst-ate.Rmd
rmst-ate.RmdRMST
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)
Average treatment effect
Computes average treatment effect for restricted mean survival and years lost in competing risks situation
dfactor(bmt) <- tcell~tcell
bmt$event <- (bmt$cause!=0)*1
out <- resmeanATE(Event(time,event)~tcell+platelet,data=bmt,time=40,treat.model=tcell~platelet)
summary(out)
#>
#> n events
#> 408 241
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 2.8637403 0.0756653 2.7154390 3.0120416 0.0000
#> tcell1 0.0185112 0.1981777 -0.3699100 0.4069324 0.9256
#> platelet 0.2753483 0.1452274 -0.0092922 0.5599888 0.0580
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 17.52696 15.11124 20.3289
#> tcell1 1.01868 0.69080 1.5022
#> platelet 1.31699 0.99075 1.7507
#>
#> Average Treatment effects (G-formula) :
#> Estimate Std.Err 2.5% 97.5% P-value
#> treat0 19.26997 1.05138 17.20931 21.33064 0.0000
#> treat1 19.63001 3.43091 12.90554 26.35447 0.0000
#> treat:1-0 0.36003 3.87963 -7.24391 7.96397 0.9261
#>
#> Average Treatment effects (double robust) :
#> Estimate Std.Err 2.5% 97.5% P-value
#> treat0 19.3253 1.0516 17.2642 21.3864 0.0000
#> treat1 21.5622 3.8017 14.1111 29.0133 0.0000
#> treat:1-0 2.2369 4.1991 -5.9932 10.4670 0.5942
out1 <- resmeanATE(Event(time,cause)~tcell+platelet,data=bmt,cause=1,outcome="rmst-cause",
time=40,treat.model=tcell~platelet)
summary(out1)
#>
#> n events
#> 408 157
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 2.807099 0.069703 2.670483 2.943715 0.0000
#> tcell1 -0.376884 0.247936 -0.862829 0.109061 0.1285
#> platelet -0.494384 0.165213 -0.818194 -0.170573 0.0028
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 16.56180 14.44695 18.9862
#> tcell1 0.68600 0.42197 1.1152
#> platelet 0.60995 0.44123 0.8432
#>
#> Average Treatment effects (G-formula) :
#> Estimate Std.Err 2.5% 97.5% P-value
#> treat0 14.53514 0.95673 12.65999 16.41029 0.0000
#> treat1 9.97105 2.37568 5.31479 14.62730 0.0000
#> treat:1-0 -4.56409 2.57161 -9.60437 0.47618 0.0759
#>
#> Average Treatment effects (double robust) :
#> Estimate Std.Err 2.5% 97.5% P-value
#> treat0 14.5202792 0.9577035 12.6432148 16.3973436 0.0000
#> treat1 9.4567098 2.4051143 4.7427723 14.1706473 0.0001
#> treat:1-0 -5.0635694 2.5856565 -10.1313629 0.0042241 0.0502
out2 <- resmeanATE(Event(time,cause)~tcell+platelet,data=bmt,cause=2,outcome="rmst-cause",
time=40,treat.model=tcell~platelet)
summary(out2)
#>
#> n events
#> 408 84
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 1.8287103 0.1312246 1.5715149 2.0859057 0.0000
#> tcell1 0.4681132 0.2432252 -0.0085996 0.9448259 0.0543
#> platelet -0.0109542 0.2178062 -0.4378464 0.4159380 0.9599
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 6.22585 4.81394 8.0519
#> tcell1 1.59698 0.99144 2.5724
#> platelet 0.98911 0.64542 1.5158
#>
#> Average Treatment effects (G-formula) :
#> Estimate Std.Err 2.5% 97.5% P-value
#> treat0 6.20457 0.71391 4.80534 7.60381 0.0000
#> treat1 9.90857 2.10922 5.77458 14.04256 0.0000
#> treat:1-0 3.70399 2.23512 -0.67676 8.08474 0.0975
#>
#> Average Treatment effects (double robust) :
#> Estimate Std.Err 2.5% 97.5% P-value
#> treat0 6.21823 0.71423 4.81836 7.61809 0.0000
#> treat1 10.38475 2.22282 6.02810 14.74140 0.0000
#> treat:1-0 4.16652 2.33621 -0.41237 8.74542 0.0745SessionInfo
sessionInfo()
#> R version 4.3.2 (2023-10-31)
#> Platform: x86_64-pc-linux-gnu (64-bit)
#> Running under: Ubuntu 22.04.3 LTS
#>
#> Matrix products: default
#> BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so; LAPACK version 3.10.0
#>
#> locale:
#> [1] LC_CTYPE=C.UTF-8 LC_NUMERIC=C LC_TIME=C.UTF-8
#> [4] LC_COLLATE=C.UTF-8 LC_MONETARY=C.UTF-8 LC_MESSAGES=C.UTF-8
#> [7] LC_PAPER=C.UTF-8 LC_NAME=C LC_ADDRESS=C
#> [10] LC_TELEPHONE=C LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C
#>
#> time zone: UTC
#> tzcode source: system (glibc)
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] mets_1.3.4 timereg_2.0.5 survival_3.5-7
#>
#> loaded via a namespace (and not attached):
#> [1] Matrix_1.6-1.1 future.apply_1.11.1 jsonlite_1.8.8
#> [4] highr_0.10 compiler_4.3.2 Rcpp_1.0.12
#> [7] stringr_1.5.1 parallel_4.3.2 jquerylib_0.1.4
#> [10] globals_0.16.2 splines_4.3.2 systemfonts_1.0.5
#> [13] textshaping_0.3.7 yaml_2.3.8 fastmap_1.1.1
#> [16] lattice_0.21-9 R6_2.5.1 knitr_1.45
#> [19] future_1.33.1 desc_1.4.3 bslib_0.6.1
#> [22] rlang_1.1.3 cachem_1.0.8 stringi_1.8.3
#> [25] xfun_0.42 fs_1.6.3 sass_0.4.8
#> [28] memoise_2.0.1 cli_3.6.2 pkgdown_2.0.7
#> [31] magrittr_2.0.3 digest_0.6.34 grid_4.3.2
#> [34] mvtnorm_1.2-4 lifecycle_1.0.4 lava_1.7.3
#> [37] vctrs_0.6.5 evaluate_0.23 glue_1.7.0
#> [40] listenv_0.9.1 numDeriv_2016.8-1.1 codetools_0.2-19
#> [43] ragg_1.2.7 parallelly_1.37.0 rmarkdown_2.25
#> [46] purrr_1.0.2 tools_4.3.2 htmltools_0.5.7