# 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) # 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  0.0745

# SessionInfo

sessionInfo()
#> R version 4.3.2 (2023-10-31)
#> Platform: aarch64-apple-darwin22.6.0 (64-bit)
#> Running under: macOS Ventura 13.6.2
#>
#> Matrix products: default
#> BLAS:   /Users/kkzh/.asdf/installs/R/4.3.2/lib/R/lib/libRblas.dylib
#> LAPACK: /Users/kkzh/.asdf/installs/R/4.3.2/lib/R/lib/libRlapack.dylib;  LAPACK version 3.11.0
#>
#> locale:
#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#>
#> time zone: Europe/Copenhagen
#> tzcode source: internal
#>
#> attached base packages:
#> [1] splines   stats     graphics  grDevices utils     datasets  methods
#> [8] base
#>
#> other attached packages:
#> [1] ggplot2_3.4.4  cowplot_1.1.1  mets_1.3.3     timereg_2.0.5  survival_3.5-7
#>
#> loaded via a namespace (and not attached):
#>  [1] sass_0.4.7          utf8_1.2.4          future_1.33.0
#>  [4] generics_0.1.3      lattice_0.22-5      listenv_0.9.0
#>  [7] digest_0.6.33       magrittr_2.0.3      evaluate_0.23
#> [10] grid_4.3.2          mvtnorm_1.2-3       fastmap_1.1.1
#> [13] jsonlite_1.8.7      Matrix_1.6-3        fansi_1.0.5
#> [16] scales_1.2.1        isoband_0.2.7       codetools_0.2-19
#> [19] numDeriv_2016.8-1.1 jquerylib_0.1.4     lava_1.7.4
#> [22] cli_3.6.1           rlang_1.1.1         parallelly_1.36.0
#> [25] future.apply_1.11.0 munsell_0.5.0       withr_2.5.2
#> [28] cachem_1.0.8        yaml_2.3.7          tools_4.3.2
#> [31] parallel_4.3.2      ucminf_1.2.0        dplyr_1.1.3
#> [34] colorspace_2.1-0    globals_0.16.2      vctrs_0.6.4
#> [37] R6_2.5.1            lifecycle_1.0.3     MASS_7.3-60
#> [40] pkgconfig_2.0.3     bslib_0.5.1         pillar_1.9.0
#> [43] gtable_0.3.4        glue_1.6.2          Rcpp_1.0.11.6
#> [46] tidyselect_1.2.0    xfun_0.41           tibble_3.2.1
#> [49] highr_0.10          knitr_1.45          farver_2.1.1
#> [52] htmltools_0.5.6.1   labeling_0.4.3      rmarkdown_2.25
#> [55] compiler_4.3.2