The binreg function can fit a logistic link model with IPCW adjustment for a specific time-point, and can thus be used for describing survival or competing risks data. The function can be used for large data and is completely scalable, that is, linear in the data. A nice feature is that influcence functions are computed and are available, and can thus be used for all other settings based on these parameters.
In addition and to summarize
The binreg function does direct binomial regression for one time-point, \(t\), fitting the model \[\begin{align*} P(T \leq t, \epsilon=1 | X ) & = \mbox{expit}( X^T \beta) = F_1(t,X,\beta) \end{align*}\] to an IPCW adjusted estmating equation (EE) with response \(Y(t)=I(T \leq t, \epsilon=1 )\) \[\begin{align*} U(\beta,\hat G_c) = & X ( Y(t) \frac{ \Delta(t) }{\hat G_c(T_i \wedge t)} - \mbox{expit}( X^T \beta)) = 0, \end{align*}\] with \(G_c(t)=P(C>t)\), the censoring survival distribution, and with \(\Delta(t) = I( C_i > T_i \wedge t)\) the indicator of being uncensored at time \(t\) (type=“I”). The default type=“II” is to augment with a censoring term, that is solve \[\begin{align*} & U(\beta,\hat G_c) + \int_0^t X \frac{\hat E(Y(t)| T>u)}{\hat G_c(u)} d\hat M_c(u) =0 \end{align*}\] where \(M_c(u)\) is the censoring martingale, this typically improves the performance. This is equivlent to the pseudo-value approach (see Overgaard (2025)).
The influence function for the type=“II” estimator is \[\begin{align*} U(\beta,G_c) + \int_0^t X \frac{E(Y| T>u)}{G_c(u)} d M_c(u) - \int_0^t \frac{E(X| T>u) E(Y| T>u)}{G_c(u)} d M_c(u) - \int_0^t \frac{E( X Y| T>u)}{G_c(u)} d M_c(u) \end{align*}\] and for type=“I” \[\begin{align*} & U(\beta) + \int_0^t \frac{E( X Y| T>u)}{G_c(u)} d M_c(u). \end{align*}\] The means \(E(X Y(t) | T>u)\) and \(E(Y(t)| T>u)\) are estimated by IPCW estimators among survivors to get estimates of the influence functions.
The function logitIPCW instead considers \[\begin{align*} U^{glm}(\beta,\hat G_c) = & \frac{ \Delta(t) }{\hat G_c(T_i \wedge t)} X ( Y(t) - \mbox{expit}( X^T \beta)) = 0. \end{align*}\] This score equation is quite similar to those of the binreg, and exactly the same when the censoring model is fully-nonparametric.
The logitIPCW has influence function \[\begin{align*} & U^{glm}(\beta,G_c) + \int_0^t \frac{E( X ( Y - F_1(t,\beta)) | T>u)}{G_c(u)} d M_c(u) \end{align*}\]
Which estimator performs the best depends on the censoring distribution and it seems that the binreg with type=“II” performs overall quite nicely (see Blanche et al (2023) and Overgaard (2024)). For the full estimated censoring model all estimators have the same influence function (see Blanche et al (2023)).
Additional functions logitATE, and binregATE computes the average treatment effect. We demonstrate this in another vignette.
The functions logitATE/binregATE can be used there is no censoring and we thus have simple binary outcome.
The variance is based on sandwich formula with IPCW adjustment (using the influence functions), and naive.var is the variance under known censoring model. The influence functions are stored in the output. Clusters can be specified to get cluster corrected standard errors.
library(mets)
options(warn=-1)
set.seed(1000) # to control output in random noise just below.
data(bmt)
bmt$time <- bmt$time+runif(nrow(bmt))*0.01
# logistic regresion with IPCW binomial regression
out <- binreg(Event(time,cause)~tcell+platelet,bmt,time=50)
summary(out)
#> n events
#> 408 160
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) -0.180338 0.126748 -0.428760 0.068084 0.1548
#> tcell -0.418545 0.345480 -1.095675 0.258584 0.2257
#> platelet -0.437644 0.240978 -0.909952 0.034665 0.0694
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 0.83499 0.65132 1.0705
#> tcell 0.65800 0.33431 1.2951
#> platelet 0.64556 0.40254 1.0353We can also compute predictions using the estimates
predict(out,data.frame(tcell=c(0,1),platelet=c(1,1)),se=TRUE)
#> pred se lower upper
#> 1 0.3502406 0.04847582 0.2552280 0.4452533
#> 2 0.2618207 0.06969334 0.1252217 0.3984196Further the censoring model can depend on strata
outs <- binreg(Event(time,cause)~tcell+platelet,bmt,time=50,cens.model=~strata(tcell,platelet))
summary(outs)
#> n events
#> 408 160
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) -0.180697 0.127414 -0.430424 0.069030 0.1561
#> tcell -0.365928 0.350632 -1.053154 0.321299 0.2967
#> platelet -0.433494 0.240270 -0.904415 0.037428 0.0712
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 0.83469 0.65023 1.0715
#> tcell 0.69355 0.34884 1.3789
#> platelet 0.64824 0.40478 1.0381Now for illustrations I wish to consider the absolute risk difference depending on tcell
outs <- binreg(Event(time,cause)~tcell,bmt,time=50,cens.model=~strata(tcell))
summary(outs)
#> n events
#> 408 160
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) -0.30054 0.11153 -0.51914 -0.08194 0.0070
#> tcell -0.51741 0.33981 -1.18342 0.14860 0.1278
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 0.74042 0.59503 0.9213
#> tcell 0.59606 0.30623 1.1602the risk difference is
ps <- predict(outs,data.frame(tcell=c(0,1)),se=TRUE)
ps
#> pred se lower upper
#> 1 0.4254253 0.02726306 0.3719897 0.4788609
#> 2 0.3061988 0.06819019 0.1725461 0.4398516
sum( c(1,-1) * ps[,1])
#> [1] 0.1192264Getting the standard errors are easy enough since the two-groups are independent. In the case where we in addition had adjusted for other covariates, however, we would need the to apply the delta-theorem thus using the relevant covariances along the lines of
dd <- data.frame(tcell=c(0,1))
p <- predict(outs,dd)
riskdifratio <- function(p,contrast=c(1,-1)) {
outs$coef <- p
p <- predict(outs,dd)[,1]
pd <- sum(contrast*p)
r1 <- p[1]/p[2]
r2 <- p[2]/p[1]
return(c(pd,r1,r2))
}
estimate(outs,f=riskdifratio,dd,null=c(0,1,1))
#> Estimate Std.Err 2.5% 97.5% P-value
#> [p1] 0.1192 0.07344 -0.02471 0.2632 0.10448
#> [p2] 1.3894 0.32197 0.75833 2.0204 0.22652
#> [p3] 0.7197 0.16679 0.39284 1.0467 0.09291
#>
#> Null Hypothesis:
#> [p1] = 0
#> [p2] = 1
#> [p3] = 1
#>
#> chisq = 12.0249, df = 3, p-value = 0.007298same as
Rather than using a larger censoring model we can also compute an augmentation term and then fit the binomial regression model based on this augmentation term. Here we compute the augmentation based on stratified non-parametric estimates of \(F_1(t,S(X))\), where \(S(X)\) gives strata based on \(X\) as a working model.
Computes the augmentation term for each individual as well as the sum \[\begin{align*} A & = \int_0^t H(u,X) \frac{1}{S^*(u,s)} \frac{1}{G_c(u)} dM_c(u) \end{align*}\] with \[\begin{align*} H(u,X) & = F_1^*(t,S(X)) - F_1^*(u,S(X)) \end{align*}\] using a KM for \(G_c(t)\) and a working model for cumulative baseline related to \(F_1^*(t,s)\) and \(s\) is strata, \(S^*(t,s) = 1 - F_1^*(t,s) - F_2^*(t,s)\).
Standard errors computed under assumption of correct but estimated \(G_c(s)\) model.
data(bmt)
dcut(bmt,breaks=2) <- ~age
out1<-BinAugmentCifstrata(Event(time,cause)~platelet+agecat.2+
strata(platelet,agecat.2),data=bmt,cause=1,time=40)
summary(out1)
#> n events
#> 408 157
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) -0.51295 0.17090 -0.84791 -0.17799 0.0027
#> platelet -0.63011 0.23585 -1.09237 -0.16785 0.0075
#> agecat.2(0.203,1.94] 0.55926 0.21211 0.14353 0.97500 0.0084
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 0.59873 0.42831 0.8370
#> platelet 0.53253 0.33542 0.8455
#> agecat.2(0.203,1.94] 1.74938 1.15434 2.6512
out2<-BinAugmentCifstrata(Event(time,cause)~platelet+agecat.2+
strata(platelet,agecat.2)+strataC(platelet),data=bmt,cause=1,time=40)
summary(out2)
#> n events
#> 408 157
#>
#> 408 clusters
#> coeffients:
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) -0.51346 0.17109 -0.84879 -0.17814 0.0027
#> platelet -0.63636 0.23653 -1.09996 -0.17276 0.0071
#> agecat.2(0.203,1.94] 0.56280 0.21229 0.14672 0.97889 0.0080
#>
#> exp(coeffients):
#> Estimate 2.5% 97.5%
#> (Intercept) 0.59842 0.42793 0.8368
#> platelet 0.52922 0.33288 0.8413
#> agecat.2(0.203,1.94] 1.75559 1.15803 2.6615sessionInfo()
#> R version 4.5.1 (2025-06-13)
#> Platform: aarch64-apple-darwin24.5.0
#> Running under: macOS Sequoia 15.6.1
#>
#> Matrix products: default
#> BLAS: /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRblas.dylib
#> LAPACK: /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRlapack.dylib; LAPACK version 3.12.1
#>
#> 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] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] timereg_2.0.7 survival_3.8-3 mets_1.3.7
#>
#> loaded via a namespace (and not attached):
#> [1] cli_3.6.5 knitr_1.50 rlang_1.1.6
#> [4] xfun_0.53 jsonlite_2.0.0 future.apply_1.20.0
#> [7] listenv_0.9.1 lava_1.8.1 htmltools_0.5.8.1
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#> [19] lifecycle_1.0.4 compiler_4.5.1 codetools_0.2-20
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#> [25] lattice_0.22-7 digest_0.6.37 R6_2.6.1
#> [28] parallelly_1.45.1 parallel_4.5.1 splines_4.5.1
#> [31] Matrix_1.7-4 bslib_0.9.0 tools_4.5.1
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