# Binomial Regression for Survival and Competing Risks Data

#### Klaus Holst & Thomas Scheike

#### 2025-10-21

# Binomial Regression for censored data

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 censoring weights can be strata dependent
- predictions can be computed with standard errors
- computation time linear in data
  - including standard errors
- clusters can be given and then cluster corrected standard errors are computed

# Details

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.

# Examples

     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 1.0353

We 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 0.3984196

Further 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 1.0381

# Absolute risk differences and ratio

Now 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 1.1602

the 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] 0.1192264

Getting 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 = 0.007298

same as

    run <- 0
    if (run==1) {
    library(prodlim)
    pl <- prodlim(Hist(time,cause)~tcell,bmt)
    spl <- summary(pl,times=50,asMatrix=TRUE)
    spl
    }

# Augmenting the Binomial Regression

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 2.6615

# SessionInfo

    sessionInfo()
    #> R version 4.5.1 (2025-06-13)
    #> Platform: aarch64-apple-darwin25.0.0
    #> Running under: macOS Tahoe 26.0.1
    #>
    #> 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_1.3.8
    #>
    #> 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
    #> [10] sass_0.4.10         rmarkdown_2.30      grid_4.5.1
    #> [13] evaluate_1.0.5      jquerylib_0.1.4     fastmap_1.2.0
    #> [16] numDeriv_2016.8-1.1 yaml_2.3.10         mvtnorm_1.3-3
    #> [19] lifecycle_1.0.4     compiler_4.5.1      codetools_0.2-20
    #> [22] ucminf_1.2.2        Rcpp_1.1.0          future_1.67.0
    #> [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
    #> [34] globals_0.18.0      cachem_1.1.0