# IPCW Cumulative Cost

#### Klaus Holst & Thomas Scheike

#### 2026-05-23

# Overview

We describe how to perform regression modelling for cumulative cost \[\begin{align*} {\cal U}(t) & = \int_0^t Z(s) dN(s) \end{align*}\] where \(N(s)\) is a counting process that registers the times at which costs are realised and accumulated, and \(Z(t)\) is the cost (or mark) at the event times. The counting process can be a mix of random and fixed times, and the data are represented in counting process format with the marks/costs attached to the event times. There are many additional uses of such cumulative processes; for example, when considering time lost in a recurrent events setting, which we return to below.

We can estimate the marginal mean of the cumulative process \[\begin{align*} \nu(t) & = E ( {\cal U}(t) ) \end{align*}\] possibly for strata with standard errors based on the derived influence function.

We provide semi-parametric regression modelling using the proportional model \[\begin{align*} E ( {\cal U}(t) | X) & = \Lambda_0(t) \exp( X^T \beta). \end{align*}\]

In addition for a fixed time-point \(t \in [0,\tau]\) we can estimate the mean given covariates \[\begin{align*} E ( {\cal U}(t) | X) & = \exp( X^T \beta) \end{align*}\] where \(\tau\) is some maximum follow-up time.

- These quantities are estimated under independent right-censoring given \(X\), using IPCW-adjusted estimating equations,
  - similarly to the Ghosh-Lin model for recurrent events.
- A terminal event can be specified.

We also estimate the probability of exceeding thresholds over time \[\begin{align*} P ( {\cal U}(t) > k ) & = \mu_k(t), \end{align*}\] and in the setting with a terminal event this is based on a derived competing risks data structure that keeps track of the competing terminal event.

Regression modelling of this quantity is also possible using competing risks regression models, for example via the `cifreg` function in mets.

# HF-action data

Using the HF-action data, we simulate a severity score for each event.

    library(mets)
    data(hfactioncpx12)
    hf <- hfactioncpx12
    hf$severity <- abs((5+rnorm(741)*2))[hf$id]

    ## marginal mean using formula
    outNZ <- recurrent_marginal(Event(entry,time,status)~strata(treatment)+cluster(id)
                 +marks(severity),hf,cause=1,death.code=2)
    plot(outNZ,se=TRUE)
    summary(outNZ,times=3)
    #> [[1]]
    #>     new.time     mean        se  CI-2.5% CI-97.5% strata
    #> 682        3 11.11151 0.6910881 9.836302 12.55203      0
    #>
    #> [[2]]
    #>     new.time     mean        se  CI-2.5% CI-97.5% strata
    #> 601        3 9.874025 0.6937881 8.603704 11.33191      1

    outN <- recurrent_marginal(Event(entry,time,status)~strata(treatment)+cluster(id),data=hf,
                   cause=1,death.code=2)
    plot(outN,se=TRUE,add=TRUE)

    summary(outN,times=3)
    #> [[1]]
    #>     new.time     mean        se  CI-2.5% CI-97.5% strata
    #> 682        3 2.118496 0.1138572 1.906692 2.353829      0
    #>
    #> [[2]]
    #>     new.time     mean        se  CI-2.5% CI-97.5% strata
    #> 601        3 1.924062 0.1216577 1.699801 2.177912      1

For comparison we also compute the IPCW estimates with and without marks at time 3 using the linear model, and note that they are identical. Standard errors are based on different formulae that are asymptotically equivalent, and we note that they are very similar.

    outNZ3 <- recregIPCW(Event(entry,time,status)~-1+treatment+cluster(id)+marks(severity),data=hf,
              cause=1,death.code=2,time=3,cens.model=~strata(treatment),model="lin")
    summary(outNZ3)
    #>    n events
    #>  741   1281
    #>
    #>  741 clusters
    #> coeffients:
    #>            Estimate  Std.Err     2.5%    97.5% P-value
    #> treatment0 11.11151  0.69107  9.75703 12.46598       0
    #> treatment1  9.87403  0.69372  8.51436 11.23369       0
    head(iid(outNZ3))
    #>            [,1]       [,2]
    #> 1  0.0124490201 0.00000000
    #> 2  0.0221674074 0.00000000
    #> 3  0.0000000000 0.02380208
    #> 4 -0.0223323887 0.00000000
    #> 5 -0.0006315006 0.00000000
    #> 6 -0.0371624689 0.00000000

    outN3 <- recregIPCW(Event(entry,time,status)~-1+treatment+cluster(id),data=hf,cause=1,death.code=2,time=3,
             cens.model=~strata(treatment),model="lin")
    summary(outN3)
    #>    n events
    #>  741   1281
    #>
    #>  741 clusters
    #> coeffients:
    #>            Estimate Std.Err    2.5%   97.5% P-value
    #> treatment0  2.11850 0.11385 1.89535 2.34164       0
    #> treatment1  1.92406 0.12165 1.68564 2.16248       0
    head(iid(outN3))
    #>            [,1]        [,2]
    #> 1  0.0004542472 0.000000000
    #> 2  0.0009756994 0.000000000
    #> 3  0.0000000000 0.009301496
    #> 4 -0.0029668336 0.000000000
    #> 5 -0.0001120764 0.000000000
    #> 6 -0.0070693971 0.000000000

We also apply the semiparametric proportional cost model with IPCW adjustment:

    propNZ <- recreg(Event(entry,time,status)~treatment+marks(severity)+cluster(id),data=hf,cause=1,death.code=2)
    summary(propNZ)
    #>
    #>     n events
    #>  2132   1391
    #>
    #>  741 clusters
    #> coefficients:
    #>             Estimate      S.E.   dU^-1/2 P-value
    #> treatment1 -0.138380  0.088834  0.023629  0.1193
    #>
    #> exp(coefficients):
    #>            Estimate    2.5%  97.5%
    #> treatment1  0.87077 0.73162 1.0364
    plot(propNZ,main="Baselines")

    GL <- recreg(Event(entry,time,status)~treatment+cluster(id),hf,cause=1,death.code=2)
    summary(GL)
    #>
    #>     n events
    #>  2132   1391
    #>
    #>  741 clusters
    #> coefficients:
    #>             Estimate      S.E.   dU^-1/2 P-value
    #> treatment1 -0.110404  0.078656  0.053776  0.1604
    #>
    #> exp(coefficients):
    #>            Estimate    2.5%  97.5%
    #> treatment1  0.89547 0.76754 1.0447
    plot(GL,add=TRUE,col=2)

Those treated have 14% lower cumulative severity and 11% lower expected number of events.

# Exceed threshold

Finally, we estimate the probability of exceeding cumulative severity thresholds of 1, 5, and 10:

    ooNZ <- prob_exceed_recurrent(Event(entry,time,status)~strata(treatment)+cluster(id)+marks(severity),data=hf,
                      cause=1,death.code=2,exceed=c(1,5,10,20))
    plot(ooNZ,strata=1)
    plot(ooNZ,strata=2,add=TRUE)

    summary(ooNZ,times=3)
    #> $`0`
    #> $`0`$prob
    #>            times
    #>                3 2.99865085
    #> N<1            3 0.04526295
    #> exceed>=1      3 0.95473705
    #> exceed>=5      3 0.91747371
    #> exceed>=10     3 0.80652629
    #> exceed>=20     3 0.54711578
    #>
    #> $`0`$se
    #>            times
    #>                3 2.99865085
    #> N<1            3 0.01846437
    #> exceed>=1      3 0.01846437
    #> exceed>=5      3 0.02337160
    #> exceed>=10     3 0.03591698
    #> exceed>=20     3 0.04986960
    #>
    #> $`0`$lower
    #>      times
    #> [1,]     3 2.99865085
    #> [2,]     3 0.08077514
    #> [3,]     3 0.91922486
    #> [4,]     3 0.87279095
    #> [5,]     3 0.73911503
    #> [6,]     3 0.45760655
    #>
    #> $`0`$upper
    #>      times
    #> [1,]     3 2.998650853
    #> [2,]     3 0.008378819
    #> [3,]     3 0.991621181
    #> [4,]     3 0.964444024
    #> [5,]     3 0.880085818
    #> [6,]     3 0.654133291
    #>
    #>
    #> $`1`
    #> $`1`$prob
    #>            times
    #>                3 2.99865085
    #> N<1            3 0.01766525
    #> exceed>=1      3 0.98233475
    #> exceed>=5      3 0.96103565
    #> exceed>=10     3 0.81294277
    #> exceed>=20     3 0.50657726
    #>
    #> $`1`$se
    #>            times
    #>                3 2.99865085
    #> N<1            3 0.01040641
    #> exceed>=1      3 0.01040641
    #> exceed>=5      3 0.01434812
    #> exceed>=10     3 0.03730653
    #> exceed>=20     3 0.05792904
    #>
    #> $`1`$lower
    #>      times
    #> [1,]     3 2.99865085
    #> [2,]     3 0.03785115
    #> [3,]     3 0.96214885
    #> [4,]     3 0.93332132
    #> [5,]     3 0.74301525
    #> [6,]     3 0.40486250
    #>
    #> $`1`$upper
    #>      times
    #> [1,]     3 2.9986509
    #> [2,]     3 0.0000000
    #> [3,]     3 1.0000000
    #> [4,]     3 0.9895729
    #> [5,]     3 0.8894514
    #> [6,]     3 0.6338461

# Cumulative time lost for recurrent events

The cumulative time lost for recurrent events is defined as \[\begin{align*} {\cal M}(t) = E\left[ \int_0^\tau (\tau-s) dN(s) \right] = \int_0^\tau \mu(s) ds \end{align*}\] where \(\mu(t) = E( N(t) )\) is the marginal mean of the recurrent events at time \(t\).

    hf$lost5 <- 5-hf$time

    RecLost <- recregIPCW(Event(entry,time,status)~-1+treatment+cluster(id)+marks(lost5),data=hf,
               cause=1,death.code=2,time=5,cens.model=~strata(treatment),model="lin")
    summary(RecLost)
    #>    n events
    #>  741   1391
    #>
    #>  741 clusters
    #> coeffients:
    #>            Estimate Std.Err    2.5%   97.5% P-value
    #> treatment0  8.58300 0.42951 7.74118 9.42482       0
    #> treatment1  7.66234 0.46400 6.75292 8.57177       0
    head(iid(RecLost))
    #>            [,1]       [,2]
    #> 1  0.0016920221 0.00000000
    #> 2  0.0073388996 0.00000000
    #> 3  0.0000000000 0.02120478
    #> 4 -0.0095548150 0.00000000
    #> 5 -0.0005696809 0.00000000
    #> 6 -0.0201750011 0.00000000

# SessionInfo

    sessionInfo()
    #> R version 4.6.0 (2026-04-24)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /home/kkzh/.asdf/installs/r/4.6.0/lib/R/lib/libRblas.so
    #> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0  LAPACK version 3.12.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> time zone: Europe/Copenhagen
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] timereg_2.0.7  survival_3.8-6 mets_1.3.10
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] cli_3.6.6              knitr_1.51             rlang_1.2.0
    #>  [4] xfun_0.57              otel_0.2.0             jsonlite_2.0.0
    #>  [7] listenv_0.10.1         future.apply_1.20.2    lava_1.9.1
    #> [10] htmltools_0.5.9        stats4_4.6.0           sass_0.4.10
    #> [13] rmarkdown_2.31         grid_4.6.0             evaluate_1.0.5
    #> [16] jquerylib_0.1.4        fastmap_1.2.0          numDeriv_2016.8-1.1
    #> [19] yaml_2.3.12            mvtnorm_1.3-7          lifecycle_1.0.5
    #> [22] compiler_4.6.0         codetools_0.2-20       ucminf_1.2.3
    #> [25] Rcpp_1.1.1-1.1         future_1.70.0          lattice_0.22-9
    #> [28] digest_0.6.39          R6_2.6.1               parallelly_1.47.0
    #> [31] parallel_4.6.0         splines_4.6.0          Matrix_1.7-5
    #> [34] bslib_0.11.0           tools_4.6.0            RcppArmadillo_15.2.6-1
    #> [37] globals_0.19.1         cachem_1.1.0