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Using formula to fit CIFs for time-to-event data with intercurrent events

Source: R/tteICE.R
tteICE.Rd

This function estimates the potential cumulative incidence function for time-to event data under ICH E9 (R1) to address intercurrent events. The input data can be competing or semicompeting risks data structure.

Usage

tteICE(
  formula,
  add.scr = NULL,
  data,
  strategy = "composite",
  method = "np",
  weights = NULL,
  subset = NULL,
  na.rm = FALSE,
  nboot = 0,
  seed = 0
)

Arguments

formula

An object of class "formula" (or one that can be coerced to that class). A symbolic description of the model to be fitted. For example, formula=Surv(time, status, type="mstate")~treatment | baseline.covariate. The details of model specification are given under ‘Details’.

add.scr

Required for semicompeting data. An object of class "Surv" (or one that can be coerced to that class). For example, add.scr=~Surv(time.intercurrent, status.intercurrent). The details of model specification are given under ‘Details’.

data

Data or object coercible by as.data.frame to a data frame, containing the variables in the model.

strategy

Strategy to address intercurrent events, "treatment" indicating treatment policy strategy, "composite" indicating composite variable strategy, "natural" indicating hypothetical strategy (Scenario I, controlling the hazard of intercurrent events), "removed" indicating hypothetical strategy (Scenario II, removing intercurrent events), "whileon" indicating while on treatment strategy, and "principal" indicating principal stratum strategy.

method

Estimation method, "np" indicating nonparametric estimation, "np" indicating inverse treatment probability weighting, "eff" indicating semiparametrically efficient estimation based on efficient influence functions.

weights

Weight for each subject.

subset

Subset, either numerical or logical.

na.rm

Whether to remove missing values.

nboot

Number of resamplings in the boostrapping method. If nboot is 0 or 1, then asymptotic standard error based on the explicit form is calculated instead of bootstrapping.

seed

Seed for bootstrapping.

Value

A list including the fitted object and input variables.

Details

Background

Intercurrent events refer to the events occurring after treatment initiation of clinical trials that affect either the interpretation of or the existence of the measurements associated with the clinical question of interest. The International Conference on Harmonization (ICH) E9 (R1) addendum proposed five strategies to address intercurrent events, namely, treatment policy strategy, composite variable strategy, while on treatment strategy, hypothetical strategy, and principal stratum strategy. To answer a specific scientific question, a strategy with a particular estimand is chosen before the study design.

Model

We adopt the potential outcomes framework that defines a causal estimand as the contrast between functionals of potential outcomes. Consider a randomized controlled trial with \(n\) individuals randomly assigned to one of two treatment conditions, denoted by \(w\), where \(w = 1\) represents the active treatment (a test drug) and \(w = 0\) represents the control (placebo). Assume that all patients adhere to their treatment assignments and do not discontinue treatment. Associated with individual \(i = 1, ..., n\) are two potential time-to-event primary outcomes \(T_i(1)\) and \(T_i(0)\), if any, which represent the time durations from treatment initiation to the primary outcome event under two treatment assignments respectively. Let \(R_i(1)\) and \(R_i(0)\) denote the occurrence time of potential intercurrent events, if any, under the two treatment assignments, respectively. Intercurrent events are considered as absent if no post-treatment intercurrent events occur until the end of study.

Estimand

We adopt the potential cumulative incidences under both treatment assignments as the target estimands. Potential cumulative incidences describe the probability of time-to-event outcomes occurring at each time point. We define the treatment effect as the contrast of two potential cumulative incidences. Cumulative incidences are model-free and collapsible, enjoying causal interpretations.

Formula specifications

The formula should be set as the following two ways.

When data take format of competing risk data, set the first argument formula = Surv(time, status, type="mstate") ~ treatment | covariate1+covariate2 or formula = Surv(time, status)~ A without any baseline covariates, where status=0,1,2 (1 for the primary event, 2 for the intercurrent event, and 0 for censoring).

When data take the format of semicompeting risk data, set the first argument formula = Surv(time, status) ~ treatment | covariate1+covariate2 or formula = Surv(time, status)~ A without any baseline covariates, where status=0,1 (1 for the primary event and 0 for censoring). In addition, the second argument add.scr = ~ Surv(time.intercurrent, status.intercurrent) is required.

References

Deng, Y., Han, S., & Zhou, X. H. (2025). Inference for Cumulative Incidences and Treatment Effects in Randomized Controlled Trials With Time-to-Event Outcomes Under ICH E9 (R1). Statistics in Medicine. doi:10.1002/sim.70091

See also

surv.boot, scr.tteICE

Examples

## load data
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
X = as.matrix(bmt[,c('z1','z3','z5')])
bmt$A = A

library(survival)
## Composite variable strategy,
## nonparametric estimation without covariates
## Composite variable strategy,
## nonparametric estimation without covariates

## model fitting for competing risk data without covariates
fit1 = tteICE(Surv(t2, d4, type = "mstate")~A,
 data=bmt, strategy="composite", method='eff')
print(fit1)
#> Input:
#> tteICE(formula = Surv(t2, d4, type = "mstate") ~ A, data = bmt, 
#>     strategy = "composite", method = "eff")
#> -----------------------------------------------------------------------
#> Data type: competing risks 
#> Strategy: composite variable strategy 
#> Estimation method: semiparametrically efficient estimation 
#> Observations: 137 (including 99 treated and 38 control)
#> Maximum follow-up time: 2640 
#> P-value of the average treatment effect: 0.5907 
#> -----------------------------------------------------------------------
#> The estimated cumulative incidences and treatment effects at quartiles:
#>           660    1320    1980    2640
#> CIF1   0.5323  0.5864  0.5864  0.6299
#> se1    0.0501  0.0499  0.0499  0.0617
#> CIF0   0.6087  0.6377  0.6377  0.6377
#> se0    0.0803  0.0793  0.0793  0.0793
#> ATE   -0.0764 -0.0513 -0.0513 -0.0078
#> se     0.0946  0.0937  0.0937  0.1005
#> p.val  0.4192  0.5843  0.5843  0.9384
#> 

## model fitting for competing risk data without covariates 
## with bootstrap confidence intervals
fit.bt1 = tteICE(Surv(t2, d4, type = "mstate")~A,
 data=bmt, strategy="composite", method='eff', nboot=20, seed=2)
print(fit.bt1)
#> Input:
#> tteICE(formula = Surv(t2, d4, type = "mstate") ~ A, data = bmt, 
#>     strategy = "composite", method = "eff", nboot = 20, seed = 2)
#> -----------------------------------------------------------------------
#> Data type: competing risks 
#> Strategy: composite variable strategy 
#> Estimation method: semiparametrically efficient estimation 
#> Observations: 137 (including 99 treated and 38 control)
#> Maximum follow-up time: 2640 
#> P-value of the average treatment effect: 0.5907 
#> -----------------------------------------------------------------------
#> The estimated cumulative incidences and treatment effects at quartiles:
#>           660    1320    1980    2640
#> CIF1   0.5323  0.5864  0.5864  0.6299
#> se1    0.0000  0.0000  0.0000  0.0000
#> CIF0   0.6087  0.6377  0.6377  0.6377
#> se0    0.0000  0.0000  0.0000  0.0000
#> ATE   -0.0764 -0.0513 -0.0513 -0.0078
#> se     0.0000  0.0000  0.0000  0.0000
#> p.val  0.0000  0.0000  0.0000  0.0000
#> 

## model fitting for competing risk data with covariates
fit2 = tteICE(Surv(t2, d4, type = "mstate")~A|z1+z3+z5,
 data=bmt, strategy="composite", method='eff')
print(fit2)
#> Input:
#> tteICE(formula = Surv(t2, d4, type = "mstate") ~ A | z1 + z3 + 
#>     z5, data = bmt, strategy = "composite", method = "eff")
#> -----------------------------------------------------------------------
#> Data type: competing risks 
#> Strategy: composite variable strategy 
#> Estimation method: semiparametrically efficient estimation 
#> Observations: 137 (including 99 treated and 38 control)
#> Maximum follow-up time: 2640 
#> P-value of the average treatment effect: 0.1366 
#> -----------------------------------------------------------------------
#> The estimated cumulative incidences and treatment effects at quartiles:
#>           660    1320    1980    2640
#> CIF1   0.5246  0.5836  0.5836  0.6347
#> se1    0.0514  0.0511  0.0511  0.0587
#> CIF0   0.6784  0.7010  0.7010  0.7010
#> se0    0.0662  0.0651  0.0651  0.0651
#> ATE   -0.1538 -0.1174 -0.1174 -0.0663
#> se     0.0838  0.0828  0.0828  0.0877
#> p.val  0.0667  0.1560  0.1560  0.4494
#> 

## model fitting for semicompeting risk data without covariates
fitscr1 = tteICE(Surv(t1, d1)~A, ~Surv(t2, d2),
 data=bmt, strategy="composite", method='eff')
print(fitscr1)
#> Input:
#> tteICE(formula = Surv(t1, d1) ~ A, add.scr = ~Surv(t2, d2), data = bmt, 
#>     strategy = "composite", method = "eff")
#> -----------------------------------------------------------------------
#> Data type: semicompeting risks 
#> Strategy: composite variable strategy 
#> Estimation method: semiparametrically efficient estimation 
#> Observations: 137 (including 99 treated and 38 control)
#> Maximum follow-up time: 2640 
#> P-value of the average treatment effect: 0.5907 
#> -----------------------------------------------------------------------
#> The estimated cumulative incidences and treatment effects at quartiles:
#>           660    1320    1980    2640
#> CIF1   0.5323  0.5864  0.5864  0.6299
#> se1    0.0501  0.0499  0.0499  0.0617
#> CIF0   0.6087  0.6377  0.6377  0.6377
#> se0    0.0803  0.0793  0.0793  0.0793
#> ATE   -0.0764 -0.0513 -0.0513 -0.0078
#> se     0.0946  0.0937  0.0937  0.1005
#> p.val  0.4192  0.5843  0.5843  0.9384
#> 

## model fitting for semicompeting risk data without covariates
fitscr2 = tteICE(Surv(t1, d1)~A|z1+z3+z5, ~Surv(t2, d2),
 data=bmt, strategy="composite", method='eff')
print(fitscr2)
#> Input:
#> tteICE(formula = Surv(t1, d1) ~ A | z1 + z3 + z5, add.scr = ~Surv(t2, 
#>     d2), data = bmt, strategy = "composite", method = "eff")
#> -----------------------------------------------------------------------
#> Data type: semicompeting risks 
#> Strategy: composite variable strategy 
#> Estimation method: semiparametrically efficient estimation 
#> Observations: 137 (including 99 treated and 38 control)
#> Maximum follow-up time: 2640 
#> P-value of the average treatment effect: 0.1366 
#> -----------------------------------------------------------------------
#> The estimated cumulative incidences and treatment effects at quartiles:
#>           660    1320    1980    2640
#> CIF1   0.5246  0.5836  0.5836  0.6347
#> se1    0.0514  0.0511  0.0511  0.0587
#> CIF0   0.6784  0.7010  0.7010  0.7010
#> se0    0.0662  0.0651  0.0651  0.0651
#> ATE   -0.1538 -0.1174 -0.1174 -0.0663
#> se     0.0838  0.0828  0.0828  0.0877
#> p.val  0.0667  0.1560  0.1560  0.4494
#>