Data from Section 1.3 of Klein and Moeschberger (1997)

Description

The bmt data frame has 137 rows and 22 columns.

Usage

bmt

Format

This data frame contains the following columns:

group

Disease Group 1-ALL, 2-AML Low Risk, 3-AML High Risk

t1

Time To Death Or On Study Time

t2

Disease Free Survival Time (Time To Relapse, Death Or End Of Study)

d1

Death Indicator 1-Dead 0-Alive

d2

Relapse Indicator 1-Relapsed, 0-Disease Free

d3

Disease Free Survival Indicator 1-Dead Or Relapsed, 0-Alive Disease Free)

ta

Time To Acute Graft-Versus-Host Disease

da

Acute GVHD Indicator 1-Developed Acute GVHD 0-Never Developed Acute GVHD)

tc

Time To Chronic Graft-Versus-Host Disease

dc

Chronic GVHD Indicator 1-Developed Chronic GVHD 0-Never Developed Chronic GVHD

tp

Time To Platelet Recovery

dp

Platelet Recovery Indicator 1-Platelets Returned To Normal, 0-Platelets Never Returned to Normal

z1

Patient Age In Years

z2

Donor Age In Years

z3

Patient Sex: 1-Male, 0-Female

z4

Donor Sex: 1-Male, 0-Female

z5

Patient CMV Status: 1-CMV Positive, 0-CMV Negative

z6

Donor CMVStatus: 1-CMV Positive, 0-CMV Negative

z7

Waiting Time to Transplant In Days

z8

FAB: 1-FAB Grade 4 Or 5 and AML, 0-Otherwise

z9

Hospital: 1-The Ohio State University, 2-Alferd, 3-St. Vincent, 4-Hahnemann

z10

MTX used as a Graft-Versus-Host-Prophylactic: 1-Yes 0-No

Source

Klein and Moeschberger (1997) Survival Analysis Techniques for Censored and Truncated Data, Springer.

Examples

data(bmt)

Graphical results of tteICE

Description

This function plots the estimated potential cumulative incidence functions or treatment effect curve with pointwise confidence intervals.

Usage

## S3 method for class 'tteICE'
plot(
  x,
  type = c("ate", "inc")[1],
  decrease = FALSE,
  conf.int = 0.95,
  nboot = 0,
  seed = 0,
  xlab = "Time",
  xlim = NULL,
  ylim = NULL,
  plot.configs = list(),
  ...
)

Arguments

Value

Plot the results from a tteICE object

See Also

plot_ate, plot_inc

Examples

## load data
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
## plot cumulative incidence functions with p-values
for (st in c('composite','natural','removed','whileon','principal')){
 fit = surv.tteICE(A, bmt$t2, bmt$d4, st)
 plot(fit, type="inc", decrease=TRUE, ylim=c(0,1),
      plot.configs=list(show.p.value=TRUE))
}
## plot treatment effects for semicompeting risk data
for (st in c('composite','natural','removed','whileon','principal')){
 fit = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, st)
 plot(fit, type="ate", ylim=c(-1,1), xlab="time", 
      plot.configs=list(col="red"))
}

Plot the estimated treatment effect

Description

This function plots the estimated treatment effect, defined as the difference in potential cumulative incidences under treated and control groups, along with pointwise confidence intervals.

Usage

plot_ate(
  fit,
  decrease = FALSE,
  conf.int = 0.95,
  nboot = 0,
  seed = 0,
  xlab = "Time",
  ylim = c(-1, 1),
  xlim = NULL,
  plot.configs = list(ylab = NULL, main = NULL, lty = 1, lwd = 2, col = "black",
    add.null.line = TRUE, null.line.lty = 2, ci.lty = 5, ci.lwd = 1.5, ci.col =
    "darkgrey"),
  ...
)

Arguments

Value

Plot the average treatment effect (ATE) results from a tteICE object

See Also

plot.default, points, curve, plot.tteICE

Examples

## Load data and fit the model
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
## Model with competing risk data
fit1 = surv.tteICE(A, bmt$t2, bmt$d4, 'composite')
## Plot asymptotic confidence intervals based on explicit formulas
plot_ate(fit1, ylim=c(-0.4,0.4))

## Plot bootstrap confidence intervals (may take some seconds)
plot_ate(fit1, nboot=200, ylim=c(-0.4,0.4))

## Model with semicompeting risk data
fit2 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "composite")
## Plot asymptotic confidence intervals based on explicit formulas
plot_ate(fit2, ylim=c(-0.4,0.4),
         plot.configs=list(add.null.line=FALSE))
## Plot bootstrap confidence intervals
plot_ate(fit2, nboot=200, ylim=c(-0.4,0.4),
         plot.configs=list(add.null.line=FALSE, lty=2, main=""))

Plot the estimated cumulative incidence function (CIF)

Description

This function plots the estimated potential cumulative incidence function, along with pointwise confidence intervals.

Usage

plot_inc(
  fit,
  decrease = FALSE,
  conf.int = 0.95,
  nboot = 0,
  seed = 0,
  xlab = "Time",
  xlim = NULL,
  ylim = c(0, 1),
  plot.configs = list(ylab = NULL, main = NULL, cex = 0.9, lty = 1, lwd = 2, ci.lty = 5,
    ci.lwd = 1.5, legend = c("Treated", "Control"), col = c("brown", "darkcyan"),
    legend.cex = 0.9, show.p.value = TRUE),
  ...
)

Arguments

Value

Plot the cumulative incidence function results from a tteICE object

See Also

plot.default, points, curve, plot.tteICE

Examples

## load data and fit the model
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
## Model with competing risk data
fit1 = surv.tteICE(A, bmt$t2, bmt$d4, 'treatment')
## plot asymptotic confidence intervals based on explicit formulas
plot_inc(fit1, ylim=c(0,1),
         plot.configs=list(legend=c('AML','ALL'), show.p.value=FALSE) )

## plot bootstrap confidence intervals (may take some seconds)
plot_inc(fit1, nboot=200, ylim=c(0,1),
         plot.configs=list(legend=c('AML','ALL')) )

## Model with semicompeting risk data
fit2 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "composite")
## plot asymptotic confidence intervals based on explicit formulas
plot_inc(fit2, ylim=c(0,1), plot.configs=list(add.null.line=FALSE))
## plot bootstrap confidence intervals
plot_inc(fit2, nboot=200, ylim=c(0,1),
         plot.configs=list(lty=2, lwd=3,main="Title"))

Print a short summary of the estimated treatment effect

Description

This function summarize the results

Usage

## S3 method for class 'tteICE'
print(x, digits = 3, ...)

Arguments

Value

Print the summary of a tteICE object

See Also

surv.tteICE, scr.tteICE

Examples

## load data
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
## print the results
for (st in c('composite','natural','removed','whileon','principal')){
 fit = surv.tteICE(A, bmt$t2, bmt$d4, st)
 print(fit)
}
for (st in c('composite','natural','removed','whileon','principal')){
 fit = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, st)
 print(fit, digits=2)
}

Risk prediction at specific time points

Description

This function predicts the potential cumulative incidence function and treatment effect at specific time points.

Usage

riskpredict(fit, timeset = NULL, nboot = 0, seed = 0)

Arguments

Value

A matrix. The meanings of each row are: time points, potential cumulative incidences (under treated and under control), treatment effects, standard errors, and P-values.

See Also

scr.tteICE, surv.tteICE, surv.boot

Examples

## load data
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
X = as.matrix(bmt[,c('z1','z3','z5')])
## Composite variable strategy,
## nonparametric estimation without covariates
fit1 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "composite")
riskpredict(fit1, timeset=c(670,2000))

Fit the CIF using composite variable strategy for semicompeting risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using composite variable strategy (semicompeting risks data structure). This strategy adopts the first occurrence of either the intermediate or primary event as the event of interest.

Usage

scr.composite(
  A,
  Time,
  status,
  Time_int,
  status_int,
  weights = rep(1, length(A)),
  subset = NULL
)

Arguments

Details

The composite variable strategy addresses the problem of intercurrent events by expanding the outcome variables. It aggregates the intercurrent event and the primary outcome event into a single composite outcome variable. The idea is not new in the context of progression-free survival, where the composite outcome variable is defined as the occurrence of either a non-terminal event (e.g., cancer progression) or a terminal event (e.g., death). One widely used composite outcome variable has the form Q(w) = \min\{T(w), R(w)\} for w = 1, 0. When this simple form is adopted, the difference in counterfactual cumulative incidences is \tau(t) = P( Q(1) < t ) - P( Q(0) < t ), representing the difference in probabilities of experiencing either intercurrent events or primary outcome events during (0,t) under active treatment and placebo.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

scr.composite.eff, scr.tteICE

Fit the CIF using composite variable strategy for semicompeting risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using composite variable strategy (semicompeting risks data structure). Cox models are employed for survival models. This strategy adopts the first occurrence of either the intermediate or primary event as the event of interest.

Usage

scr.composite.eff(
  A,
  Time,
  status,
  Time_int,
  status_int,
  X = NULL,
  subset = NULL
)

Arguments

Details

The composite variable strategy addresses the problem of intercurrent events by expanding the outcome variables. It aggregates the intercurrent event and the primary outcome event into a single composite outcome variable. The idea is not new in the context of progression-free survival, where the composite outcome variable is defined as the occurrence of either a non-terminal event (e.g., cancer progression) or a terminal event (e.g., death). One widely used composite outcome variable has the form Q(w) = \min\{T(w), R(w)\} for w = 1, 0. When this simple form is adopted, the difference in counterfactual cumulative incidences is \tau(t) = P( Q(1) < t ) - P( Q(0) < t ), representing the difference in probabilities of experiencing either intercurrent events or primary outcome events during (0,t) under active treatment and placebo.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

scr.composite, scr.tteICE

Fit the CIF using hypothetical strategy (I) for semicompeting risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using hypothetical strategy (semicompeting risks data structure). The intercurrent event is only permitted under treated if it would occur under control.

Usage

scr.natural(
  A,
  Time,
  status,
  Time_int,
  status_int,
  weights = rep(1, length(A)),
  subset = NULL
)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that the intercurrent events that occurred when individuals were assigned to test drugs were only permitted if these intercurrent events would have also occurred if these individuals had been assigned to the placebo. In this hypothetical scenario, when assigned to placebo, individuals would be equally likely to experience intercurrent events as they are assigned to placebo in the real-world trial in terms of the hazards; when assigned to test drug, the hazard of intercurrent events would be identical to that if assigned to placebo in the real-world trial. That is, \lambda_2'(t;0) = \lambda_2'(t;1) = \lambda_2(t;0). The treatment effect corresponds to the natural direct effect, with the hazard of intercurrent events set at the level under control. Markovness is assumed in estimation.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

scr.natural.eff, scr.tteICE

Fit the CIF using hypothetical strategy (I) for semicompeting risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using hypothetical strategy (semicompeting risks data structure). Cox models are employed for survival models. The intercurrent event is only permitted under treated if it would occur under control.

Usage

scr.natural.eff(A, Time, status, Time_int, status_int, X = NULL, subset = NULL)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that the intercurrent events that occurred when individuals were assigned to test drugs were only permitted if these intercurrent events would have also occurred if these individuals had been assigned to the placebo. In this hypothetical scenario, when assigned to placebo, individuals would be equally likely to experience intercurrent events as they are assigned to placebo in the real-world trial in terms of the hazards; when assigned to test drug, the hazard of intercurrent events would be identical to that if assigned to placebo in the real-world trial. That is, \lambda_2'(t;0) = \lambda_2'(t;1) = \lambda_2(t;0). The treatment effect corresponds to the natural direct effect, with the hazard of intercurrent events set at the level under control. Markovness is assumed in estimation.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

scr.natural, scr.tteICE

Fit the CIF using principal stratum strategy for semicompeting risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using principal stratum strategy (semicompeting risks data structure). The estimand is defined in a subpopulation where intercurrent events would never occur regardless of treatment conditions.

Usage

scr.principal(
  A,
  Time,
  status,
  Time_int,
  status_int,
  weights = rep(1, length(A)),
  subset = NULL
)

Arguments

Details

The principal stratum strategy aims to stratify the population into subpopulations based on the joint potential occurrences of intercurrent events under the two treatment assignments (R(1), R(0)). Suppose we are interested in a principal stratum comprised of individuals who would never experience intercurrent events, regardless of which treatment they receive. This principal stratum can be indicated by \{R(1)=R(0)=\infty\}. The treatment effect is now defined within this subpopulation, \tau(t) = P(T(1) < t \mid R(1)=R(0)=\infty) - P(T(0) < t \mid R(1)=R(0)=\infty), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo in the subpopulation that will not experience intercurrent events regardless of treatment during (0,t). A principal ignorability assumption is made for identification. If the size of the target principal stratum is small, the results could be highly variable.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect, which is not available under this strategy.

See Also

scr.principal.eff, scr.tteICE

Fit the CIF using principal stratum strategy for semicompeting risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using principal stratum strategy (semicompeting risks data structure). Cox models are employed for survival models. The estimand is defined in a subpopulation where intercurrent events would never occur regardless of treatment conditions.

Usage

scr.principal.eff(
  A,
  Time,
  status,
  Time_int,
  status_int,
  X = NULL,
  subset = NULL
)

Arguments

Details

The principal stratum strategy aims to stratify the population into subpopulations based on the joint potential occurrences of intercurrent events under the two treatment assignments (R(1), R(0)). Suppose we are interested in a principal stratum comprised of individuals who would never experience intercurrent events, regardless of which treatment they receive. This principal stratum can be indicated by \{R(1)=R(0)=\infty\}. The treatment effect is now defined within this subpopulation, \tau(t) = P(T(1) < t \mid R(1)=R(0)=\infty) - P(T(0) < t \mid R(1)=R(0)=\infty), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo in the subpopulation that will not experience intercurrent events regardless of treatment during (0,t). A principal ignorability assumption is made for identification. If the size of the target principal stratum is small, the results could be highly variable.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

scr.principal, scr.tteICE

Fit the CIF using hypothetical strategy (II) for semicompeting risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using hypothetical strategy (semicompeting risks data structure). The intercurrent event is assumed to be absent in the hypothetical scenario.

Usage

scr.removed(
  A,
  Time,
  status,
  Time_int,
  status_int,
  weights = rep(1, length(A)),
  subset = NULL
)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that intercurrent events are absent in the hypothetical scenario for all individuals, so \lambda_2'(t;0) = \Lambda_2'(t;1) = 0. This hypothetical scenario leads to an estimand called the marginal cumulative incidence. The treatment effect corresponds to the controlled direct effect with the intercurrent events removed.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

scr.removed.eff, scr.tteICE

Fit the CIF using hypothetical strategy (II) for semicompeting risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using hypothetical strategy (semicompeting risks data structure). Cox models are employed for survival models. The intercurrent event is assumed to be absent in the hypothetical scenario.

Usage

scr.removed.eff(A, Time, status, Time_int, status_int, X = NULL, subset = NULL)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the cause-specific hazard specific to the primary outcome event under no intercurrent events \lambda_1(t; w) remains unchanged. Specifically, we envision that intercurrent events are absent in the hypothetical scenario for all individuals, so \lambda_2'(t;0) = \Lambda_2'(t;1) = 0. This hypothetical scenario leads to an estimand called the marginal cumulative incidence. The treatment effect corresponds to the controlled direct effect with the intercurrent events removed.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

scr.removed, scr.tteICE

Fit the CIF using treatment policy strategy for semicompeting risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using treatment policy strategy (semicompeting risks data structure). This strategy ignores the intercurrent event and uses the time to the primary event as it was recorded.

Usage

scr.treatment(
  A,
  Time,
  status,
  Time_int,
  status_int,
  weights = rep(1, length(A)),
  subset = NULL
)

Arguments

Details

The treatment policy strategy addresses the problem of intercurrent events by expanding the initial treatment conditions to a treatment policy. This strategy is applicable only if intercurrent events do not hinder primary outcome events. The treatments under comparison are now two treatment policies: (w, R(w)), where w = 1, 0. One policy (1,R(1)) involves administering the test drug, along with any naturally occurring intercurrents, whereas the other policy (0,R(0)) involves administering a placebo, along with any naturally occurring intercurrents. Thus, the potential outcomes are T(1,R(1)) and T(0,R(0)). Instead of comparing the test drug and placebo themselves, the contrast of interest is made between the two treatment policies. The difference in cumulative incidences under the two treatment policies is then \tau(t) = P(T(1, R(1)) < t) - P(T(0, R(0)) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo.
The average treatment effect \tau^{\text{tp}}(t) has a meaningful causal interpretation only when T(1, R(1)) and T(0, R(0)) are well defined. Because the treatment policy includes the occurrence of the intercurrent event as natural, the entire treatment policy is determined by manipulating the initial treatment condition $w$ only. Therefore, we can simplify the notations T(w, R(w)) = T(w) in defining estimands. As such, \tau(t) = P(T(1)) < t) - P(T(0) < t) as the intention-to-treat analysis.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

scr.treatment.eff, scr.tteICE

Fit the CIF using treatment policy strategy for semicompeting risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using treatment policy strategy (semicompeting risks data structure). Cox models are employed for the survival model. This strategy ignores the intercurrent event and uses the time to the primary event as it was recorded.

Usage

scr.treatment.eff(
  A,
  Time,
  status,
  Time_int,
  status_int,
  X = NULL,
  subset = NULL
)

Arguments

Details

The treatment policy strategy addresses the problem of intercurrent events by expanding the initial treatment conditions to a treatment policy. This strategy is applicable only if intercurrent events do not hinder primary outcome events. The treatments under comparison are now two treatment policies: (w, R(w)), where w = 1, 0. One policy (1,R(1)) involves administering the test drug, along with any naturally occurring intercurrents, whereas the other policy (0,R(0)) involves administering a placebo, along with any naturally occurring intercurrents. Thus, the potential outcomes are T(1,R(1)) and T(0,R(0)). Instead of comparing the test drug and placebo themselves, the contrast of interest is made between the two treatment policies. The difference in cumulative incidences under the two treatment policies is then \tau(t) = P(T(1, R(1)) < t) - P(T(0, R(0)) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo.
The average treatment effect \tau^{\text{tp}}(t) has a meaningful causal interpretation only when T(1, R(1)) and T(0, R(0)) are well defined. Because the treatment policy includes the occurrence of the intercurrent event as natural, the entire treatment policy is determined by manipulating the initial treatment condition $w$ only. Therefore, we can simplify the notations T(w, R(w)) = T(w) in defining estimands. As such, \tau(t) = P(T(1)) < t) - P(T(0) < t) as the intention-to-treat analysis.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

scr.treatment, scr.tteICE

Fit the CIF for time-to-event data with intercurrent events for semicompeting risks data

Description

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

Usage

scr.tteICE(
  A,
  Time,
  status,
  Time_int,
  status_int,
  strategy = "composite",
  cov1 = NULL,
  method = "np",
  weights = NULL,
  subset = NULL,
  na.rm = FALSE
)

Arguments

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.

Value

A list including the fitted object and input variables.

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, surv.tteICE

Examples

## load data
data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
X = as.matrix(bmt[,c('z1','z3','z5')])
## Composite variable strategy,
## nonparametric estimation without covariates
fit1 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "composite")
fit10 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "aa")
## Hypothetical strategy (natural effects),
## nonparametric estimation with inverse probability weighting
fit2 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "natural", X, method='ipw')
## nonparametric estimation with weights as non-standardized inverse probability score
ps = predict(glm(A ~ X, family='binomial'), type='response')
w = A/ps + (1-A)/(1-ps)
fit2 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "natural", weights=w)
## Hypothetical strategy (removing intercurrent events),
## semiparametrically efficient estimation with covariates
fit3 = scr.tteICE(A, bmt$t1, bmt$d1, bmt$t2, bmt$d2, "removed", X, method='eff')

Fit the CIF using while on treatment strategy for semicompeting risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using while on treatment strategy (semicompeting risks data structure). This strategy can be understood as the competing risks model, which gives the subdistribution of the primary event.

Usage

scr.whileon(
  A,
  Time,
  status,
  Time_int,
  status_int,
  weights = rep(1, length(A)),
  subset = NULL
)

Arguments

Details

The while on treatment strategy considers the measure of outcome variables taken only up to the occurrence of intercurrent events. The failures of primary outcome events should not be counted in the cumulative incidences if intercurrent events occurred. The difference in counterfactual cumulative incidences under this strategy is \tau(t) = P(T(1) < t, R(1) \geq t) - P(T(0) < t, R(0) \geq t), representing the difference in probabilities of experiencing primary outcome events without intercurrent events during (0,t) under active treatment and placebo. The cumulative incidence function is also known as the cause-specific cumulative incidence or subdistribution function.
The while on treatment strategy is closely related to the competing risks model. However, for causal interpretations, it is worth emphasizing that the hazard of R(1) may differ from that of R(0), leading to vast difference in the underlying features of individuals who have not experienced the primary outcome event between treatment conditions at any time t \in (0,t^*), where t^* is the end of study. When the scientific question of interest is the impact of treatment on the primary outcome event, the estimand \tau(t) is hard to interpret if a systematic difference in the risks of intercurrent events between two treatment conditions under comparison is anticipated.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on Gray test.

See Also

scr.whileon.eff, scr.tteICE

Fit the CIF using while on treatment strategy for semicompeting risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using while on treatment strategy (semicompeting risks data structure). Cox models are employed for survival models. This strategy can be understood as the competing risks model, which gives the subdistribution of the primary event.

Usage

scr.whileon.eff(A, Time, status, Time_int, status_int, X = NULL, subset = NULL)

Arguments

Details

The while on treatment strategy considers the measure of outcome variables taken only up to the occurrence of intercurrent events. The failures of primary outcome events should not be counted in the cumulative incidences if intercurrent events occurred. The difference in counterfactual cumulative incidences under this strategy is \tau(t) = P(T(1) < t, R(1) \geq t) - P(T(0) < t, R(0) \geq t), representing the difference in probabilities of experiencing primary outcome events without intercurrent events during (0,t) under active treatment and placebo. The cumulative incidence function is also known as the cause-specific cumulative incidence or subdistribution function.
The while on treatment strategy is closely related to the competing risks model. However, for causal interpretations, it is worth emphasizing that the hazard of R(1) may differ from that of R(0), leading to vast difference in the underlying features of individuals who have not experienced the primary outcome event between treatment conditions at any time t \in (0,t^*), where t^* is the end of study. When the scientific question of interest is the impact of treatment on the primary outcome event, the estimand \tau(t) is hard to interpret if a systematic difference in the risks of intercurrent events between two treatment conditions under comparison is anticipated.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

scr.whileon, scr.tteICE

Estimate the hazard ratio with intercurrent events

Description

This function estimates the hazard ratio for time-to event data under ICH E9 (R1) to address intercurrent events. Multiple strategies except the principal stratum strategy are allowed.

Usage

surv.HR(
  A,
  Time,
  cstatus,
  strategy = "composite",
  cov1 = NULL,
  conf.int = 0.95,
  weights = NULL,
  subset = NULL
)

Arguments

Details

For the treatment policy and hypothetical strategies, the hazard ratio (HR) is given by the Cox regression regarding intercurrent events as censoring. For the composite variable strategy, the hazard ratio is given by the Cox regression regarding the first occurrence of either intercurrent event or primary event as the event of interest. For the while on treatment strategy, the hazard ratio is given by the Fine-Gray subdistribution model. There is no existing method to estimate the hazard ratio using principal stratum strategy.
The weakness of using hazard ratio to infer treatment effects is critical. First, the hazard ratio relies on model specification. Second, the hazard ratio is not collapsible. Therefore, the hazard ratio should only be treated as a descriptive or exploratory measure of the treatment effect.

Value

A list including

logHR

Estimated log hazard ratio (logHR) of the treatment effect on the primary event.

se

Standard error of the estimated log hazard ratio (logHR).

CI

Confidence interval of the hazard ratio (HR).

p.val

P value of the hazard ratio.

Examples

data(bmt)
bmt = transform(bmt, d4=d2+d3)
A = as.numeric(bmt$group>1)
## composite variable strategy
fit = surv.HR(A, bmt$t2, bmt$d4, "composite")
## while on treatment strategy
X = bmt[,c('z1','z3','z5')]
fit = surv.HR(A, bmt$t2, bmt$d4, "whileon", cov1=X)

Calculate the standard error for the estimated CIF and treatment effect

Description

This function calculates the standard error for the estimated potential cumulative incidence function and treatment effect. Two methods to calculate the standard error are considered: the asymptotic standard error based on the explicit formula and bootstrapping.

Usage

surv.boot(fit, nboot = 0, seed = 0)

Arguments

Value

A list including

time

Time points in both groups.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

strategy

Strategy used.

method

Estimation method used.

See Also

surv.tteICE, scr.tteICE

Fit the CIF using composite variable strategy for competing risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using composite variable strategy (competing risks data structure). This strategy adopts the first occurrence of either the intermediate or primary event as the event of interest.

Usage

surv.composite(A, Time, cstatus, weights = rep(1, length(A)), subset = NULL)

Arguments

Details

The composite variable strategy addresses the problem of intercurrent events by expanding the outcome variables. It aggregates the intercurrent event and the primary outcome event into a single composite outcome variable. The idea is not new in the context of progression-free survival, where the composite outcome variable is defined as the occurrence of either a non-terminal event (e.g., cancer progression) or a terminal event (e.g., death). One widely used composite outcome variable has the form Q(w) = \min\{T(w), R(w)\} for w = 1, 0. When this simple form is adopted, the difference in counterfactual cumulative incidences is \tau(t) = P( Q(1) < t ) - P( Q(0) < t ), representing the difference in probabilities of experiencing either intercurrent events or primary outcome events during (0,t) under active treatment and placebo.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

surv.composite.eff, surv.tteICE

Fit the CIF using composite variable strategy for competing risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using composite variable strategy (competing risks data structure). Cox models are employed for survival models. This strategy adopts the first occurrence of either the intermediate or primary event as the event of interest.

Usage

surv.composite.eff(A, Time, cstatus, X = NULL, subset = NULL)

Arguments

Details

The composite variable strategy addresses the problem of intercurrent events by expanding the outcome variables. It aggregates the intercurrent event and the primary outcome event into a single composite outcome variable. The idea is not new in the context of progression-free survival, where the composite outcome variable is defined as the occurrence of either a non-terminal event (e.g., cancer progression) or a terminal event (e.g., death). One widely used composite outcome variable has the form Q(w) = \min\{T(w), R(w)\} for w = 1, 0. When this simple form is adopted, the difference in counterfactual cumulative incidences is \tau(t) = P( Q(1) < t ) - P( Q(0) < t ), representing the difference in probabilities of experiencing either intercurrent events or primary outcome events during (0,t) under active treatment and placebo.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

surv.composite, surv.tteICE

Fit the CIF using hypothetical strategy (I) for competing risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using hypothetical strategy (competing risks data structure). The intercurrent event is only permitted under treated if it would occur under control.

Usage

surv.natural(A, Time, cstatus, weights = rep(1, length(A)), subset = NULL)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that the intercurrent events that occurred when individuals were assigned to test drugs were only permitted if these intercurrent events would have also occurred if these individuals had been assigned to the placebo. In this hypothetical scenario, when assigned to placebo, individuals would be equally likely to experience intercurrent events as they are assigned to placebo in the real-world trial in terms of the hazards; when assigned to test drug, the hazard of intercurrent events would be identical to that if assigned to placebo in the real-world trial. That is, \lambda_2'(t;0) = \lambda_2'(t;1) = \lambda_2(t;0). The treatment effect corresponds to the natural direct effect with the hazard of intercurrent events set at the level under control.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

surv.natural.eff, surv.tteICE

Fit the CIF using hypothetical strategy (I) for competing risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using hypothetical strategy (competing risks data structure). Cox models are employed for survival models. The intercurrent event is only permitted under treated if it would occur under control.

Usage

surv.natural.eff(A, Time, cstatus, X = NULL, subset = NULL)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that the intercurrent events that occurred when individuals were assigned to test drugs were only permitted if these intercurrent events would have also occurred if these individuals had been assigned to the placebo. In this hypothetical scenario, when assigned to placebo, individuals would be equally likely to experience intercurrent events as they are assigned to placebo in the real-world trial in terms of the hazards; when assigned to test drug, the hazard of intercurrent events would be identical to that if assigned to placebo in the real-world trial. That is, \lambda_2'(t;0) = \lambda_2'(t;1) = \lambda_2(t;0). The treatment effect corresponds to the natural direct effect with the hazard of intercurrent events set at the level under control.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

surv.natural, surv.tteICE

Fit the CIF using principal stratum strategy for competing risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using principal stratum strategy (competing risks data structure). The estimand is defined in a subpopulation where intercurrent events would never occur regardless of treatment conditions.

Usage

surv.principal(A, Time, cstatus, weights = rep(1, length(A)), subset = NULL)

Arguments

Details

The principal stratum strategy aims to stratify the population into subpopulations based on the joint potential occurrences of intercurrent events under the two treatment assignments (R(1), R(0)). Suppose we are interested in a principal stratum comprised of individuals who would never experience intercurrent events, regardless of which treatment they receive. This principal stratum can be indicated by \{R(1)=R(0)=\infty\}. The treatment effect is now defined within this subpopulation, \tau(t) = P(T(1) < t \mid R(1)=R(0)=\infty) - P(T(0) < t \mid R(1)=R(0)=\infty), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo in the subpopulation that will not experience intercurrent events regardless of treatment during (0,t). A principal ignorability assumption is made for identification. If the size of the target principal stratum is small, the results could be highly variable.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect, which is not available under this strategy.

See Also

surv.principal.eff, surv.tteICE

Fit the CIF using principal stratum strategy for competing risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using principal stratum strategy (competing risks data structure). Cox models are employed for survival models. The estimand is defined in a subpopulation where intercurrent events would never occur regardless of treatment conditions.

Usage

surv.principal.eff(A, Time, cstatus, X = NULL, subset = NULL)

Arguments

Details

The principal stratum strategy aims to stratify the population into subpopulations based on the joint potential occurrences of intercurrent events under the two treatment assignments (R(1), R(0)). Suppose we are interested in a principal stratum comprised of individuals who would never experience intercurrent events, regardless of which treatment they receive. This principal stratum can be indicated by \{R(1)=R(0)=\infty\}. The treatment effect is now defined within this subpopulation, \tau(t) = P(T(1) < t \mid R(1)=R(0)=\infty) - P(T(0) < t \mid R(1)=R(0)=\infty), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo in the subpopulation that will not experience intercurrent events regardless of treatment during (0,t). A principal ignorability assumption is made for identification. If the size of the target principal stratum is small, the results could be highly variable.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

surv.principal, surv.tteICE

Fit the CIF using hypothetical strategy (II) for competing risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using hypothetical strategy (competing risks data structure). The intercurrent event is assumed to be absent in the hypothetical scenario.

Usage

surv.removed(A, Time, cstatus, weights = rep(1, length(A)), subset = NULL)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that intercurrent events are absent in the hypothetical scenario for all individuals, so \lambda_2'(t;0) = \Lambda_2'(t;1) = 0. This hypothetical scenario leads to an estimand called the marginal cumulative incidence. The treatment effect corresponds to the controlled direct effect with the intercurrent events removed.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

surv.removed.eff, surv.tteICE

Fit the CIF using hypothetical strategy (II) for competing risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using hypothetical strategy (competing risks data structure). Cox models are employed for survival models. The intercurrent event is assumed to be absent in the hypothetical scenario.

Usage

surv.removed.eff(A, Time, cstatus, X = NULL, subset = NULL)

Arguments

Details

The hypothetical strategy envisions a hypothetical clinical trial condition where the occurrence of intercurrent events is restricted in certain ways. By doing so, the distribution of potential outcomes under the hypothetical scenario can capture the impact of intercurrent events explicitly through a pre-specified criterion. We use T'(w), w = 1, 0 to denote the time to the primary outcome event in the hypothetical scenario. The time-dependent treatment effect specific to this hypothetical scenario is written as \tau(t) = P(T'(1) < t) - P(T'(0) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) in the pre-specified hypothetical scenario under active treatment and placebo.
The key question is how to envision T'(w). We manipulate the hazard specific to intercurrent event \lambda_2(t; w) while assuming the hazard specific to the primary outcome event \lambda_1(t; w) remains unchanged. Specifically, we envision that intercurrent events are absent in the hypothetical scenario for all individuals, so \lambda_2'(t;0) = \Lambda_2'(t;1) = 0. This hypothetical scenario leads to an estimand called the marginal cumulative incidence. The treatment effect corresponds to the controlled direct effect with the intercurrent events removed.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

surv.removed, surv.tteICE

Fit the CIF using treatment policy strategy for competing risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using treatment policy strategy (competing risks data structure). This strategy ignores the intercurrent event and uses the time to the primary event as it was recorded.

Usage

surv.treatment(A, Time, cstatus, weights = rep(1, length(A)), subset = NULL)

Arguments

Details

The treatment policy strategy addresses the problem of intercurrent events by expanding the initial treatment conditions to a treatment policy. This strategy is applicable only if intercurrent events do not hinder primary outcome events. The treatments under comparison are now two treatment policies: (w, R(w)), where w = 1, 0. One policy (1,R(1)) involves administering the test drug, along with any naturally occurring intercurrents, whereas the other policy (0,R(0)) involves administering a placebo, along with any naturally occurring intercurrents. Thus, the potential outcomes are T(1,R(1)) and T(0,R(0)). Instead of comparing the test drug and placebo themselves, the contrast of interest is made between the two treatment policies. The difference in cumulative incidences under the two treatment policies is then \tau(t) = P(T(1, R(1)) < t) - P(T(0, R(0)) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo.
The average treatment effect \tau^{\text{tp}}(t) has a meaningful causal interpretation only when T(1, R(1)) and T(0, R(0)) are well defined. Because the treatment policy includes the occurrence of the intercurrent event as natural, the entire treatment policy is determined by manipulating the initial treatment condition $w$ only. Therefore, we can simplify the notations T(w, R(w)) = T(w) in defining estimands. As such, \tau(t) = P(T(1)) < t) - P(T(0) < t) as the intention-to-treat analysis.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on logrank test.

See Also

surv.treatment.eff, surv.tteICE

Fit the CIF using treatment policy strategy for competing risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using treatment policy strategy (competing risks data structure). Cox models are employed for the survival model. This strategy ignores the intercurrent event and uses the time to the primary event as it was recorded.

Usage

surv.treatment.eff(A, Time, cstatus, X = NULL, subset = NULL)

Arguments

Details

The treatment policy strategy addresses the problem of intercurrent events by expanding the initial treatment conditions to a treatment policy. This strategy is applicable only if intercurrent events do not hinder primary outcome events. The treatments under comparison are now two treatment policies: (w, R(w)), where w = 1, 0. One policy (1,R(1)) involves administering the test drug, along with any naturally occurring intercurrents, whereas the other policy (0,R(0)) involves administering a placebo, along with any naturally occurring intercurrents. Thus, the potential outcomes are T(1,R(1)) and T(0,R(0)). Instead of comparing the test drug and placebo themselves, the contrast of interest is made between the two treatment policies. The difference in cumulative incidences under the two treatment policies is then \tau(t) = P(T(1, R(1)) < t) - P(T(0, R(0)) < t), representing the difference in probabilities of experiencing primary outcome events during (0,t) under active treatment and placebo.
The average treatment effect \tau^{\text{tp}}(t) has a meaningful causal interpretation only when T(1, R(1)) and T(0, R(0)) are well defined. Because the treatment policy includes the occurrence of the intercurrent event as natural, the entire treatment policy is determined by manipulating the initial treatment condition $w$ only. Therefore, we can simplify the notations T(w, R(w)) = T(w) in defining estimands. As such, \tau(t) = P(T(1)) < t) - P(T(0) < t) as the intention-to-treat analysis.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

surv.treatment, surv.tteICE

Fit the CIF for time-to-event with intercurrent events for competing risks data

Description

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

Usage

surv.tteICE(
  A,
  Time,
  cstatus,
  strategy = "composite",
  cov1 = NULL,
  method = "np",
  weights = NULL,
  subset = NULL,
  na.rm = FALSE
)

Arguments

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.

Value

A list including the fitted object and input variables.

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$group>1)
X = as.matrix(bmt[,c('z1','z3','z5')])
## Composite variable strategy, 
## nonparametric estimation without covariates
fit1 = surv.tteICE(A, bmt$t2, bmt$d4, "composite")
## Hypothetical strategy (natural effects), 
## nonparametric estimation with inverse probability weighting
fit2 = surv.tteICE(A, bmt$t2, bmt$d4, "natural", X, method='ipw')
## nonparametric estimation with weights as inverse propensity score
ps = predict(glm(A ~ X, family='binomial'), type='response')
w = A/ps + (1-A)/(1-ps)
fit2 = surv.tteICE(A, bmt$t2, bmt$d4, "natural", weights=w)
## Hypothetical strategy (removing intercurrent events),
## semiparametrically efficient estimation with covariates
fit3 = surv.tteICE(A, bmt$t2, bmt$d4, "removed", X, method='eff')

Fit the CIF using while on treatment strategy for competing risks data

Description

This function nonparametrically estimates the potential cumulative incidence function using while on treatment strategy (competing risks data structure). This strategy can be understood as the competing risks model, which gives the subdistribution of the primary event.

Usage

surv.whileon(A, Time, cstatus, weights = rep(1, length(A)), subset = NULL)

Arguments

Details

The while on treatment strategy considers the measure of outcome variables taken only up to the occurrence of intercurrent events. The failures of primary outcome events should not be counted in the cumulative incidences if intercurrent events occurred. The difference in counterfactual cumulative incidences under this strategy is \tau(t) = P(T(1) < t, R(1) \geq t) - P(T(0) < t, R(0) \geq t), representing the difference in probabilities of experiencing primary outcome events without intercurrent events during (0,t) under active treatment and placebo. The cumulative incidence function is also known as the cause-specific cumulative incidence or subdistribution function.
The while on treatment strategy is closely related to the competing risks model. However, for causal interpretations, it is worth emphasizing that the hazard of R(1) may differ from that of R(0), leading to vast difference in the underlying features of individuals who have not experienced the primary outcome event between treatment conditions at any time t \in (0,t^*), where t^* is the end of study. When the scientific question of interest is the impact of treatment on the primary outcome event, the estimand \tau(t) is hard to interpret if a systematic difference in the risks of intercurrent events between two treatment conditions under comparison is anticipated.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on Gray test.

See Also

surv.whileon.eff, surv.tteICE

Fit the CIF using while on treatment strategy for competing risks data, based on efficient influence functions

Description

This function estimates the potential cumulative incidence function based on efficient influence functions using while on treatment strategy (competing risks data structure). Cox models are employed for survival models. This strategy can be understood as the competing risks model, which gives the subdistribution of the primary event.

Usage

surv.whileon.eff(A, Time, cstatus, X = NULL, subset = NULL)

Arguments

Details

The while on treatment strategy considers the measure of outcome variables taken only up to the occurrence of intercurrent events. The failures of primary outcome events should not be counted in the cumulative incidences if intercurrent events occurred. The difference in counterfactual cumulative incidences under this strategy is \tau(t) = P(T(1) < t, R(1) \geq t) - P(T(0) < t, R(0) \geq t), representing the difference in probabilities of experiencing primary outcome events without intercurrent events during (0,t) under active treatment and placebo. The cumulative incidence function is also known as the cause-specific cumulative incidence or subdistribution function.
The while on treatment strategy is closely related to the competing risks model. However, for causal interpretations, it is worth emphasizing that the hazard of R(1) may differ from that of R(0), leading to vast difference in the underlying features of individuals who have not experienced the primary outcome event between treatment conditions at any time t \in (0,t^*), where t^* is the end of study. When the scientific question of interest is the impact of treatment on the primary outcome event, the estimand \tau(t) is hard to interpret if a systematic difference in the risks of intercurrent events between two treatment conditions under comparison is anticipated.

Value

A list including

time1

Time points in the treated group.

time0

Time points in the control group.

cif1

Estimated cumulative incidence function in the treated group.

cif0

Estimated cumulative incidence function in the control group.

se1

Standard error of the estimated cumulative incidence function in the treated group.

se0

Standard error of the estimated cumulative incidence function in the control group.

time

Time points in both groups.

ate

Estimated treatment effect (difference in cumulative incidence functions).

se

Standard error of the estimated treatment effect.

p.val

P value of testing the treatment effect based on the efficient influence function of the restricted mean survival time lost by the end of study.

See Also

surv.whileon, surv.tteICE

Shiny app for tteICE

Description

This function opens the Rshiny app for tteICE. R Shiny application can be used for generating plots and basic analysis results. It provides a point-and-click interface, so users can obtain results without writing R code directly.

Usage

tteICEShiny()

Value

Rshiny interface

Examples

if(interactive()){
  tteICEShiny()
}