simulateTVcureData2: Simulation of survival data with a cure fraction and...

In tvcure: Additive Cure Survival Model with Time-Varying Covariates

Simulation of survival data with a cure fraction and time-varying covariates.

Description

Simulation of cure survival data in a counting process format with time-varying covariates. The population hazard at time t underlying the tvcure model is, for given covariate values,

h_p(t|{\bf v}(t),\tilde{\bf v}(t)) = \mathrm{e}^{\eta_\vartheta({\bf v}(t))+\eta_F(\tilde{\bf v}(t))} f_0(t)S_0(t)^{\exp(\eta_F(\tilde{\bf v}(t)))-1}

with linear predictors

\eta_\vartheta({\bf v}(t)) = \beta_0 + \beta_1 z_1(t) + \beta_2 z_2 + \beta_{f_1} f_1(x_1(t)) + \beta_{f_2} f_2(x_2(t))

\eta_F(\tilde{{\bf v}}(t)) = \gamma_1 z_3(t) + \gamma_2 z_4 + \gamma_{f_1} \tilde{f}_1(x_3(t)) + \gamma_{f_2} \tilde{f}_2(x_4(t))

where {\bf v}(t)=(z_1(t),z_2,x_1(t),x_2(t)), \tilde{\bf v}(t)=(z_3(t),z_4,x_3(t),x_4(t)), with time-varying covariates x_1(t), x_3(t) assumed identical and shared by the 2 submodels when shared.cov is TRUE.

The density f_0(t) governing the reference cumulative hazard dynamic is, by default, a Weibull with shape parameter 2.65 and scale parameter 100, ensuring that all susceptible units will experience the monitored event by time Tmax=300.

The functions defining the additive terms are

f_1(x_1)= -.63 + .57\arctan(4x_1) ~;~f_2(x_2)= -.5 \cos(2\pi x_2)

\tilde{f}_1(\tilde{x}_3) = .15 - .5 \cos\big(\pi(\tilde{x}_3-.75)\big)~;~ \tilde{f}_2(\tilde{x}_4) = .8\big(\tilde{x}_4-.5\big)

Covariates are generated as follows:

• z_1(t), z_3(t) are recentered time-varying Bernoulli(0.5) on (0,T_{max}) ;

• z_2, z_4 \sim N(0,1) ;

• x_1(t), x_2(t), x_3(t), x_4(t) follow random cubic polynomial trajectories on (0,T_{max}).

More details can be found in Lambert and Kreyenfeld (2024).

Usage

simulateTVcureData2(n, seed, Tmax=300,
       f0F0 = list(f0=function(x) dweibull(x, 2.65, 100),
                   F0=function(x) pweibull(x, 2.65, 100)),
       beta, gam, beta.f = rep(1,2), gam.f = rep(1,2), shared.cov=TRUE,
       RC.dist=c("uniform","exponential","Tmax"),
       tRC.min = 120, mu.cens=40, get.details=TRUE)

Arguments

n	Number of units.
seed	Seed (integer) for the random TVcure data generator.
Tmax	Maximum follow-up time after which a unit is considered cured in the absence of a previous event. (Default: 300).
f0F0	List of length 2 providing the density f_0(t) and associated cdf F_0(t) governing the bounded hazard dynamic on (0,Tmax), with F_0(Tmax)=1.0. (Default: f0F0 = list(f0=function(x) dweibull(x, 2.65, 100), F0=function(x) pweibull(x, 2.65, 100))).
beta	3-vector with the regression coefficients <beta> in the long-term (cure) survival (or quantum) submodel.
gam	2-vector with the regression coefficients <gamma> in the short-term (cure) survival (or timing) submodel.
beta.f	2-vector with the multiplying coefficients of the additive terms in long-term (cure) survival (or quantum) submodel.
gam.f	2-vector with the multiplying coefficients of the additive terms in the short-term (cure) survival (or timing) submodel.
shared.cov	Logical indicating whether shared covariates for both submodels are assumed, with then x_1(t)=x_3(t). (Default: TRUE).
RC.dist	Right-censoring distribution: "uniform" (Uniform on (tRC.min,Tmax)),"exponential" (with mean mu.cens) or "Tmax" (when right-censoring occurs at Tmax)
tRC.min	Minimum right-censoring time value if the right-censoring time distribution is Uniform. (Default: 120).
mu.cens	Mean of the right-censoring time distribution if it is Exponential. (Default: 40).
get.details	Logical indicating if a detailed data frame rawdata including the sequence of time-varying covariate values should also be reported. (Default: TRUE).

Value

A list with following elements:

• seeds : Seeds used to generate the data for each of the n units.

• tRC.min : Minimum right-censoring time value if the right-censoring time distribution is Uniform.

• RC.dist : Right-censoring distribution ("Uniform", "Exponential" or "Tmax").

• cure.rate : Underlying proportion of cured units (i.e. without an observed event by Tmax if the follow-up is not interrupted by that time due to right-censoring).

• RC.rate : Observed right-censoring rate.

• rawdata : Data frame containing the generated data in a counting process format with the detailed follow-up for each unit until the event or right-censoring occurs:

  • id : Unit identificator for each row.

  • time : Discrete observation times, starting at 1 for a given unit, until the end of its follow-up. The number of rows associated to a given unit corresponds to the follow-up duration.

  • event : Event indicator (1 if it occured, 0 otherwise) for given unit at a given time.

  • z1, z2, z3, z4, x1, x2, x3, x4 : Covariate values for a given unit at a given time.

• data.summary : Data frame with n rows containing summarized information on the generated data for each unit:

  • id : Unit identificator (the ith row corresponding to the ith unit).

  • t.obs : Observed event or right-censoring time.

  • delta : Event indicator (1 if it occured, 0 otherwise).

  • t.true : True (possibly unobserved) event time (Inf for a cured unit).

  • t.cens : True (possibly unobserved) right-censoring time.

  • cured : True (possibly unobserved) cure status.

• parameters : List containing the defining elements of the tvcure model:

  • beta : The regression parameters in the long-term survival (or quantum) submodel.

  • gam : The regression parameters in the short-term survival (or timing) submodel.

  • beta.f : The multiplying coefficients of the additive terms in the long-term survival (or quantum) submodel.

  • gam.f : The multiplying coefficients of the additive terms in the short-term survival (or timing) submodel.

  • f.theta : A list of length 2 containing the functions defining the additive terms in the long-term survival (or quantum) submodel.

  • f.gam : A list of length 2 containing the functions defining the additive terms in the short-term survival (or timing) submodel.

  • f0 : Density function governing the dynamic of the reference cumulative hazard on (0,Tmax).

  • F0 : CDF governing the dynamic of the reference cumulative hazard on (0,Tmax).

• call : Function call.

Author(s)

Philippe Lambert [e-mail]

References

Lambert, P. and Kreyenfeld, M. (2025). Time-varying exogenous covariates with frequently changing values in double additive cure survival model: an application to fertility. Journal of the Royal Statistical Society, Series A. <doi:10.1093/jrsssa/qnaf035>

Examples

require(tvcure)
## Regression parameters
beta = c(beta0=.4, beta1=-.2, beta2=.15) ##  beta0 tunes the cure rate
gam = c(gam1=.2, gam2=.2)
beta.f = gam.f = rep(1,2) ## Multiply additive terms by 1.0 (by default)
## Data simulation
temp = simulateTVcureData2(n=500, seed=123,
                          beta=beta, gam=gam,
                          beta.f=beta.f, gam.f=gam.f,
                          RC.dist="exponential",mu.cens=550)
head(temp$rawdata) ## Overview of the simulated raw data
head(temp$data.summary) ## Overview of the summarized data
with(temp, c(cure.rate=cure.rate,RC.rate=RC.rate)) ## Cure and RC rates

tvcure documentation built on April 12, 2025, 1:58 a.m.