Weibull.simu {joint.Cox}

Simulating data from the Weibull joint frailty-copula model

Description

This function generate clustered (grouped) bivariate event times from the joint frailty-copula model with the Weibull baseline hazard functions. Simulating (X_ij,D_ij,C_ij), i=1,2,...,G, and j=1,2,...,N, where G is the number of studies (groups), and N is the number of individuals (patients) within each study. X_ij is time-to-event, D_ij is time-to-death, and C_ij is time-to-censoring. (X_ij, D_ij) and C_ij are independent. Dependence structure on (X_ij, D_ij) is modeled by a copula, which can be the Clayton (default), Frank, Gumbel, or BB1. Covariate effects are specified by the Cox models given a frailty term.

Usage

Weibull.simu(G,N,scale1,scale2,shape1,shape2,beta1,beta2,
 eta,copula="Clayton",theta,d=0,alpha,beta12=0,C.max,
 cmprsk=FALSE,tau=FALSE,Z.dist=runif,...)

Arguments

G

The number of studies or groups

N

The number of patients within each study

scale1

scale parameter related to the baseline hazard for progression

scale2

scale parameter related to the baseline hazard for death

shape1

shape parameter related to the baseline hazard for progression

shape2

shape parameter related to the baseline hazard for death

beta1

regression coefficients for progression

beta2

regression coefficients for death

eta

frailty variance

copula

copula function; "Clayton" (default), "Gumbel", "Frank", or "BB1"

theta

copula parameter

d

BB1 copula's departure parameter from the Clayton (d=0 is the default)

alpha

parameter related to frailty, e.g., alpha=1

beta12

regression coefficients for copula

C.max

the upper bound for the censoring distribution

cmprsk

if TRUE, simulated data follow the competing risks setting

tau

if TRUE, conditional Kendall's tau given Z is shown

Z.dist

the distribution of a covariate Z

...

parameters for Z.dist

Details

See Wu et al. (2020) for the algorithms for the Clayton copula. The method was later extended by including covariate effects on a copula (beta12) via the conditional copula model of Emura et al. (2021), The available copulas are the Frank, Gumbel, and BB1 copulas. For the BB1 copula, please see Supplementary Material:Additional simulation studies under the copula misspecification in Emura et al. (2021),

Value

X

: time to event

D

: time to death

C

: time to independent censoring

t.event

: time to event (=min(X,D,C))

event

: event indicator (=I(X<=D,X<=C))

event1

: indicator for Event 1 (=I(X<=D,X<=C))

t.death

: time to death (=min(D,C))

death

: death indicator (=I(D<=C))

event2

: indicator for Event 2 (=I(D<X,D<=C))

group

: study ID (=1,2,...,G)

Z

: covariate

tau

: Conditional Kendall's tau given Z

Author(s)

Takeshi Emura

References

Wu BH, Michimae H, Emura T (2020), Meta-analysis of individual patient data with semi-competing risks under the Weibull joint frailty-copula model. Comp Stat 35(4):1525-52

Emura T, Shih JH, Ha ID, Wilke RA (2020), Comparison of the marginal hazard model and the sub-distribution hazard model for competing risks under an assumed copula, Stat Methods Med Res 29(8):2307-27

Emura T, Sofeu C, Rondeau V (2021), Conditional copula models for correlated survival endpoints: individual patient data meta-analysis of randomized controlled trials, Stat Methods Med Res 30(12):2634-50

Supplementary Material:Additional simulation studies under the copula misspecification in "Emura T, Sofeu C, Rondeau V (2021), Conditional copula models for correlated survival endpoints: individual patient data meta-analysis of randomized controlled trials, Stat Methods Med Res 30(12):2634-50"

Examples

Weibull.simu(G=5,N=2,scale1=1,scale2=1,shape1=1,shape2=1,
        beta1=1,beta2=1,eta=0.5,theta=2,alpha=1,C.max=5)

Weibull.simu(G=5,N=2,scale1=1,scale2=1,shape1=1,shape2=1,
        beta1=1,beta2=1,eta=0.5,copula="Gumbel",theta=2,alpha=1,C.max=5)

Weibull.simu(G=5,N=2,scale1=1,scale2=1,shape1=1,shape2=1,
        beta1=1,beta2=1,eta=0.5,theta=2,alpha=1,C.max=5,Z.dist=rbinom,size=1,prob=0.5)

## simulated data follow the competing risks setting
Weibull.simu(G=5,N=2,scale1=1,scale2=1,shape1=1,shape2=1,
        beta1=1,beta2=1,eta=0.5,theta=2,alpha=1,C.max=5,cmprsk=TRUE)