PWEALL (version 1.3.0.1)

pwecxpwuforvar: calculate the utility function used for varaince calculation

Description

This is a utility function to calculate the overall variance accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

pwecxpwuforvar(tfix=10,t=seq(0,10,by=0.5),taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),
    rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),rate4=rate2,rate5=rate2,ratec=c(0.5,0.6),
         tchange=c(0,1),type=1,rp2=0.5,eps=1.0e-2)

Value

f0

the integral when \(l=0\)

f1

the integral when \(l=1\)

Arguments

tfix

The upper point where the integral is computed.

t

A vector of lower bounds where the integral is computed.

taur

Recruitment time

u

Piecewise constant recuitment rate

ut

Recruitment intervals

rate1

Hazard before crossover

rate2

Hazard after crossover

rate3

Hazard for time to crossover

rate4

Hazard after crossover for complex case

rate5

Hazard after crossover for complex case

ratec

Hazard for time to censoring

tchange

A strictly increasing sequence of time points at which the event rates changes. The first element of tchange must be zero. It must have the same length as rate1, rate2, rate3, etc.

type

Type of crossover

rp2

re-randomization prob

eps

A small number representing the error tolerance when calculating the utility function $$\Phi_l(x)=\frac{\int_0^x s^l e^{-s}ds}{x^{l+1}}$$ with \(l=0,1,2\).

Author

Xiaodong Luo

Details

This is to calculate the function $$B_l(t,s)=\int_0^s x^l G_E(t-x)\widetilde{f}(x)S_C(x)dx,$$ where \(G_E\) is the accrual function defined by taur, u and ut, \(S_C\) is the piecewise exponential survival function of the censoring time, defined by tchange and ratec, and \(\widetilde{f}\) is the density for the event distribution subject to crossover defined by tchange, rate1 to rate5 and type. This function is useful when calculating the overall varaince and covariance.

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,ovbeta,innervar

Examples

taur<-1.2
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
r11<-c(1,0.5)
r21<-c(0.5,0.8)
r31<-c(0.7,0.4)
r41<-r51<-r21
rc1<-c(0.5,0.6)
getf<-pwecxpwuforvar(tfix=3,t=seq(0,3,by=1),taur=taur,u=u,ut=ut,
                 rate1=r11,rate2=r21,rate3=r31,rate4=r41,rate5=r51,ratec=rc1,
                 tchange=c(0,1),type=1,eps=1.0e-2)
getf