overallvar {PWEALL}

calculate the overall variance

Description

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

Usage

overallvar(tfix=2.0,taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),pi1=0.5,
                     rate11=c(1,0.5),rate21=rate11,rate31=c(0.7,0.4),
                     rate41=rate21,rate51=rate21,ratec1=c(0.5,0.6),
                     rate10=rate11,rate20=rate10,rate30=rate31,
                     rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
                     tchange=c(0,1),type1=1,type0=1,
                     rp21=0.5,rp20=0.5,
                     eps=1.0e-2,veps=1.0e-2,beta=0)

Arguments

tfix

The time point where the overall variance is computed.

taur

Recruitment time

u

Piecewise constant recuitment rate

ut

Recruitment intervals

pi1

Allocation probability for the treatment group

rate11

Hazard before crossover for the treatment group

rate21

Hazard after crossover for the treatment group

rate31

Hazard for time to crossover for the treatment group

rate41

Hazard after crossover for the treatment group for complex case

rate51

Hazard after crossover for the treatment group for complex case

ratec1

Hazard for time to censoring for the treatment group

rate10

Hazard before crossover for the control group

rate20

Hazard after crossover for the control group

rate30

Hazard for time to crossover for the control group

rate40

Hazard after crossover for the control group for complex case

rate50

Hazard after crossover for the control group for complex case

ratec0

Hazard for time to censoring for the control group

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 rate11, rate21, rate31, etc.

type1

Type of crossover in the treatment group

type0

Type of crossover in the control group

rp21

re-randomization prob in the treatment group

rp20

re-randomization prob in the control group

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.

veps

A small number representing the error tolerance when calculating the Fisher information.

beta

The value at which the varaince is computed.

Details

The hazard functions corresponding to rate11,...,rate51,ratec1, rate10,...,rate50,ratec0 are all piecewise constant function taking the form \lambda(t)=\sum_{j=1}^m \lambda_j I(t_{j-1}\le t<t_j), where \lambda_1,\ldots,\lambda_m are the corresponding elements of the rates and t_0,\ldots,t_{m-1} are the corresponding elements of tchange, t_m=\infty. Note that all the rates must have the same tchange.

Value

vbeta

The variance of the overall log hazard ratio at the specified beta

vs

The variance of the score function at the specified beta

xdenom

Fisher information at the specified beta

EA

value of the score function

EA2

The first part of the variance

AB

Half of the second part of the variance

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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)
r10<-c(1,0.7)
r20<-c(0.5,1)
r30<-c(0.3,0.4)
r40<-r50<-r20
rc0<-c(0.2,0.4)
###variance with beta=0, calculate log-rank variance under the alternative
vbeta0<-overallvar(tfix=2.0,taur=taur,u=u,ut=ut,pi1=0.5,
        rate11=r11,rate21=r21,rate31=r31,rate41=r41,rate51=r51,ratec1=rc1,
        rate10=r10,rate20=r20,rate30=r30,rate40=r40,rate50=r50,ratec0=rc0,
        tchange=c(0,1),type1=1,type0=1,eps=1.0e-2,veps=1.0e-2,beta=0)

###variance with beta=0, calculate log-rank variance under the alternative
###Estimate the overall beta
getbeta<-ovbeta(tfix=2.0,taur=taur,u=u,ut=ut,pi1=0.5,
        rate11=r11,rate21=r21,rate31=r31,rate41=r41,rate51=r51,ratec1=rc1,
        rate10=r10,rate20=r20,rate30=r30,rate40=r40,rate50=r50,ratec0=rc0,
        tchange=c(0,1),type1=1,type0=1,eps=1.0e-2,veps=1.0e-2,beta0=0,
        epsbeta=1.0e-4,iterbeta=25)
vbeta<-overallvar(tfix=2.0,taur=taur,u=u,ut=ut,pi1=0.5,
        rate11=r11,rate21=r21,rate31=r31,rate41=r41,rate51=r51,ratec1=rc1,
        rate10=r10,rate20=r20,rate30=r30,rate40=r40,rate50=r50,ratec0=rc0,
      tchange=c(0,1),type1=1,type0=1,eps=1.0e-2,veps=1.0e-2,beta=getbeta$b1)
cbind(vbeta0$vs,vbeta$vs)