PWEALL (version 1.3.0.1)

overallcov: calculate the overall covariance

Description

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

Usage

overallcov(tfix=2.0,tfix0=1.0,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=c(1,0.7),rate20=rate10,rate30=rate31,
              rate40=rate20,rate50=rate20,ratec0=ratec1,
              tchange=c(0,1),type1=1,type0=1,
              rp21=0.5,rp20=0.5,
              eps=1.0e-2,veps=1.0e-2,beta=0,beta0=0)

Value

covbeta

The covariance the score functions

covbeta1

The first part of the cov

covbeta2

The second part of the cov

covbeta3

The third part of the cov

covbeta4

The fourth part of the cov

EA1

The first score function

EA2

The second score function

Arguments

tfix

The upper point where the overall covariance is computed.

tfix0

The lower point where the overall covariance 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 covaraince is computed, upper bound

beta0

The value at which the covaraince is computed, lower bound

Author

Xiaodong Luo

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.

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)
getcov<-overallcov(tfix=2.0,tfix0=1.0,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,beta0=0)
getcov$covbeta