overallcovp1: calculate the first part of the overall covariance

Description

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

Usage

overallcovp1(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=rate51,ratec1=c(0.5,0.6),
                    rate10=rate11,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

covbeta1: The first part of the covariance
EA1: The first 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 $Φ_{l} (x) = \frac{\int_{0}^{x} s^{l} e^{- s} d s}{x^{l + 1}}$ 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 $λ (t) = \sum_{j = 1}^{m} λ_{j} I (t_{j - 1} \leq t < t_{j})$ , where $λ_{1}, \dots, λ_{m}$ are the corresponding elements of the rates and $t_{0}, \dots, t_{m - 1}$ are the corresponding elements of tchange, $t_{m} = \infty$ . Note that all the rates must have the same tchange.

References

Luo, et al. (2017)

Examples

Run this code

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)
getcov1<-overallcovp1(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)
getcov1$covbeta1

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