pwefvplus: A utility functon

Description

This will calculate the more complex integration accounting for crossover

Usage

pwefvplus(t=seq(0,5,by=0.5),rate1=c(0,5,0.8),rate2=rate1,
                   rate3=c(0.1,0.2),rate4=rate2,rate5=rate2,
                   rate6=c(0.5,0.3),tchange=c(0,3),type=1,
                   rp2=0.5,eps=1.0e-2)

Value

f0: values when $k = 0$
f1: values when $k = 1$
f2: values when $k = 2$

Arguments

t: A vector of time points
rate1: piecewise constant event rate
rate2: piecewise constant event rate
rate3: piecewise constant event rate
rate4: additional piecewise constant
rate5: additional piecewise constant
rate6: piecewise constant event rate for censoring
tchange: a strictly increasing sequence of time points starting from zero at which event rate changes. The first element of tchange must be zero. The above rates and tchange must have the same length.
type: type of the crossover, markov, semi-markov and hybrid
rp2: re-randomization prob
eps: tolerance

Author

Xiaodong Luo

Details

Let $h_{1}, \dots, h_{6}$ correspond to rate1,...,rate6, and $H_{1}, \dots, H_{6}$ be the corresponding survival functions. Also let $π_{2} = \code r p 2$ . when type=1, we calculate $\int_{0}^{t} s^{k} h_{2} (s) H_{2} (s) H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) / H_{2} (u) d u d s;$ when type=2, we calculate $\int_{0}^{t} s^{k} H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) h_{2} (s - u) H_{2} (s - u) d u d s;$ when type=3, we calculate the sum of $π_{2} \int_{0}^{t} s^{k} H_{4}^{1 - π_{2}} (s) H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) h_{2} (s - u) H_{2}^{π_{2}} (s - u) / H_{4}^{1 - π_{2}} (u) d u d s$ and $(1 - π_{2}) \int_{0}^{t} s^{k} h_{4} (s) H_{4}^{1 - π_{2}} (s) H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) H_{2}^{π_{2}} (s - u) / H_{4}^{1 - π_{2}} (u) d u d s;$ when type=4, we calculate the sum of $π_{2} \int_{0}^{t} s^{k} H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) h_{2} (s - u) H_{2} (s - u) d u d s$ and $(1 - π_{2}) \int_{0}^{t} s^{k} h_{4} (s) H_{4} (s) H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) / H_{4} (u) d u d s;$ when type=5, we calculate the sum of $π_{2} \int_{0}^{t} s^{k} H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) h_{2} (s - u) H_{2} (s - u) d u d s$ and $(1 - π_{2}) \int_{0}^{t} s^{k} H_{6} (s) \int_{0}^{s} h_{3} (u) H_{1} (u) H_{3} (u) h_{4} (s - u) H_{4} (s - u) d u d s .$

References

Luo et al. (2018) Design and monitoring of survival trials in complex scenarios, Statistics in Medicine <doi: https://doi.org/10.1002/sim.7975>.

Examples

Run this code

r1<-c(0.6,0.3)
r2<-c(0.6,0.6)
r3<-c(0.1,0.2)
r4<-c(0.5,0.4)
r5<-c(0.4,0.5)
r6<-c(0.4,0.5)
tchange<-c(0,1.75)
pwefun<-pwefvplus(t=seq(0,5,by=0.5),rate1=r1,rate2=r2,rate3=r3,
                 rate4=r4,rate5=r5,rate6=r6,
                 tchange=c(0,3),type=1,eps=1.0e-2)
pwefun

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