pwefvplus {PWEALL}

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)

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

Details

Let h_1,\ldots,h_6 correspond to rate1,...,rate6, and H_1,\ldots,H_6 be the corresponding survival functions. Also let \pi_2=\code{rp2}. 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)duds;

when type=2, we calculate

\int_0^t s^kH_6(s)\int_0^s h_3(u)H_1(u)H_3(u)h_2(s-u)H_2(s-u)duds;

when type=3, we calculate the sum of

\pi_2\int_0^t s^kH_4^{1-\pi_2}(s)H_6(s)\int_0^s h_3(u)H_1(u)H_3(u)h_2(s-u)H_2^{\pi_2}(s-u)/H_4^{1-\pi_2}(u)duds

and

(1-\pi_2)\int_0^t s^kh_4(s)H_4^{1-\pi_2}(s)H_6(s)\int_0^s h_3(u)H_1(u)H_3(u)H_2^{\pi_2}(s-u)/H_4^{1-\pi_2}(u)duds;

when type=4, we calculate the sum of

\pi_2\int_0^t s^kH_6(s)\int_0^s h_3(u)H_1(u)H_3(u)h_2(s-u)H_2(s-u)duds

and

(1-\pi_2)\int_0^t s^kh_4(s)H_4(s)H_6(s)\int_0^s h_3(u)H_1(u)H_3(u)/H_4(u)duds;

when type=5, we calculate the sum of

\pi_2\int_0^t s^kH_6(s)\int_0^s h_3(u)H_1(u)H_3(u)h_2(s-u)H_2(s-u)duds

and

(1-\pi_2)\int_0^t s^kH_6(s)\int_0^s h_3(u)H_1(u)H_3(u)h_4(s-u)H_4(s-u)duds.

Value

f0

values when k=0

f1

values when k=1

f2

values when k=2

Note

This provides the result of the complex integration

Author(s)

Xiaodong Luo

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>.

See Also

rpwe

Examples

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