pwefvplus: A utility functon

In PWEALL: Design and Monitoring of Survival Trials Accounting for Complex Situations

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

PWEALL documentation built on Aug. 9, 2023, 9:08 a.m.