Design and Monitoring of Survival Trials Accounting for Complex Situations

Description

Calculates various functions needed for design and monitoring survival trials accounting for complex situations such as delayed treatment effect, treatment crossover, non-uniform accrual, and different censoring distributions between groups. The event time distribution is assumed to be piecewise exponential (PWE) distribution and the entry time is assumed to be piecewise uniform distribution. As compared with Version 1.2.1, two more types of hybrid crossover are added. A bug is corrected in the function "pwecx" that calculates the crossover-adjusted survival, distribution, density, hazard and cumulative hazard functions. Also, to generate the crossover-adjusted event time random variable, a more efficient algorithm is used and the output includes crossover indicators.

Details

The DESCRIPTION file:

Index of help topics:

PWEALL-package          Design and Monitoring of Survival Trials
                        Accounting for Complex Situations
cp                      Conditional power given observed log hazard
                        ratio
cpboundary              The stopping boundary based on the conditional
                        power criteria
cpstop                  The stopping probability based on the stopping
                        boundary
fourhr                  A utility functon
hxbeta                  A function to calculate the beta-smoothed
                        hazard rate
innercov                A utility function to calculate the inner
                        integration of the overall covariance
innervar                A utility function to calculate the inner
                        integration of the overall variance
instudyfindt            calculate the timeline in study when some or
                        all subjects have entered
ovbeta                  calculate the overall log hazard ratio
overallcov              calculate the overall covariance
overallcovp1            calculate the first part of the overall
                        covariance
overallcovp2            calculate the other parts of the overall
                        covariance
overallvar              calculate the overall variance
pwe                     Piecewise exponential distribution: hazard,
                        cumulative hazard, density, distribution,
                        survival
pwecx                   Various function for piecewise exponential
                        distribution with crossover effect
pwecxcens               Integration of the density of piecewise
                        exponential distribution with crossover effect
                        and the censoring function
pwecxpwu                Integration of the density of piecewise
                        exponential distribution with crossover effect,
                        censoring and recruitment function
pwecxpwufindt           calculate the timeline when certain number of
                        events accumulates
pwecxpwuforvar          calculate the utility function used for
                        varaince calculation
pwefv2                  A utility function
pwefvplus               A utility functon
pwepower                Calculating the powers of various the test
                        statistics for superiority trials
pwepowereq              Calculating the powers of various the test
                        statistics for equivalence trials
pwepowerfindt           Calculating the timepoint where a certain power
                        of the specified test statistics is obtained
pwepowerni              Calculating the powers of various the test
                        statistics for non-inferiority trials
pwesim                  simulating the test statistics
pwu                     Piecewise uniform distribution: distribution
qpwe                    Piecewise exponential distribution: quantile
                        function
qpwu                    Piecewise uniform distribution: quantile
                        function
rmstcov                 Calculation of the variance and covariance of
                        estimated restricted mean survival time
rmsth                   Estimate the restricted mean survival time
                        (RMST) and its variance from data
rmstpower               Calculate powers at different cut-points based
                        on difference of restricted mean survival times
                        (RMST)
rmstpowerfindt          Calculating the timepoint where a certain power
                        of mean difference of RMSTs is obtained
rmstsim                 simulating the restricted mean survival times
rmstutil                A utility function to calculate the true
                        restricted mean survival time (RMST) and its
                        variance account for delayed treatment,
                        discontinued treatment and non-uniform entry
rpwe                    Piecewise exponential distribution: random
                        number generation
rpwecx                  Piecewise exponential distribution with
                        crossover effect: random number generation
rpwu                    Piecewise uniform distribution: random number
                        generation
spf                     A utility function
wlrcal                  A utility function to calculate the weighted
                        log-rank statistics and their varainces given
                        the weights
wlrcom                  A function to calculate the various weighted
                        log-rank statistics and their varainces
wlrutil                 A utility function to calculate some common
                        functions in contructing weights

There are 5 types of crossover considered in the package: (1) Markov crossover, (2) Semi-Markov crosover, (3) Hybrid crossover-1, (4) Hybrid crossover-2 and (5) Hybrid crossover-3. The first 3 types are described in Luo et al. (2018). The fourth and fifth types are added for Version 1.3.0. The crossover type is determined by the hazard function after crossover \lambda_2^{\bf x}(t\mid u). For Type (1), the Markov crossover,

\lambda_2^{\bf x}(t\mid u)=\lambda_2(t).

For Type (2), the Semi-Markov crossover,

\lambda_2^{\bf x}(t\mid u)=\lambda_2(t-u).

For Type (3), the hybrid crossover-1,

\lambda_2^{\bf x}(t\mid u)=\pi_2\lambda_2(t-u)+(1-\pi_2)\lambda_4(t).

For Type (4), the hazard after crossover is

\lambda_2^{\bf x}(t\mid u)=\frac{\pi_2\lambda_2(t-u)S_2(t-u)+(1-\pi_2)\lambda_4(t)S_4(t)/S_4(u)}{\pi_2 S_2(t-u)+(1-\pi_2)S_4(t)/S_4(u)}.

For Type (5), the hazard after crossover is

\lambda_2^{\bf x}(t\mid u)=\frac{\pi_2\lambda_2(t-u)S_2(t-u)+(1-\pi_2)\lambda_4(t-u)S_4(t-u)}{\pi_2 S_2(t-u)+(1-\pi_2)S_4(t-u)}.

The types (4) and (5) are more closely related to "re-randomization", i.e. when a patient crosses, (s)he will have probability \pi_2 to have hazard \lambda_2 and probability 1-\pi_2 to have hazard \lambda_4. The types (4) and (5) differ in having \lambda_4 as Markov or Semi-markov.

Author(s)

Xiaodong Luo [aut, cre], Xuezhou Mao [ctb], Xun Chen [ctb], Hui Quan [ctb], Sanofi [cph]

Maintainer: Xiaodong Luo <Xiaodong.Luo@sanofi.com>

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>.

Conditional power given observed log hazard ratio

Description

This will calculate the conditional power given the observed log hazard ratio based on Cox model

Usage

cp(Dplan=300,alpha=0.05,two.sided=1,pi1=0.5,Obsbeta=log(seq(1,0.6,by=-0.01)),
   BetaD=log(0.8),Beta0=log(1),prop=seq(0.1,0.9,by=0.1))

Arguments

Details

This is to calculated conditional power at time point when certain percent of target number of event has been observed and an observed log hazard ratio is provided.

Value

Note

This will calculate the conditional power given the observed log hazard ratio based on Cox model

Author(s)

Xiaodong Luo

References

Halperin, Lan, Ware, Johnson and DeMets (1982). Controlled Clinical Trials.

See Also

cpboundary,cpstop

Examples

###Calculate the CP at 10-90 percent of the target 300 events when the observed HR 
###are seq(1,0.6,by=-0.01) with 2:1 allocation 
###ratio between the treatment group and the control group
cp(pi1=2/3)

The stopping boundary based on the conditional power criteria

Description

This will calculate the stopping boundary based on the conditional power criteria, i.e. if observed HR is above the boundary, the conditional power will be lower than the designated level. All the calculation is based on the proportional hazards assumption and the Cox model.

Usage

cpboundary(Dplan=300,alpha=0.05,two.sided=1,pi1=0.5,cpcut=c(0.2,0.3,0.4),
            BetaD=log(0.8),Beta0=log(1),prop=seq(0.1,0.9,by=0.1))

Arguments

Details

This will calculate the stopping boundary based on the conditional power criteria, i.e. if observed HR is above the boundary, the conditional power will be lower than the designated level. All the calculation is based on the proportional hazards assumption and the Cox model.

Value

Note

This will calculate the stopping boundary based on the conditional power criteria

Author(s)

Xiaodong Luo

References

Halperin, Lan, Ware, Johnson and DeMets (1982). Controlled Clinical Trials.

See Also

cp,cpstop

Examples

###Calculate the stopping boundary at 10-90 percent of the target 300 events 
###when the condition power are c(0.2,0.3,0.4) with 
###2:1 allocation ratio between the treatment group and the control group
cpboundary(pi1=2/3)

The stopping probability based on the stopping boundary

Description

This will calculate the stopping probability given the stopping boundary. All the calculation is based on the proportional hazards assumption and the Cox model.

Usage

cpstop(Dplan=300,pi1=0.5,Beta1=log(0.8),Beta0=log(1),
        prop=seq(0.1,0.9,by=0.1),HRbound=rep(0.85,length(prop)))

Arguments

Details

This will calculate the stopping probability given the stopping boundary. All the calculation is based on the proportional hazards assumption and the Cox model.

Value

Note

This will calculate the stopping probability given the stopping boundary

Author(s)

Xiaodong Luo

References

Halperin, Lan, Ware, Johnson and DeMets (1982). Controlled Clinical Trials.

See Also

cp,cpboundary

Examples

###Calculate the stopping boundary at 10-90 percent of the target 300 events 
###when the condition power are c(0.2,0.3,0.4) with 2:1 allocation ratio 
###between the treatment group and the control group, we pick the boundary 
###based on 20 percent conditional power according to design, i.e. under alternative
targetD<-800 ###target number of events at study end
#############Allocation prob for the treatment group#############
pi1<-2/3
propevent<-seq(0.1,0.9,by=0.1) ###proportion of events at interim
HRbound<-cpboundary(Dplan=targetD,pi1=pi1,prop=propevent)$CPDbound[,1]  ###picking a boundary
pa<-cpstop(pi1=pi1,HRbound=HRbound)    ###stopping probabilities under null and alternative  
pa

###Calculate the stopping probability under non-constant hazard ratio
n1<-length(propevent)

####time point at which hazard rates and hazard ratios change
tchange<-c(0,6,12,24)                       
###annual event rates=0.09(1st yr), 0.07(2nd yr) and 0.05(2+yr) for control
ratet<-c(0.09/12,0.09/12,0.07/12,0.05/12)   
###annual censoring rate=0%(1st yr) and 1.5%(after) for control and treatment
ratec0<-c(0/12,0/12,0.015/12,0.015/12)      
ratec1<-ratec0                              
###annual treatment discontinuation rate=4% (1st yr) and 3% (after)
rate31<-c(0.04/12,0.04/12,0.03/12,0.03/12)  
rate30<-rep(0,length(tchange))              

############Recruitment curve##################
oa<-c(100,200,300,300,400,400,400,400,400,400,400,400,300,200)
ntotal<-sum(oa)
ntotal

taur<-length(oa)
ut<-seq(1,taur,by=1)
u<-oa/ntotal


#############Type-1 error rate#############
alpha<-0.05

####null hypothesis
eta0<-log(1)

####constant HR
etac<-log(0.8)

####non-constant HR
eta<-c(log(1),log(0.75),log(0.75),log(0.75)) ###6-m delayed 


####target number of events where calculations are performed##############
sevent<-propevent*targetD
nse<-length(sevent)
xtimeline<-xbeta<-xvar<-pxstop<-matrix(0,ncol=2,nrow=nse)
xtimeline[,1]<-xbeta[,1]<-xvar[,1]<-pxstop[,1]<-sevent
i<-1
tbegin<-proc.time()
for (i in 1:nse){
###find timeline
xtimeline[i,2]<-pwecxpwufindt(target=sevent[i],ntotal=ntotal,
                taur=taur,u=u,ut=ut,pi1=0.5,
                rate11=exp(eta)*ratet,rate21=exp(eta)*ratet,rate31=rate31,ratec1=ratec1,
                rate10=ratet,rate20=ratet,rate30=rate30,ratec0=ratec0,
                tchange=tchange,eps=0.001,init=taur,epsilon=0.000001,maxiter=100)$tau1

#Overall hazard ratio and varaince
xbeta[i,2]<-ovbeta(tfix=xtimeline[i,2],taur=taur,u=u,ut=ut,pi1=pi1,
                rate11=exp(eta)*ratet,rate21=exp(eta)*ratet,rate31=rate31,ratec1=ratec1,
                rate10=ratet,rate20=ratet,rate30=rate30,ratec0=ratec0,
                tchange=tchange,eps=0.001,veps=0.001,epsbeta=1.0e-10)$b1
xvar[i,2]<-overallvar(tfix=xtimeline[i,2],taur=taur,u=u,ut=ut,pi1=pi1,
                rate11=exp(eta)*ratet,rate21=exp(eta)*ratet,rate31=rate31,ratec1=ratec1,
                rate10=ratet,rate20=ratet,rate30=rate30,ratec0=ratec0,
                tchange=tchange,eps=0.001,veps=0.001,beta=xbeta[i,2])$vbeta
}
##stopping prob
pxstop[,2]<-1-pnorm(sqrt(ntotal)*(log(HRbound)-xbeta[,2])/sqrt(xvar[,2]))
tend<-proc.time()

xout<-cbind(xtimeline[,1],xtimeline[,2],xbeta[,2],xvar[,2]/ntotal,
            1/pi1/(1-pi1)/xtimeline[,1],pxstop[,2],pa$pstop0,pa$pstop1)
xnames<-c("# of events", "Time", "Estbeta", "TrueV", "ApproxV", "NCHR", "Null", "CHR")
colnames(xout)<-xnames
options(digits=2)
xout

A utility functon

Description

This will calculate the more complex integration

Usage

fourhr(t=seq(0,5,by=0.5),rate1=c(0,5,0.8),rate2=rate1,
                   rate3=c(0.1,0.2),rate4=rate2,tchange=c(0,3),eps=1.0e-2)

Arguments

Details

Let h_1,\ldots,h_4 correspond to rate1,...,rate4, and H_1,\ldots,H_4 be the corresponding survival functions. We calculate

\int_0^t h_1(s)H_2(s)h_3(t-s)H_4(t-s)ds.

Value

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)
tchange<-c(0,1.75)
fourhrfun<-fourhr(t=seq(0,5,by=0.5),rate1=r1,rate2=r2,rate3=r3,
                 rate4=r4,tchange=c(0,3),eps=1.0e-2)
fourhrfun

A function to calculate the beta-smoothed hazard rate

Description

A function to calculate the beta-smoothed hazard rate

Usage

hxbeta(x=c(0.5,1),y=seq(.1,1,by=0.01),d=rep(1,length(y)),
           tfix=2,K=20,eps=1.0e-06)

Arguments

Details

V1:3/21/2018

Value

Author(s)

Xiaodong Luo

Examples

n<-200
taur<-2.8
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
tfix<-taur+2
tseq<-seq(0,tfix,by=0.1)
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)
tchange<-c(0,1.873)

E<-T<-C<-d<-rep(0,n)
E<-rpwu(nr=n,u=u,ut=ut)$r
C<-rpwe(nr=n,rate=rc1,tchange=tchange)$r
T<-rpwecx(nr=n,rate1=r11,rate2=r21,rate3=r31,
               rate4=r41,rate5=r51,tchange=tchange,type=1)$r
y<-pmin(pmin(T,C),tfix-E)
y1<-pmin(C,tfix-E)
d[T<=y]<-1

lambda=hxbeta(x=tseq,y=y,d=d,tfix=tfix,K=20,eps=1.0e-06)$lambda
lambda

A utility function to calculate the inner integration of the overall covariance

Description

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

Usage

innercov(tupp=seq(0,10,by=0.5),tlow=tupp-0.1,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=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)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

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

pwe,rpwe,qpwe,pwecx,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)
getinner<-innercov(tupp=rep(5,times=11),tlow=seq(0,5,by=0.5),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=0.5)
cbind(getinner$qf1,getinner$qf0)

A utility function to calculate the inner integration of the overall variance

Description

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

Usage

innervar(t=seq(0,10,by=0.5),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=rate11,rate20=rate10,rate30=rate31,
                     rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
                     tchange=c(0,1),type1=1,type0=1,
                     rp21=0.5,rp20=0.5,
                     eps=1.0e-2,veps=1.0e-2,beta=0)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

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

pwe,rpwe,qpwe,pwecx,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)
getinner<-innervar(t=seq(0,10,by=0.5),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=0.5)
cbind(getinner$qf1,getinner$qf0)

calculate the timeline in study when some or all subjects have entered

Description

This will calculate the timeline from some timepoint in study when some/all subjects have entered accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

instudyfindt(target=400,y=exp(rnorm(300)),z=rbinom(300,1,0.5),
                  d=rep(c(0,1,2),each=100),
                  tcut=2,blinded=1,type0=1,type1=type0,
                  rp20=0.5,rp21=0.5,tchange=c(0,1),
                  rate10=c(1,0.7),rate20=c(0.9,0.7),rate30=c(0.4,0.6),rate40=rate20,
                  rate50=rate20,ratec0=c(0.3,0.3),
                  rate11=rate10,rate21=rate20,rate31=rate30,
                  rate41=rate40,rate51=rate50,ratec1=ratec0,
                  withmorerec=1,
                  ntotal=1000,taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),pi1=0.5,
                  ntype0=1,ntype1=1,
                  nrp20=0.5,nrp21=0.5,ntchange=c(0,1),
                  nrate10=rate10,nrate20=rate20,nrate30=rate30,nrate40=rate40,
                  nrate50=rate50,nratec0=ratec0,
                  nrate11=rate10,nrate21=rate20,nrate31=rate30,nrate41=rate40,
                  nrate51=rate50,nratec1=ratec0,
                  eps=1.0e-2,init=tcut*1.1,epsilon=0.001,maxiter=100)

Arguments

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. The hazard functions corresponding to nrate11,...,nrate51,nratec1, nrate10,...,nrate50,nratec0 are all piecewise constant functions and all must have the same ntchange.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,pwecxpwufindt

Examples

n<-1000
target<-550
ntotal<-1000
pi1<-0.5
taur<-2.8
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)
tchange<-c(0,1.873)
tcut<-2

####generate the data
E<-T<-C<-Z<-delta<-rep(0,n)
E<-rpwu(nr=n,u=u,ut=ut)$r
Z<-rbinom(n,1,pi1)
n1<-sum(Z)
n0<-sum(1-Z)
C[Z==1]<-rpwe(nr=n1,rate=rc1,tchange=tchange)$r
C[Z==0]<-rpwe(nr=n0,rate=rc0,tchange=tchange)$r
T[Z==1]<-rpwecx(nr=n1,rate1=r11,rate2=r21,rate3=r31,
                rate4=r41,rate5=r51,tchange=tchange,type=1)$r
T[Z==0]<-rpwecx(nr=n0,rate1=r10,rate2=r20,rate3=r30,
                rate4=r40,rate5=r50,tchange=tchange,type=1)$r
y<-pmin(pmin(T,C),tcut-E)
y1<-pmin(C,tcut-E)
delta[T<=y]<-1
delta[C<=y]<-0
delta[tcut-E<=y & tcut-E>0]<-2
delta[tcut-E<=y & tcut-E<=0]<--1

ys<-y[delta>-1]
Zs<-Z[delta>-1]
ds<-delta[delta>-1]

nplus<-sum(delta==-1)
nd0<-sum(ds==0)
nd1<-sum(ds==1)
nd2<-sum(ds==2)


ntaur<-taur-tcut
nu<-c(1/ntaur,1/ntaur)
nut<-c(ntaur/2,ntaur)

###calculate the timeline at baseline
xt<-pwecxpwufindt(target=target,ntotal=n,taur=taur,u=u,ut=ut,pi1=pi1,
              rate11=r11,rate21=r21,rate31=r31,ratec1=rc1,
              rate10=r10,rate20=r20,rate30=r30,ratec0=rc0,
              tchange=tchange,eps=0.001,init=taur,epsilon=0.000001,maxiter=100)
###calculate the timeline in study
yt<-instudyfindt(target=target,y=ys,z=Zs,d=ds,
                       tcut=tcut,blinded=0,type1=1,type0=1,tchange=tchange,
                       rate10=r10,rate20=r20,rate30=r30,ratec0=rc0,
                       rate11=r11,rate21=r21,rate31=r31,ratec1=rc1,
                       withmorerec=1,
                       ntotal=nplus,taur=ntaur,u=nu,ut=nut,pi1=pi1,
                       ntype1=1,ntype0=1,ntchange=tchange,
                       nrate10=r10,nrate20=r20,nrate30=r30,nratec0=rc0,
                       nrate11=r11,nrate21=r21,nrate31=r31,nratec1=rc1,
                       eps=1.0e-2,init=2,epsilon=0.001,maxiter=100)
##timelines                       
c(yt$t1,xt$t1)
##standard errors of the timeline estimators 
c(sqrt(yt$tvar/yt$ny),sqrt(xt$tvar/n))
###95 percent CIs
c(yt$t1-1.96*sqrt(yt$tvar/yt$ny),yt$t1+1.96*sqrt(yt$tvar/yt$ny))
c(xt$t1-1.96*sqrt(xt$tvar/n),xt$t1+1.96*sqrt(xt$tvar/n))

calculate the overall log hazard ratio

Description

This will calculate the overall (log) hazard ratio accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

ovbeta(tfix=2.0,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=rate11,rate20=rate10,rate30=rate31,rate40=rate20,
       rate50=rate20,ratec0=c(0.4,0.3),
       tchange=c(0,1),type1=1,type0=1,
       rp21=0.5,rp20=0.5,
       eps=1.0e-2,veps=1.0e-2,
       beta0=0,epsbeta=1.0e-4,iterbeta=25)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe

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)
getbeta<-ovbeta(tfix=2.0,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,beta0=0,epsbeta=1.0e-4,iterbeta=25)
getbeta$b1

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)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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

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)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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)
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

calculate the other parts of the overall covariance

Description

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

Usage

overallcovp2(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)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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)
getcov2<-overallcovp2(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)
getcov2

calculate the overall variance

Description

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

Usage

overallvar(tfix=2.0,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=rate11,rate20=rate10,rate30=rate31,
                     rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
                     tchange=c(0,1),type1=1,type0=1,
                     rp21=0.5,rp20=0.5,
                     eps=1.0e-2,veps=1.0e-2,beta=0)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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)
###variance with beta=0, calculate log-rank variance under the alternative
vbeta0<-overallvar(tfix=2.0,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)

###variance with beta=0, calculate log-rank variance under the alternative
###Estimate the overall beta
getbeta<-ovbeta(tfix=2.0,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,beta0=0,
        epsbeta=1.0e-4,iterbeta=25)
vbeta<-overallvar(tfix=2.0,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=getbeta$b1)
cbind(vbeta0$vs,vbeta$vs)

Piecewise exponential distribution: hazard, cumulative hazard, density, distribution, survival

Description

This will provide the related functions of the specified piecewise exponential distribution.

Usage

pwe(t=seq(0,5,by=0.5),rate=c(0,5,0.8),tchange=c(0,3))

Arguments

Details

Let \lambda(t)=\sum_{j=1}^m \lambda_j I(t_{j-1}\le t<t_j) be the hazard function, where \lambda_1,\ldots,\lambda_m are the corresponding elements of rate and t_0,\ldots,t_{m-1} are the corresponding elements of tchange, t_m=\infty. The cumulative hazard function

\Lambda(t)=\sum_{j=1}^m \lambda_j(t\wedge t_j-t\wedge t_{j-1}),

the survival function S(t)=\exp\{-\Lambda(t)\}, the distribution function F(t)=1-S(t) and the density function f(t)=\lambda(t)S(t).

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

rpwe,qpwe

Examples

t<-seq(0,3,by=0.1)
rate<-c(0.6,0.3)
tchange<-c(0,1.75)
pwefun<-pwe(t=t,rate=rate,tchange=tchange)
pwefun

Various function for piecewise exponential distribution with crossover effect

Description

This will calculate the functions according to the piecewise exponential distribution with crossover

Usage

pwecx(t=seq(0,10,by=0.5),rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),
      rate4=rate2,rate5=rate2,tchange=c(0,1),type=1,rp2=0.5,eps=1.0e-2)

Arguments

Details

More details

Value

Note

This provides a random number generator of the piecewise exponetial distribution with crossover

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)
pwecxfun<-pwecx(t=seq(0,10,by=0.5),rate1=r1,rate2=r2,rate3=r3,rate4=r4,
                rate5=r5,tchange=c(0,1),type=1,eps=1.0e-2)
pwecxfun$surv

Integration of the density of piecewise exponential distribution with crossover effect and the censoring function

Description

This will calculate the functions according to the piecewise exponential distribution with crossover

Usage

pwecxcens(t=seq(0,10,by=0.5),rate1=c(1,0.5),rate2=rate1,
                rate3=c(0.7,0.4),rate4=rate2,rate5=rate2,ratec=c(0.2,0.3),
                tchange=c(0,1),type=1,rp2=0.5,eps=1.0e-2)

Arguments

Details

This is to calculate the function (and its derivative)

\xi(t)=\int_0^t \widetilde{f}(s)S_C(s)ds,

where S_C is the piecewise exponential survival function of the censoring time, defined by tchange and ratec, and \widetilde{f} is the density for the event distribution subject to crossover defined by tchange, rate1 to rate5 and type.

Value

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

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)
rc<-c(0.5,0.6)
exu<-pwecxcens(t=seq(0,10,by=0.5),rate1=r1,rate2=r2,
               rate3=r3,rate4=r4,rate5=r5,ratec=rc,
               tchange=c(0,1),type=1,eps=1.0e-2)
c(exu$du,exu$duprime)

Integration of the density of piecewise exponential distribution with crossover effect, censoring and recruitment function

Description

This will calculate the functions according to the piecewise exponential distribution with crossover

Usage

pwecxpwu(t=seq(0,10,by=0.5),taur=5,
        u=c(1/taur,1/taur),ut=c(taur/2,taur),
        rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),
        rate4=rate2,rate5=rate2,ratec=c(0.5,0.6),
        tchange=c(0,1),type=1,rp2=0.5,eps=1.0e-2)

Arguments

Details

This is to calculate the function (and its derivative)

\xi(t)=\int_0^t G_E(t-s)\widetilde{f}(s)S_C(s)ds,

where G_E is the accrual function defined by taur, u and ut, S_C is the piecewise exponential survival function of the censoring time, defined by tchange and ratec, and \widetilde{f} is the density for the event distribution subject to crossover defined by tchange, rate1 to rate5 and type.

Value

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

rpwe

Examples

taur<-2
u<-c(0.6,0.4)
ut<-c(1,2)
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)
rc<-c(0.5,0.6)
exu<-pwecxpwu(t=seq(0,10,by=0.5),taur=taur,u=u,ut=ut,
        rate1=r1,rate2=r2,rate3=r3,rate4=r4,rate5=r5,ratec=rc,
        tchange=c(0,1),type=1,eps=1.0e-2)
c(exu$du,exu$duprime)

calculate the timeline when certain number of events accumulates

Description

This will calculate the timeline from study inception accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

pwecxpwufindt(target=400,ntotal=1000,taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),pi1=0.5,
                         rate11=c(1,0.5),rate21=c(0.8,0.9),rate31=c(0.7,0.4),
                         rate41=rate21,rate51=rate21,ratec1=c(0.5,0.6),
                         rate10=c(1,0.7),rate20=c(0.9,0.7),rate30=c(0.4,0.6),
                         rate40=rate20,rate50=rate20,ratec0=c(0.3,0.3),
                         tchange=c(0,1),type1=1,type0=1,
                         rp21=0.5,rp20=0.5,eps=1.0e-2,
                         init=taur,epsilon=0.000001,maxiter=100)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

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

pwe,rpwe,qpwe,instudyfindt

Examples

target<-400
ntotal<-2000
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)
gettimeline<-pwecxpwufindt(target=target,ntotal=ntotal,
                taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),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,init=taur,epsilon=0.000001,maxiter=100)
gettimeline$t1

calculate the utility function used for varaince calculation

Description

This is a utility function to calculate the overall variance accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

pwecxpwuforvar(tfix=10,t=seq(0,10,by=0.5),taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),
    rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),rate4=rate2,rate5=rate2,ratec=c(0.5,0.6),
         tchange=c(0,1),type=1,rp2=0.5,eps=1.0e-2)

Arguments

Details

This is to calculate the function

B_l(t,s)=\int_0^s x^l G_E(t-x)\widetilde{f}(x)S_C(x)dx,

where G_E is the accrual function defined by taur, u and ut, S_C is the piecewise exponential survival function of the censoring time, defined by tchange and ratec, and \widetilde{f} is the density for the event distribution subject to crossover defined by tchange, rate1 to rate5 and type. This function is useful when calculating the overall varaince and covariance.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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)
getf<-pwecxpwuforvar(tfix=3,t=seq(0,3,by=1),taur=taur,u=u,ut=ut,
                 rate1=r11,rate2=r21,rate3=r31,rate4=r41,rate5=r51,ratec=rc1,
                 tchange=c(0,1),type=1,eps=1.0e-2)
getf

A utility function

Description

This will $int_0^t s^k lambda_1(s)S_2(s)ds$ where k=0,1,2 and rate1=lambda_1 and S_2 has hazard rate2

Usage

pwefv2(t=seq(0,5,by=0.5),rate1=c(0,5,0.8),
      rate2=rate1,tchange=c(0,3),eps=1.0e-2)

Arguments

Details

Let h_1,h_2 correspond to rate1,rate2, and H_1,H_2 be the corresponding survival functions. This function will calculate

\int_0^t s^k h_1(s)H_2(s)ds,\hspace{1cm} k=0,1,2.

Value

Note

This will provide the number of events.

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)
tchange<-c(0,1.75)
pwefun<-pwefv2(t=seq(0,5,by=0.5),rate1=r1,rate2=r2,
              tchange=tchange,eps=1.0e-2)
pwefun

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

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

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

Calculating the powers of various the test statistics for superiority trials

Description

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

Usage

pwepower(t=seq(0.1,3,by=0.5),alpha=0.05,twosided=1,taur=1.2,
             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=rate11,rate20=rate10,rate30=rate31,
             rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
             tchange=c(0,1),type1=1,type0=1,rp21=0.5,rp20=0.5,
             eps=1.0e-2,veps=1.0e-2,epsbeta=1.0e-4,iterbeta=25,
             n=1000)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,ovbeta,innervar, pwepowerni,pwepowereq

Examples

t<-seq(3,6,by=1)
taur<-1.2
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
r11<-c(0.2,0.1)
r21<-r11
r31<-c(0.03,0.02)
r41<-r51<-r21
rc1<-c(0.01,0.02)
r10<-c(0.2,0.2)
r20<-r10
r30<-c(0.02,0.01)
r40<-r50<-r20
rc0<-c(0.02,0.01)
getpower<-pwepower(t=t,alpha=0.05,twosided=1,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,n=1000)
###powers at each time point
cbind(t,getpower$power[,c(2:4,12:14)])

Calculating the powers of various the test statistics for equivalence trials

Description

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

Usage

pwepowereq(t=seq(0.1,3,by=0.5),uppermargin=1.3,lowermargin=1/uppermargin,
           alpha=0.05,taur=1.2,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=rate11,rate20=rate10,rate30=rate31,
             rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
             tchange=c(0,1),type1=1,type0=1,
             rp21=0.5,rp20=0.5,eps=1.0e-2,veps=1.0e-2,
             epsbeta=1.0e-4,iterbeta=25,n=1000)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,ovbeta,innervar, pwepower,pwepowerni

Examples

t<-seq(3,6,by=1)
taur<-1.2
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
r11<-c(0.2,0.1)
r21<-r11
r31<-c(0.03,0.02)
r41<-r51<-r21
rc1<-c(0.01,0.02)
r10<-c(0.2,0.2)
r20<-r10
r30<-c(0.02,0.01)
r40<-r50<-r20
rc0<-c(0.02,0.01)
getpowereq<-pwepowereq(t=t,uppermargin=1.3,lowermargin=0.8,alpha=0.05,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,n=1000)
###powers at each time point
cbind(t,getpowereq$power[,1:3])

Calculating the timepoint where a certain power of the specified test statistics is obtained

Description

This will calculate the timepoint where a certain power of the specified test statistics is obtained accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

pwepowerfindt(power=0.9,alpha=0.05,twosided=1,tupp=5,tlow=1,taur=1.2,
                     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=rate11,rate20=rate10,rate30=rate31,
                     rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
                     tchange=c(0,1),type1=1,type0=1,
                     rp21=0.5,rp20=0.5,eps=1.0e-2,veps=1.0e-2,
                     epsbeta=1.0e-04,iterbeta=25,
                     n=1000,testtype=1,maxiter=20,itereps=0.001)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,ovbeta,innervar

Examples

t<-seq(3,6,by=1)
taur<-1.2
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
r11<-c(0.2,0.1)
r21<-r11
r31<-c(0.03,0.02)
r41<-r51<-r21
rc1<-c(0.01,0.02)
r10<-c(0.2,0.2)
r20<-r10
r30<-c(0.02,0.01)
r40<-r50<-r20
rc0<-c(0.02,0.01)
getpower<-pwepower(t=t,alpha=0.05,twosided=1,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,n=1000)
###powers at each time point
cbind(t,getpower$power[,1:3])

###90% power should be in (3,3.5)
getpwtime<-pwepowerfindt(power=0.9,alpha=0.05,twosided=1,tupp=3.5,tlow=3,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,n=1000,testtype=1,maxiter=30)
getpwtime

Calculating the powers of various the test statistics for non-inferiority trials

Description

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

Usage

pwepowerni(t=seq(0.1,3,by=0.5),nimargin=1.3,alpha=0.05,twosided=0,taur=1.2,
           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=rate11,rate20=rate10,rate30=rate31,
           rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
           tchange=c(0,1),type1=1,type0=1,
           rp21=0.5,rp20=0.5,eps=1.0e-2,veps=1.0e-2,
           epsbeta=1.0e-4,iterbeta=25,n=1000)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,ovbeta,innervar, pwepower,pwepowereq

Examples

t<-seq(3,6,by=1)
taur<-1.2
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
r11<-c(0.2,0.1)
r21<-r11
r31<-c(0.03,0.02)
r41<-r51<-r21
rc1<-c(0.01,0.02)
r10<-c(0.2,0.2)
r20<-r10
r30<-c(0.02,0.01)
r40<-r50<-r20
rc0<-c(0.02,0.01)
getpowerni<-pwepowerni(t=t,nimargin=1.3,alpha=0.05,twosided=1,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,n=1000)
###powers at each time point
cbind(t,getpowerni$power[,1:3])

simulating the test statistics

Description

This will simulate the test statistics accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

pwesim(t=seq(1,2,by=0.1),taur=1.2,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=rate11,rate20=rate10,rate30=rate31,
                     rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
                     tchange=c(0,1),type1=1,type0=1,
                     rp21=0.5,rp20=0.5,
                     n=1000,rn=200,testtype=c(1,2,3,4))

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

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)
ar<-pwesim(t=seq(1,2,by=0.1),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,
        n=300,rn=10)

Piecewise uniform distribution: distribution

Description

This will calculate the distribution function of the piecewise uniform distribution

Usage

pwu(t=seq(0,1,by=0.1),u=c(0,5,0.5),ut=c(1,2))

Arguments

Details

Let f(t)=\sum_{j=1}^m u_j I(t_{j-1}<t\le t_j) be the density function, where u_1,\ldots,u_m are the corresponding elements of u and t_1,\ldots,t_{m} are the corresponding elements of ut and t_0=0. The distribution function

F(t)=\sum_{j=1}^m u_j(t\wedge t_j-t\wedge t_{j-1}).

User must make sure that \sum_{j=1}^m u_j (t_j-t_{j-1})=1 before using this function.

Value

Note

This provides distribution of the piecewise uniform distribution

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe

Examples

t<-seq(-1,3,by=0.5)
u<-c(0.6,0.4)
ut<-c(1,2)
pwud<-pwu(t=t,u=u,ut=ut)
pwud

Piecewise exponential distribution: quantile function

Description

This will provide the quantile function of the specified piecewise exponential distribution

Usage

qpwe(p=seq(0,1,by=0.1),rate=c(0,5,0.8),tchange=c(0,3))

Arguments

Details

More details

Value

Note

This provides the quantile function related to the piecewise exponetial distribution

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

piecewise exponential

Examples

p<-seq(0,1,by=0.1)
rate<-c(0.6,0.3)
tchange<-c(0,1.75)
pweq<-qpwe(p=p,rate=rate,tchange=tchange)
pweq

Piecewise uniform distribution: quantile function

Description

This will provide the quantile function of the specified piecewise uniform distribution

Usage

qpwu(p=seq(0,1,by=0.1),u=c(0,5,0.5),ut=c(1,2))

Arguments

Details

Let f(t)=\sum_{j=1}^m u_j I(t_{j-1}<t\le t_j) be the density function, where u_1,\ldots,u_m are the corresponding elements of u and t_1,\ldots,t_{m} are the corresponding elements of ut and t_0=0. The distribution function

F(t)=\sum_{j=1}^m u_j(t\wedge t_j-t\wedge t_{j-1}).

User must make sure that \sum_{j=1}^m u_j (t_j-t_{j-1})=1 before using this function.

Value

Note

This provides the quantile function related to the piecewise uniform distribution

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

piecewise uniform

Examples

p<-seq(0,1,by=0.1)
u<-c(0.6,0.4)
ut<-c(1,2)
pwuq<-qpwu(p=p,u=u,ut=ut)
pwuq

Calculation of the variance and covariance of estimated restricted mean survival time

Description

A function to calculate the variance and covariance of estimated restricted mean survival time using data from different cut-off points accounting for delayed treatment, discontinued treatment and non-uniform entry

Usage

rmstcov(t1cut=2.0,t1study=2.5,t2cut=3.0,t2study=3.5,taur=5,
        u=c(1/taur,1/taur),ut=c(taur/2,taur),
        rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),
        rate4=rate2,rate5=rate2,ratec=c(0.5,0.6),
        tchange=c(0,1),type=1,rp2=0.5,
        eps=1.0e-2,veps=1.0e-2)

Arguments

Details

More details

Value

Note

This calculates the "true" variance and covariance of restricted mean survival times

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>.

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)
rc<-c(0.1,0.1)
rmcov<-rmstcov(t1cut=2.0,t1study=2.5,t2cut=3.0,t2study=3.5,taur=5,
        rate1=r1,rate2=r2,rate3=r3,rate4=r4,rate5=r5,ratec=rc,
        tchange=c(0,1),type=1)
rmcov

Estimate the restricted mean survival time (RMST) and its variance from data

Description

A function to estimate the restricted mean survival time (RMST) and its variance from data

Usage

rmsth(y=c(1,2,3),d=c(1,1,0),tcut=2.0,eps=1.0e-08)

Arguments

Details

More details

Value

Note

This estimates the restricted mean survival time and its asymptotic variance

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

Examples

lamt<-0.8
lamc<-0.4
n<-3000
tcut<-2.0
truermst<-(1-exp(-lamt*tcut))/lamt
tt<-rexp(n)/lamt
cc<-rexp(n)/lamc
yy<-pmin(tt,cc)
dd<-rep(1,n)
dd[tt>cc]<-0
aest<-rmsth(y=yy,d=dd,tcut=tcut)
aest

Calculate powers at different cut-points based on difference of restricted mean survival times (RMST)

Description

A function to calculate powers at different cut-points based on difference of restricted mean survival times (RMST) account for delayed treatment, discontinued treatment and non-uniform entry

Usage

rmstpower(tcut=2,tstudy=seq(tcut,tcut+2,by=0.5),alpha=0.05,twosided=1,
          taur=1.2,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=rate11,rate20=rate10,rate30=rate31,
          rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
          tchange=c(0,1),type1=1,type0=1,rp21=0.5,rp20=0.5,
          eps=1.0e-2,veps=1.0e-2,n=1000)

Arguments

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.

Value

Note

This calculates the restricted mean survival times between the treatment and control groups and their standard errors

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

Examples

tcut<-3.0
tstudy<-seq(3,6,by=1)
taur<-1.2
u<-c(1/taur,1/taur)
ut<-c(taur/2,taur)
r11<-c(0.2,0.1)
r21<-r11
r31<-c(0.03,0.02)
r41<-r51<-r21
rc1<-c(0.01,0.02)
r10<-c(0.2,0.2)
r20<-r10
r30<-c(0.02,0.01)
r40<-r50<-r20
rc0<-c(0.02,0.01)
getrmst<-rmstpower(tcut=tcut,tstudy=tstudy,alpha=0.05,twosided=1,
          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,rp21=0.5,rp20=0.5,n=1000)
###powers at each time point
cbind(tstudy,getrmst$power)

Calculating the timepoint where a certain power of mean difference of RMSTs is obtained

Description

This will calculate the timepoint where a certain power of the mean difference of RMSTs is obtained accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

rmstpowerfindt(power=0.9,alpha=0.05,twosided=1,tcut=2,tupp=5,tlow=3.0,taur=1.2,
           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=rate11,rate20=rate10,rate30=rate31,
           rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
           tchange=c(0,1),type1=1,type0=1,
           rp21=0.5,rp20=0.5,eps=1.0e-2,veps=1.0e-2,
           n=1000,maxiter=20,itereps=0.001)

Arguments

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.

Value

Note

Version 1.0 (8/8/2017)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

pwe,rpwe,qpwe,ovbeta,innervar

Examples

tcut<-3.0
tstudy<-seq(3,6,by=0.2)
taur<-2
u<-c(0.3,0.7)
ut<-c(taur/2,taur)
r11<-c(0.2,0.1)
r21<-r11
r31<-c(0.03,0.02)
r41<-r51<-r21
rc1<-c(0.05,0.04)
r10<-c(0.22,0.22)
r20<-r10
r30<-c(0.02,0.01)
r40<-r50<-r20
rc0<-c(0.04,0.05)
ntotal<-1200
getrmst<-rmstpower(tcut=tcut,tstudy=tstudy,alpha=0.05,twosided=1,
        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,rp21=0.5,rp20=0.5,n=ntotal)
###powers at each time point
cbind(tstudy,getrmst$power)

###90 percent power should be in (3,4)
gettime<-rmstpowerfindt(power=0.9,alpha=0.05,twosided=1,tcut=tcut,tupp=4,tlow=3.0,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,rp21=0.5,rp20=0.5,eps=1.0e-2,veps=1.0e-2,
          n=ntotal,maxiter=20,itereps=0.0001)
gettime

simulating the restricted mean survival times

Description

This will simulate the test statistics accouting for staggered entry, delayed treatment effect, treatment crossover and loss to follow-up.

Usage

rmstsim(tcut=c(1,2),tstudy=tcut+0.2,taur=1.2,
        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=rate11,rate20=rate10,rate30=rate31,
        rate40=rate20,rate50=rate20,ratec0=c(0.6,0.5),
        tchange=c(0,1),type1=1,type0=1,rp21=0.5,rp20=0.5,
        n=1000,rn=200,eps=1.0E-08)

Arguments

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.

Value

Note

Version 1.0 (7/19/2016)

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

pwe,rpwe,qpwe,ovbeta

Examples

tcuta<-c(2,3)
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(1.5,0.7)
r20<-c(0.5,1)
r30<-c(0.3,0.4)
r40<-r50<-r20
rc0<-c(0.2,0.4)
ar<-rmstsim(tcut=tcuta,tstudy=tcuta+0.1,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,
            n=300,rn=200)
##Empirical power
apply(ar$outr>1.96,2,mean)        

A utility function to calculate the true restricted mean survival time (RMST) and its variance account for delayed treatment, discontinued treatment and non-uniform entry

Description

A utility function to calculate the true restricted mean survival time (RMST) and its variance account for delayed treatment, discontinued treatment and non-uniform entry

Usage

rmstutil(tcut=2.0,tstudy=5.0,taur=5,u=c(1/taur,1/taur),ut=c(taur/2,taur),
        rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),
        rate4=rate2,rate5=rate2,ratec=c(0.5,0.6),
        tchange=c(0,1),type=1,rp2=0.5,
        eps=1.0e-2,veps=1.0e-2)

Arguments

Details

More details

Value

Note

This calculates the "true" variance of restricted mean survival times

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>.

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)
rc<-c(0.1,0.1)
rmt<-rmstutil(tcut=2.0,tstudy=5.0,taur=5,
        rate1=r1,rate2=r2,rate3=r3,
        rate4=r4,rate5=r5,ratec=rc,
        tchange=c(0,1),type=1,rp2=0.5,
        eps=1.0e-2,veps=1.0e-2)
rmt

Piecewise exponential distribution: random number generation

Description

This will generate random numbers according to the specified piecewise exponential distribution

Usage

rpwe(nr=10,rate=c(0,5,0.8),tchange=c(0,3))

Arguments

Details

More details

Value

Note

This provides a random number generator of the piecewise exponetial distribution

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

piecewise exponential

Examples

nr<-10
rate<-c(0.6,0.3)
tchange<-c(0,1.75)
pwer<-rpwe(nr=nr,rate=rate,tchange=tchange)
pwer

Piecewise exponential distribution with crossover effect: random number generation

Description

This will generate random numbers according to the piecewise exponential distribution with crossover

Usage

rpwecx(nr=1,rate1=c(1,0.5),rate2=rate1,rate3=c(0.7,0.4),
rate4=rate2,rate5=rate2,tchange=c(0,1),type=1,rp2=0.5)

Arguments

Details

More details

Value

Note

This provides a random number generator of the piecewise exponetial distribution with crossover

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)
pwecxr<-rpwecx(nr=10,rate1=r1,rate2=r2,rate3=r3,rate4=r4,rate5=r5,tchange=c(0,1),type=1)
pwecxr$r

Piecewise uniform distribution: random number generation

Description

This will generate random numbers according to the specified piecewise uniform distribution

Usage

rpwu(nr=10,u=c(0,6,0.4),ut=c(1,2))

Arguments

Details

Let f(t)=\sum_{j=1}^m u_j I(t_{j-1}<t\le t_j) be the density function, where u_1,\ldots,u_m are the corresponding elements of u and t_1,\ldots,t_{m} are the corresponding elements of ut and t_0=0. The distribution function

F(t)=\sum_{j=1}^m u_j(t\wedge t_j-t\wedge t_{j-1}).

User must make sure that \sum_{j=1}^m u_j (t_j-t_{j-1})=1 before using this function.

Value

Note

This provides a random number generator of the piecewise uniform distribution

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

See Also

rpwe

Examples

nr<-10
u<-c(0.6,0.4)
ut<-c(1,2)
pwur<-rpwu(nr=nr,u=u,ut=ut)
pwur

A utility function

Description

A utility function to calculate a ratio.

Usage

spf(x=seq(-1,1,by=0.2),eps=1.0e-3)

Arguments

Details

This is to calculate

\Phi_l(x)=\frac{\int_0^x s^le^{-s}ds}{x^{l+1}},\hspace{0.5cm}l=0,1,2.

This function is well defined even when x=0. However, it is numerical chanllenging to calculate it when x is small. So when |x|\le \code{eps} we approximate this function and the absolute error is \code{eps}^5.

Value

Note

Version 1.0 (7/19/2016)

Author(s)

Xiaodong Luo

References

Luo, et al. (2017)

Examples

fun<-spf(x=seq(-1,1,by=0.2),eps=1.0e-3)
fun

A utility function to calculate the weighted log-rank statistics and their varainces given the weights

Description

A utility function to calculate the weighted log-rank statistics and their varainces given the weights

Usage

wlrcal(n=10,te=c(1,2,3),tfix=2.0,dd1=c(1,0,1),dd0=c(0,1,0),r1=c(1,2,3),r0=c(1,2,3),
       weights=matrix(1,nrow=length(te),ncol=1),eps=1.0e-08)

Arguments

Details

More details

Value

Author(s)

Xiaodong Luo

Examples

lr<-wlrcal(n=10,te=c(1,2,3),tfix=2.0,dd1=c(1,0,1),dd0=c(0,1,0),r1=c(1,2,3),r0=c(1,2,3))
lr

A function to calculate the various weighted log-rank statistics and their varainces

Description

A function to calculate the weighted log-rank statistics and their varainces given the weights including log-rank, gehan, Tarone-Ware, Peto-Peto, mPeto-Peto and Fleming-Harrington

Usage

wlrcom(y,d,z,tfix=max(y),p=c(1),q=c(1),eps=1.0e-08)

Arguments

Details

V1:3/21/2018

Value

Author(s)

Xiaodong Luo

Examples

n<-1000
pi1<-0.5
taur<-2.8
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)
tchange<-c(0,1.873)
tcut<-2

E<-T<-C<-z<-delta<-rep(0,n)
E<-rpwu(nr=n,u=u,ut=ut)$r
z<-rbinom(n,1,pi1)
n1<-sum(z)
n0<-sum(1-z)
C[z==1]<-rpwe(nr=n1,rate=rc1,tchange=tchange)$r
C[z==0]<-rpwe(nr=n0,rate=rc0,tchange=tchange)$r
T[z==1]<-rpwecx(nr=n1,rate1=r11,rate2=r21,rate3=r31,
                rate4=r41,rate5=r51,tchange=tchange,type=1)$r
T[z==0]<-rpwecx(nr=n0,rate1=r10,rate2=r20,rate3=r30,
                rate4=r40,rate5=r50,tchange=tchange,type=1)$r
y<-pmin(pmin(T,C),tcut-E)
y1<-pmin(C,tcut-E)
d<-rep(0,n);
d[T<=y]<-1

wlr4<-wlrcom(y=y,d=d,z=z,p=c(1,1),q=c(0,1))
wlr4

A utility function to calculate some common functions in contructing weights

Description

A utility function to calculate some common functions in contructing weights

Usage

wlrutil(y=c(1,2,3),d=c(1,0,1),z=c(1,0,0),te=c(1,3),eps=1.0e-08)

Arguments

Details

More details

Value

Author(s)

Xiaodong Luo

Examples

ww<-wlrutil(y=c(1,2,3),d=c(1,0,1),z=c(1,0,0),te=c(1,3),eps=1.0e-08)
ww