Download S2 File. Scripts of R functions. 1. R code to compute Optimal

S2 File. Scripts of R functions.
1. R code to compute Optimal Simon design
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#######
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### Optimal.Simon.Plan allow to compute an Optimal two stage Simon plan
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### ARGUMENTS :
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### Here we denote
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# pi0 :
the minimum expected efficacy of the treatment
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# pi1 :
the desirable target level of treatment efficacy
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# alpha:
the type I error rate
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# beta:
the type II error rate
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# Nmax :
trial
the maximum number of patients to be included in the Phase II
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### VALUES :
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# a list :
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# $risk :
# $j :
PLOS
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final Simon Optimal design
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stage
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# $n:
the number of patients included at stage j (N1, N2)
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# $r:
the stopping boundaries at stage j (r1, r2)
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# $alpha :
the spending function of type I error rate
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# $beta :
the spending function of type II error rate
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# $pet0 :
probability of early termination under the null hypothesis
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# $pet1 : probability of early termination under the alternative
hypothesis
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# $design: all the selected design satisfing A(pi0)<alpha and
A(pi1)>1-beta
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Optimal.Simon.Plan<-function(pi0, pi1, alpha, beta, Nmax) {
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r12<-expand.grid(list(r1=0:Nmax, r2=0:Nmax))
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n12<-r12;names(n12)<-c("n1", "n2")
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n12<-n12[n12$n1+n12$n2<=Nmax & n12$n1>0 & n12$n2>0,]
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dim(n12)
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r12<-r12[r12$r1<r12$r2,]
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dim(r12)
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id<-0
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N<-Nmax
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enh0C<-Nmax
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n12<-n12[order(runif(dim(n12)[1])),]
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n<-dim(n12)[1]
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nid<-0
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while (n>0) {
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n1<-n12$n1[1]
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n2<-n12$n2[1]
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pi12<-expand.grid(list(k1=0:n1, k2=0:n2))
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pi12$pk1.0<-dbinom(pi12$k1, n1, pi0)
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pi12$pk2.0<-dbinom(pi12$k2, n2, pi0)
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pi12$pk1.1<-dbinom(pi12$k1, n1, pi1)
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pi12$pk2.1<-dbinom(pi12$k2, n2, pi1)
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pi12$pk12.0<-pi12$pk1.0*pi12$pk2.0
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pi12$pk12.1<-pi12$pk1.1*pi12$pk2.1
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r12c<-r12[r12$r1<n1 & r12$r2<n1+n2,]
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r12c<-r12c[order(runif(dim(r12c)[1])),]
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dim(r12c)[1]
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r12c$beta1<-pbinom(r12c$r1, n1, pi1)
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r12c<-r12c[r12c$beta1<beta,]
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dim(r12c)[1]
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r12c$pet1<-pbinom(r12c$r1, n1, pi0)
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r12c$enh0C<-n1+(1-r12c$pet1)*n2
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r12c<-r12c[r12c$enh0C<=enh0C,]
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dim(r12c)[1]
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nr12c<-dim(r12c)[1]
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while (nr12c>0) {
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r1<-r12c$r1[1]
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r2<-r12c$r2[1]
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beta1<-r12c$beta1[1]
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enh0<-r12c$enh0C[1]
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ind<-(pi12$k1>r1) & (pi12$k2>r2-pi12$k1)
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alpha2<-sum(pi12$pk12.0[ind])
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ind<-(pi12$k1>r1) & (pi12$k2<=r2-pi12$k1)
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beta2<-sum(pi12$pk12.1[ind])
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c(beta1, alpha2, beta2)
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PLOS
if (alpha2<=alpha & beta1+beta2<=beta) {
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if (enh0<enh0C) {
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id<-id+1
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enh0C<-enh0
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N<-n1+n2
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d<-c(n1, n2, r1, r2, 0, beta1, alpha2, beta2, enh0, N)
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if (id==1) {
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design<-d
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} else {
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design<-rbind(design, d)
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}
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} else {
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if (n1+n2<N) {
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id<-id+1
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N<-n1+n2
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d<-c(n1, n2, r1, r2, 0, beta1, alpha2, beta2, enh0, N)
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if (id==1) {
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design<-d
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} else {
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design<-rbind(design, d)
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}
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}
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}
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r12c<-r12c[-1,];nr12c<-nr12c-1
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} else {
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if (alpha2>alpha) {
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r12c<-r12c[r12c$r1!=r1 | (r12c$r1==r1 & r12c$r2>r2),]
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}
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if (beta1+beta2>beta) {
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r12c<-r12c[r12c$r1!=r1 | (r12c$r1==r1 & r12c$r2<r2),]
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}
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nr12c<-dim(r12c)[1]
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}
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}
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if (id>nid) {
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n12<-n12[n12$n1<=floor(enh0C),]
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n<-dim(n12)[1]
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nid<-id
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} else {
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n12<-n12[-1,];n<-n-1
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}
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}
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risk<-NULL
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if (id>0) {
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d<-design[id,]
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design<-data.frame(design)
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names(design)<-c("n1", "n2","r1", "r2", "alpha1", "beta1","alpha2",
"beta2", "EN", "n_cum")
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# Loading the Calcul.Simon.Risk function is required
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risk<-Calculation.Simon.Risk(d[1], d[3], d[2], d[4], pi0, pi1)
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} else{
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stop("No design found")
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}
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res<-list(risk=risk, design=design)
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return(res)
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}
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#######
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### Calculation.Simon. Risk allow to calculate Simon risk depending on
the quadruplet and pi0 and pi1
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### ARGUMENTS :
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# n1:
the number of patients included at stage 1
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# n2:
the additional number of patients included at stage 2
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# r1 :
# r2:
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the stopping boundary at stage 1
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the stopping boundary at stage 2
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# pi0 :
the minimum expected efficacy of the treatment
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# pi1 :
the desirable target level of treatment efficacy
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### VALUES :
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# a dataframe :
# $j :
:
final Simon Optimal design
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stage
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# $n:
the number of patients included at stage j (N1, N2)
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# $r:
the stopping boundaries at stage j (r1, r2)
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# $alpha :
PLOS
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the spending function of type I error rate
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# $beta :
the spending function of type II error rate
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# $pet0 :
probability of early termination under the null hypothesis
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# $pet1 : probability of early termination under the alternative
hypothesis
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Calculation.Simon.Risk<-function(n1, r1, n2, r2, pi0, pi1) {
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k<-(r1+1):n1
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pk1<-dbinom(k, n1, pi1)
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pk0<-dbinom(k, n1, pi0)
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beta1<-pbinom(r1, n1, pi1)
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beta2<-sum(pk1*pbinom(r2-k, n2, pi1))
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alpha2<-sum(pk0*(1-pbinom(r2-k, n2, pi0)))
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pet01<-pbinom(r1, n1, pi0)
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pet02a<-sum(pk0*pbinom(r2-k, n2, pi0))
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pet02b<-sum(pk0*(1-pbinom(r2-k, n2, pi0)))
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pet11<-pbinom(r1, n1, pi1)
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pet12a<-sum(pk1*pbinom(r2-k, n2, pi1))
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pet12b<-sum(pk1*(1-pbinom(r2-k, n2, pi1)))
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res<-data.frame(j=1:2, n=c(n1, n2+n1), r=c(r1, r2), alpha=c(0, alpha2),
alphacum=c(0, alpha2), beta=c(beta1, beta2), betacum=cumsum(c(beta1,
beta2)))
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res$pet0<-c(pet01, pet02a+pet02b)
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res$pet1<-c(pet11, pet12a+pet12b)
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return(res)
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}
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2. R code to simulate data
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##########################
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####### Simul.data :
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function which allow to simulate data
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### ARGUMENTS :
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# N: Total number of patients to simulate
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# pi :
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the theorical response rate
# theta :
# t0 :
the unevaluable patients rate
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the evaluation time point
# Tdistrib :
PLOS
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the distribution of the latent failure times T;
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# W correspond to a Weibull distribution
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# E correspond to an Exponential distribution
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# L correspond to a Log logistic distribution
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# Cdistrib :
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the distribution of the censoring times C;
# U correspond to an Uniform distribution
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# E correspond to an Exponential distribution
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### VALUES :
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# a data frame
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# $time: the time between inclusion and response evaluation or last
known contact
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# $X : a dummy variable which indicates if patients respond to the
therapy (X=1) ,
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# do not respond (X=0) or is unevaluable (X=NA)
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Simul.data<-function(N=100, pi=0.2, theta=0, t0=1, Tdistrib=c("W",
"L","E"), Cdistrib=c("U", "E")){
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Tdistrib<-Tdistrib[1]
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Cdistrib<-Cdistrib[1]
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# Define the cumulative probability and the random generation for the C
distribution
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if(Cdistrib=="U"){
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G<-function(c, C){ifelse(c<C, c/C ,1)}
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rg<-function(n, C){runif(n, 0, C)}
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}
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if(Cdistrib=="E"){
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G<-function(c, C){1-exp(-C*c)}
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rg<-function(n, C){rexp(n,C)}
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}
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# Define the cumulative probability and the random generation for the T
distribution, specify the distribution
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if(Tdistrib=="L"){
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scale<-2
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shape<-t0*((pi/(1-pi))^(1/scale))
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f<-function(t, shape, scale){
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y<-(t/shape)^scale
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y<-(y/((1+y)^2))*(scale/t)
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return(y)
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}
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rf<-function(n, shape, scale){
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U<-runif(n, 0,1)
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return(((U/(1-U))^(1/scale))*shape)
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}
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}
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if(Tdistrib=="W"){
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scale<-2
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shape<-t0*((-log(pi))^(-1/scale))
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f<-function(t, shape, scale){
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y<-dweibull(t, shape=scale, scale=shape)
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return(y)
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}
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rf<-function(n, shape, scale){rweibull(n, shape=scale, scale=shape)}
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}
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if(Tdistrib=="E"){
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shape<-(-log(pi))/t0
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scale<-1
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f<-function(t, shape, scale){
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y<-dexp(t, rate=shape)
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return(y)
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}
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rf<-function(n, shape, scale){rexp(n, rate=shape)}
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}
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#If there is no unevaluable patients :
PLOS
do not introduce censoring
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if(theta==0){
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lambda<-Inf
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}else{
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#to determine the lambda parameter
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fct<-function(C, t0, shape, scale, theta){ PdV(C, t0, shape, scale, f,
G)-theta}
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lambda<-uniroot(fct,interval=c(0.01, 1000),t0=t0, shape=shape,
scale=scale, theta=theta)$root
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}
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#Random generation
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T<-rf(N, shape, scale)
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if(lambda==0){
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C<-T*Inf
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}else{
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C<-rg(N, lambda)
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}
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#calculate the time
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time<-pmin(T, C)
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#classify each simulated patient as responder (X=1), non responder (X=0)
or unevaluable patients (X=NA)
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X<-ifelse(T>t0, 1,0)
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X<-ifelse(C<T & C<t0, NA,X)
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data<-data.frame(time=time, X=X)
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return(data)
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}
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#############
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#### analyze_data : count the number of unevaluable patients and the
actuarial survival at lt0 and t0 at each stage
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### ARGUMENTS :
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# N1:
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number of patients included at the first stage
# N2 :
number of patients included at the second stage
# data :
# t0 :
# l :
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response, X=0 non response, X=NA
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the evaluation time point
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the time ratio (as described in the article)
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### VALUES :
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# a list
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# $stage1 :
# $Z1 :
result at stage 1
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number of unevaluable patient at stage 1
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# $AC : actuarial survival estimated at stage 1
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# $S1 : the number of response at stage 1 among the N1-Z1 evaluable
patients
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# $stage2 :
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# $Z2 :
PLOS
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the time from inclusion to response evaluation
# data$X : a dummy variable with X=1:
: unevaluable
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a data frame with
# data$time :
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result at stage 2
number of unevaluable patient at stage 2
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# $AC : actuarial survival estimated at stage 2
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# $S2 : the number of response at stage 2 among the N2-Z2 evaluable
patients
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analyze_data<-function(N1, N2, data, t0, l=NULL){
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#Select the appropriate number patient to analyze
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N2<-min(N2, dim(data)[1])
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data1<-data[1:N1,]
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data2<-data[1:N2,]
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#calculate the unevaluable patients
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Z1<-as.numeric(table(data1$X, exclude=NULL)[3])
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Z2<-as.numeric(table(data2$X, exclude=NULL)[3] )
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S1<-as.numeric(table(data1$X, exclude=NULL)[2])
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S2<-as.numeric(table(data2$X, exclude=NULL)[2] )
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#Estimate the actuarial survival at l.t0 and t0
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if(is.null(l)){
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AC1<-NULL
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AC2<-NULL
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} else {
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AC1<- AC.fct(data1$time, data1$X, t0, l)$Scum
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AC2<- AC.fct(data2$time, data2$X, t0, l)$Scum
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}
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res1<-list(Z1=Z1, AC=AC1, S1=S1)
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res2<-list(Z2=Z2, AC=AC2, S2=S2)
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res<-list( stage1=res1, stage2=res2)
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return(res)
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}
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#############
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#### AC: function wich estimate the actuarial survivate rate at l.t0 and
t0
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### ARGUMENTS :
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# time:
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time from inclusion to response evaluation
# response : response=1: responder, response=0 non responder and
response=NA : uneavluable patients
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# t0 :
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# l :
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the evaluation time point
the time ratio (as described in the article)
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### VALUES :
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# a dataframe
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# $ti :
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beginning of the time interval
# $t(i+1) :
PLOS
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end of the time interval
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# $ci :
number of censored subjects
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# $di:
number of event (1-response)
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# $ni:
number of patients at risk during the interval (ti , t(i+1))
# $Scond :
# $Scum :
conditional survival during the interval (ti , t(i+1))
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actuarial survival estimated at t(i+1)
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AC.fct<-function(time, response, t0, l) {
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#time<-DATA$time; response<-DATA$X ; l<-1/2
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event<-ifelse(is.na(response) , 0, 1-response)
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inter<-unique(c(0, l*t0, t0))
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n<-length(time)
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nbi<-length(l)+1
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#define time interval
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i<-cut(c(-1,-1,time), c(-1,inter, Inf), right=F)
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x<-table(i, c(0,1,event))
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x<-x[-1,]
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tab<-data.frame(ti=inter)
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tab$"t(i+1)"<-c(inter[-1], "Inf")
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#number of censored
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tab$ci<-x[,1]
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#number of event
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tab$di<-x[,2]
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#Patients at risk
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tab$ni<-c(n, n-cumsum(tab$ci+tab$di)[-dim(tab)[1]])
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#Conditional survival
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tab$Scond<-(tab$ni-tab$ci*0.5-tab$di)/(tab$ni-tab$ci*0.5)
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tab$Scond<-ifelse(is.na(tab$Scond) , 1 ,tab$Scond)
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#Cumulative survival
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tab$Scum<-cumprod(tab$Scond)
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tab<-tab[-dim(tab)[1],]
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return(tab)
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}
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3. R code to use rescue strategy because Z1 and Z2 unevaluable
patients appear
PLOS
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#######
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#### PdV function calculate the probability to be unevaluable at t0 when
F and G are specified
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### ARGUMENTS :
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10/16
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PLOS
# Here, C is the censoring distribution parameter
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# t0 :
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time point for therapeutic evaluation
# shape :
the shape parameter of the distribution of T
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# scale :
the scale parameter of the distribution of T
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# f is the probability density function of T
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# G is the cumulative probability function of C
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### VALUES : the probability of being unevaluable at t0
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PdV<-function(C, t0, shape, scale, f, G){
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f1<-function(t, C, shape, scale){f(t, shape, scale)*G(t, C)}
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f2<-function(t, t0,C, shape, scale){f(t, shape, scale)*G(t0, C)}
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I1<-integrate(f1,0,t0, shape=shape, scale=scale, C=C)$value
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I2<-integrate(f2,t0,Inf, shape=shape, scale=scale, C=C, t0=t0)$value
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I<-I1+I2
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return(I)
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}
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###############################################
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##### Function tau.fct : calculate the probability of not responding to
the therapy and being evaluable at t0
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### ARGUMENTS :
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# C: is the censoring distribution(C) parameter
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# t0 :
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time point for therapeutic evaluation
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#shape :
the shape parameter of the distribution of T
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#scale :
the scale parameter of the distribution of T
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#F et G the cumulative distribution function of T and C respectively
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#g:
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the probability density function of C
### VALUES :the probability of not responding to the therapy and being
evaluable at t0
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tau.fct<-function( C, t0,shape, scale, F, G, g){
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f1<-function(c,C,shape, scale){F(c, shape=shape, scale=scale)*g(c, C=C)}
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I1<-(integrate(f1,0, t0, shape=shape, scale=scale, C=C)$value )
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tau<-I1 + F(t0, shape=shape, scale=scale)*(1-G(t0, C=C))
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return(tau)
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}
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#####Calculate type I error rate
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### ARGUMENTS
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# a:
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vector of the stopping boundary at stage 1 and 2
11/16
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# n:
vector of the number of patients included
# p :
the treatment efficacy
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## VALUES : probability of rejecting the null hypothesis when treatment
efficacy is p
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binom_alpha=function(r, n=c(10, 10), p=0.2){
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sum(dbinom((r[1]+1):n[1] , n[1], p) * (1-pbinom (r[2] - ((r[1]+1):n[1]),
n[2] , p)))
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}
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#####Calculate type II error rate
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### ARGUMENTS
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# r:
vector of the stopping boundary at stage 1 and 2
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# n:
vector of the number of patients included
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# p :
the treatment efficacy
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## VALUES : probability of not rejecting the null hypothesis when
treatment efficacy is p
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binom_beta=function(r, n=c(10, 10), p=0.2){
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sum(dbinom((r[1]+1):n[1] , n[1], p) * (pbinom (r[2] - ((r[1]+1):n[1]),
n[2] , p)))
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}
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#####
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#### Adapt: furnish an the adaptive design denote (N1, a1, N2, a2) AD
as decsribed in the article
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### ARGUMENTS :
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#Z1 :
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the number of unevaluable patients observed at stage 1
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#Z2: the number of unevaluable patients observed at stage 2, if stage 2
is not completed yet Z2 is equal to Z1.
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#N1 :
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#N2:
PLOS
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the planned sample size at stage 1
the planned sample size at stage 2
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#pi0 :
the minimu expected efficacy of the treatment
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#pi1 :
the desirable target level of treatment efficacy
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#alpha:
the type I error rate
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#beta:
the type II error rate
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#bound: for stage 2 adaptation, bound is s=the stopping boundary used
at the first stage
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#l :
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the time ratio (as described in the article)
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#AC : vector of actuarial estimation of response rate at l.t0 and t0
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# (if length of AC is 1: exponential distribution is assumed) otherwise
weibull distribution is assumed)
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#error_fct :
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# phi1 :respect the ratio between alpha and beta, increase the type I
and type II error rates
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# phi2:
rates
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preserv the type I error rate and increase the type II error
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### VALUES
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#a list object
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# $param :
design
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design parameter used in order to establish the adapted
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# pi0a : the pi0* probability of response when being evaluable at t0
and under H0
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# pi1a : the pi0* probability of response when being evaluable at t0
and under H1
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# alpha :
the initial type I error rate
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# beta :
the initial type II error rate
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# Nmax:
PLOS
the error rate function
the maximal number of patient to be included
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# Z1 :
the number of unevaluable patients at the first stage
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# Z2 :
the number of unevaluable patients at the second stage
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# $desAD : the adapted design defined by the adapted quadruplet and
final the type I and type II error rate
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# N1AD: number of evaluable patient at stage 1
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# r1AD: the stopping boundary at first stage with Z1 unevaluable
patients
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# N2AD: number of evaluable patient at stage 2
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# r2AD: the stopping boundary at second stage with Z2 unevaluable
patients
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# alphaAD : final type I error rate to establish the adapted design
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# betaAD : final type II error rate to establish the adapted design
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Adapt<-function( Z1, Z2=Z1,N1, N2, pi0,pi1, alpha, beta,t0, bound=NULL
,l=l, AC=NULL, error_fct=c("phi1", "phi2")){
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##gamma=scale
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#h00=shape0
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#h01=shape1
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Z2<-max(Z1, Z2)
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if(is.null(bound)){bound<-c(N1, N2)
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}else{
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if(length(bound)==1){
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bound<-c(bound, N2)
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}
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13/16
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PLOS
}
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# Adaptation at stage 1
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if(Z2-Z1==0){
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n1<-N1-Z1
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n2<-N2-N1
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theta<-Z1/N1
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a<-expand.grid(list(r1=0:n1 , r2=0:n2))
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}
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# Adaptation at stage 2
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if (Z2-Z1>0){
479
n1<-N1-Z1
480
n2<-N2-N1-(Z2-Z1)
481
theta<-Z2/N2
482
a<-expand.grid(list(r1=bound[1] , r2=0:n2))
483
}
484
G<-function(c, C){punif(c, 0, C)}
485
g<-function(c, C){dunif(c, 0,C)}
486
#Weibull adaptation
487
if(length(AC)==2){
488
f<-function(t,shape, scale){dweibull(t, shape=scale, scale=shape)}
489
F<-function(t,shape, scale){pweibull(t, shape=scale, scale=shape)}
490
#Find weibull parameter (h0 and gamma) (shape and scale respectively)
491
scale<- log(log(AC[2])/log(AC[1]))/(-log(l))
492
shape0<- t0/(-log(pi0))^(1/scale)
493
shape1<- t0/(-log(pi1))^(1/scale)
494
#Find uniform parameter (lambda)
495
fct<-function(C, t0, shape, scale, theta){ PdV(C, t0, shape, scale, f,
G)-theta}
496
lambda0<-uniroot(fct,interval=c(0.01, 1000),t0=t0, shape=shape0,
scale=scale, theta=theta)$root
498
lambda1<-uniroot(fct,interval=c(0.01, 1000),t0=t0, shape=shape1,
scale=scale, theta=theta)$root
500
#Calculate adapted pi1 and pi0 :
502
497
499
501
equation 5
pi1a<-1-(tau.fct( lambda1, t0,shape1, scale, F, G, g)/(1-theta))
503
pi0a<-1-(tau.fct( lambda0, t0,shape0, scale, F, G, g)/(1-theta))
504
}
505
#Exponential adaptation
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PLOS
if(is.null(AC)){
507
f<-function(t,shape, scale){dexp(t, rate=shape)}
508
F<-function(t,shape, scale){1-exp(-(t*shape))}
509
#Find weibull parameter (h0 and gamma) (shape and scale respectively)
510
scale<- 1
511
shape0<- (-log(pi0))/t0
512
shape1<- (-log(pi1))/t0
513
#Find uniform parameter (lambda)
514
fct<-function(C, t0, shape, scale, theta){ PdV(C, t0, shape, scale, f,
G)-theta}
515
lambda0<-uniroot(fct,interval=c(0.01, 1000),t0=t0, shape=shape0,
scale=scale, theta=theta)$root
517
lambda1<-uniroot(fct,interval=c(0.01, 1000),t0=t0, shape=shape1,
scale=scale, theta=theta)$root
519
#Calculate adapted pi1 and pi0 :
521
516
518
520
equation 5
pi1a<-1-(tau.fct( lambda1, t0,shape1, scale, F, G, g)/(1-theta))
522
pi0a<-1-(tau.fct( lambda0, t0,shape0, scale, F, G, g)/(1-theta))
523
}
524
#### Détermination of the new stopping boundaries
525
a$r2<-a$r1 + a$r2
526
a<-a[a$r2<=bound[2] & a$r1<=bound[1],]
527
a$alpha1<- 0
528
a$beta1<-pbinom(a$r1, n1, pi1a)
529
a$alpha2<-apply(a[, c(1,2)], 1, binom_alpha, n=c(n1, n2), p=pi0a)
530
a$beta2<-apply(a[, c(1,2)], 1, binom_beta, n=c(n1, n2), p=pi1a)
531
a$enho<-n1+(1-pbinom(a$r1 , n1, pi0a))*n2
532
a$alpha<-a$alpha1+ a$alpha2
533
a$beta<-a$beta1+a$beta2
534
#error rate function to control type I and type II error rates
535
if(error_fct=="phi1"){
536
coef<-beta/alpha
537
a$alphacout<-a$beta/coef
538
a$ma<-pmax(a$alpha, a$alphacout)
539
a$ma[a$ma<alpha]<-2
540
a<-a[order(a$ma, a$enho),] #minimize the increase error rates then
E(N|H0)
541
desAD<-data.frame(N1AD = N1-Z1 , r1AD=a$r1[1], N2AD=N2-Z2 ,
r2AD=a$r2[1], alphaAD=a$ma[1] , betaAD=a$ma[1]*coef)
543
542
544
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PLOS
}
545
if(error_fct=="phi2"){
546
a<-a[a$alpha<=alpha,]
547
a$beta[a$beta<beta]<-2
548
a<-a[order(a$beta, a$enho),] #minimize the increase error rates then
E(N|H0)
549
desAD<-data.frame(N1AD = N1-Z1 , r1AD=a$r1[1], N2AD=N2-Z2 ,
r2AD=a$r2[1], alphaAD=a$alpha[1] , betaAD=a$beta[1])
551
}
553
param<-data.frame(pi0a=pi0a, pi1a=pi1a, alpha=alpha , beta=beta,
Nmax=N2-Z2 , Z1=Z1, Z2=Z2)
554
res<-list(param=param, desAD=desAD)
556
return(res)
557
}
558
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552
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