Adaptive Biomarker Population Selection in Phase III Confirmatory Trials with Time-to-Event Endpoints

Abstract

A key component of modern drug development is to identify the patient population(s) that benefit most from the therapy. Subjects with different biomarker/genetic profiles may respond to a therapy differently. At the same time, data of populations with best efficacy may not be available due to lack of randomized data or biomarker assay when designing the phase III confirmatory trials. In this manuscript, we propose an adaptive design that performs biomarker-informed population selection at the interim analysis, so as to refine the population for primary hypothesis at the final analysis. Unlike most of the previous research work, where the same endpoint is used for interim population selection and final analysis, we propose to use a sensitive intermediate endpoint (whenever available) for population selection. Treatment effect of the intermediate endpoint is treated as a nuisance parameter to ensure type I error control. The use of a sensitive intermediate endpoint for biomarker population selection further improves study power. Simulations were conducted to evaluate the control of overall type I error, probabilities of population selection, and power.

Appendix

R codes of type I error calculation and power when using intermediate endpoint:

################################################# ###########

###################

# Use Intermediate endpoint to select/deselect biomarker subpopulations at IA (assume treatment effects are ordered)

###########################################################

######################################

#Type I error #####

###################

#function mf1: Probability of dropping first m subgroups and has a positive study on the last (k-m) subgroups at the final analysis under H0#

#k: total number of biomarker subgroups

#m: number of deselected biomarker subgroups

#delta1: treatment effect of intermediate endpoint

#rho: correlation between intermediate endpoint and clinical endpoint

#D_int: A vector with number of events using intermediate endpoint at IA

#D_final: A vector with number of events using final clinical endpoint at IA

#Assume enrollment completed at IA

#alphat: nominal alpha level at IA

#alphastar: nominal alpha level at final analysis

# deselect m subgroup (m>=1)

mf1i<-function(alphastar, alphat, m, k, delta1, rho, D_int){

low<-c(rep(-Inf,m), qnorm(1-alphat), qnorm(1-alphastar))

up<-c(rep(qnorm(1-alphat),m), Inf, Inf)

test<-matrix(0, ncol=m+2, nrow=m+2)

diag(test)<-1;

test[m+2,m+1]<-test[m+1,m+2]<-rho*sqrt(D_final[m+1]/

sum(D_final[(m+1):k]))

# subgroups are disjoint, but Z(-m) includes Y(m+1)

altmeani<-rep(0,m+2)

altmeani[1:(m+1)]<-sqrt(D_int[1:(m+1)]/4)*delta11:(m+1)]

temp<-pmvnorm(lower=low, upper=up, mean=altmeani,corr=test)[1]

return(temp)

}

#Overall type I error:

type1upi<-function(alphastar, alphat, k, delta1 rho, D_int){

# probability of including all at the final analysis i.e., m=0

low <- c(qnorm(1-alphat), qnorm(1-alphastar))

up <- rep(Inf, 2)

test1 <- matrix(0, ncol=2,nrow=2);

diag(test1) <- 1; test1[1,2] <- test1[2,1] <-rho*sqrt(D_final[1]/sum(D_final))

altmeani<-rep(0,2)

altmeani[1]<-sqrt(D_int[1]/4)*delta1[1]

temp <- pmvnorm(lower=low, upper= up, mean=altmeani, corr=test1)[1]

#Probability of including m=1 to k-1 groups at the final analysis

for (m in 1:(k-1)){

temp<-temp+mf1i(alphastar, alphat,m=m,k,delta1, rho, D_int)

}

return(temp-0.025)

}

##################################################################

# Grid search delta to find alpha* for final analysis

##################################################################

length<-100

deltatemp<-c(0:length)*0.1

typ1temp<-matrix(rep(1,length*length),nrow=length)

minalphastar<-function(alphat, k,rho, D_int){

for (i in(1: length)){

for (j in (1:length)){

typ1temp[i,j]<-uniroot(type1upi, c(0,1), alphat, k, delta1=c(deltatemp[i],

deltatemp[j]), rho,D_int)$root

}

}

minalphastar<-min(typ1temp)

return(minalphastar)

}

##### Power ######################

# delta1: treatment effect of intermediate endpoint

# delta: treatment effect of final clinical endpoint

powerupi<-function(delta,D_final, k,alphat, delta1, rho, D_int, alphastar){

# alphastar<- minalphastar(alphat, t, k,rho, D)

# probability of including all under alternative hypotheses, m=0

mean1 <- c(sqrt(D_int[1]/4)*delta1[1], sum(delta*D_final/4)/sqrt(sum(D_final/4)))

corr1 <- matrix(0, ncol=2, nrow=2)

diag(corr1) <- 1; corr1[1,2] <- corr1[2,1] <- rho*sqrt(D_final[1]/sum(D_final))

low1 <- c(qnorm(1-alphat), qnorm(1-alphastar))

up1 <- rep(Inf, 2)

temp <- pmvnorm(lower=low1, upper=up1, mean=mean1, corr=corr1)[1]

# probability of excluding the first m hypotheses, with m=1,to k-1.

for (m in 1:(k-1)){

mean2<-c(sqrt(D_int[1:(m+1)]/4)*delta1[1:(m+1)], sum(delta[(m+1):k]*D_final

[(m+1):k]/4)/sqrt(sum(D_final[(m+1):k]/4)))

corr2 <- matrix(0, ncol=m+2, nrow=m+2); diag(corr2) <- 1;

corr2[m+1,m+2] <- corr2[m+2, m+1] <- rho*sqrt(D_final[m+1]/sum(D_final

[(m+1):k]))

low2 <- c(rep(-Inf, m),qnorm(1-alphat), qnorm(1-alphastar))

up2 <- c(rep(qnorm(1-alphat), m), Inf, Inf)

temp <- temp+pmvnorm(lower=low2, upper=up2, mean=mean2, corr=corr2)[1]

}