Web-based Supporting information for “Confidence intervals based on some weighting functions for
the difference of two binomial proportions” by K. Maruo PhD and N. Kawai PhD.

Description about R function “riskdif_ci_weight” for calculating proposed CI estimation
methods (NewL and NewB)
Usage
riskdif_ci_weight (n1, n2, x1, x2, alpha, weight)
Arguments

n1:
sample size of group1

n2:
sample size of group2,

x1:
number of event for group 1

x2:
number of event for group 2,

alpha:
100(1-alpha)% CI

weight: 1:likelihood, 2:truncated beta pdf
Value
Returns 3 x 1 vector: {lower confidence limit, MLE of delta, upper confidence limit}
R code
################################################################################
# MLE of p1 and p2 under null hypothesis
# (Farrington and Manning(1990), Stat. Med., 9, 1447-1454.)
mle_p1_h0<- function(n1,n2,x1,x2,d){
theta <- n2/n1
p1 <- x1/n1
p2 <- x2/n2
a1 <- 1+theta
a2 <- -(1+theta+p1+theta*p2+d*(theta+2))
a3 <- d^2+d*(2*p1+theta+1)+p1+theta*p2
a4 <- -p1*d*(1+d)
v <- a2^3/(3*a1)^3-a2*a3/(6*a1^2)+a4/(2*a1)
u <- sign(v)*sqrt(a2^2/(3*a1)^2-a3/(3*a1))+1e-10
uv <- v/u^3
uv[uv < -1] <- -1
uv[uv > 1] <- 1
uv[u-1e-10 == 0] <- 0
w <- (pi+acos(uv))/3
p1t <- 2*u*cos(w)-a2/(3*a1)
p1t[p1t<0] <- 0
p1t[p1t>1] <- 1
p2t <- p1t-d
p2t[p2t<0] <- 0
p2t[p2t>1] <- 1
return(list(p1t,p2t))
}
mle_se_h0<- function(n1,n2,x1,x2,d){
n0 <- matrix(0,length(x1),length(x2))
p12 <- mle_p1_h0(n0+n1,n0+n2,t(numeric(length(x2))+1)%x%x1,
(numeric(length(x1))+1)%x%t(x2),d)
se<- sqrt(p12[[1]]*(1-p12[[1]])/n1+p12[[2]]*(1-p12[[2]])/n2)
if (abs(d)==1) se[,] <- 0
if (length(x1)==1|length(x2)==1) se <- c(se)
return(se)
}
#-----------------------------------------------------------------------------
# CI for risk difference (Maruo and Kawai's method)-----------riskdif_ci_weight<-function(n1,n2,x1,x2,alpha,weight){
# CI for risk difference (Quasi exact methods based on weight functions)
# n1:sample size of group1, n2:sample size of group2,
# x1:number of event for group 1, x2:number of event for group 2,
# alpha: 100(1-alpha)% CI,
# weight: 1:likelihood, 2:truncated beta pdf
if (alpha <= 0|alpha>=1) stop('alpha is not given appropriately!')
if (x1>n1) stop('x1 must be equal to or lower than n1!')
if (x2>n2) stop('x2 must be equal to or lower than n2!')
if (weight!=1 & weight!=2) stop('weight must be 1(likelihood) or 2(truncated beta pdf)!')
if (x1<0|x2<0|abs(x1-as.integer(x1))>1e-10|abs(x2-as.integer(x2))>1e-10){
stop('x1 and x2 must be non-negative integer!')
}
if (n1<=0|n2<=0|abs(n1-as.integer(n1))>1e-10|abs(n2-as.integer(n2))>1e-10){
stop('n1 and n2 must be positive integer!')
}
if (n1>50|n2>50) print('n1 and/or n2 are large number. Calculation time may be long.')
dig <- 3
prec <- 1e-3
# precision of estimation
flg <- 0
if (n1<n2){ # for symmetry property of CI
n3 <- n1
x3 <- x1
n1 <- n2
x1 <- x2
n2 <- n3
x2 <- x3
flg <- 1
}
Y1 <- 0:n1
Y2 <- 0:n2
on1 <- numeric(n1+1)+1
on2 <- numeric(n2+1)+1
cy1 <- choose(n1,Y1)
cy2 <- choose(n2,Y2)
cx1 <- choose(n1,x1)
cx2 <- choose(n2,x2)
# calculation of (p-value - alpha/2)^2 for given delta(d)
prob_d <- function(d,up){
if (abs(d)==1){
s <- ((t(on2)%x%(Y1/n1))-(on1%x%t(Y2/n2))-d)
st <- (x1/n1-x2/n2-d)
}
else {
s <- ((t(on2)%x%(Y1/n1))-(on1%x%t(Y2/n2))-d)/(mle_se_h0(n1,n2,Y1,Y2,d))
st <- (x1/n1-x2/n2-d)/(mle_se_h0(n1,n2,x1,x2,d))
}
p12t <- mle_p1_h0(n1,n2,x1,x2,d)
# calculation of tail probability for given P and delta(d)
prob_p_d <- function(p){
ps<-numeric(length(p))
for (ip in 1:length(p)){
pn1 <- cy1*p[ip]^Y1*(1-p[ip])^(n1-Y1)
pn2 <- cy2*(p[ip]-d)^Y2*(1+d-p[ip])^(n2-Y2)
prob <- (t(on2)%x%pn1)*(on1%x%t(pn2))
if (up==1) ps[ip] <- sum((s<=st)*prob)
else ps[ip] <- sum((s>=st)*prob)
}
return(ps)
}
# weighted p-value based on likelihood
if (weight==1){
lik_wgt <- function(p){
p2=p-d
return(cx1*p^x1*(1-p)^(n1-x1)*cx2*p2^x2*(1-p2)^(n2-x2))
}
p_lik_wgt <- function(p){
return(prob_p_d(p)*lik_wgt(p))
}
par <- prob_p_d(min(c(1,1+d)))
if (min(c(1,1+d))-max(c(0,d))>1e-10){
par1 <- integrate(p_lik_wgt,max(c(0,d)),min(c(1,1+d)))$value
par2 <- integrate(lik_wgt,max(c(0,d)),min(c(1,1+d)))$value
par <- par1/par2
}
}
# weighted p-value based on truncated beta pdf
if (weight==2) {
p_beta_wgt <- function(p){
return(prob_p_d(p)*dbeta(p,n1*p12t[[1]]+1,n1*(1-p12t[[1]])+1))
}
par <- prob_p_d(min(c(1,1+d)))
if (min(c(1,1+d))-max(c(0,d))>1e-10){
par1 <- integrate(p_beta_wgt,max(c(0,d)),min(c(1,1+d)))$value
par2 <- pbeta(min(c(1,1+d)),n1*p12t[[1]]+1,n1*(1-p12t[[1]])+1)pbeta(max(c(0,d)),n1*p12t[[1]]+1,n1*(1-p12t[[1]])+1)
par <- par1/par2
}
}
return((par-alpha/2)^2)
}
dobs <- x1/n1-x2/n2
# optimization process based on grid search
opt2 <- function(up){
if (up==1) d0 <- seq(dobs,1,by=prec*50)
else d0 <- seq(-1,dobs,by=prec*50)
cl0 <- numeric(length(d0))
j <- 1
for (d0i in d0){
cl0[j] <- prob_d(d0i,up)
j <- j+1
}
m0 <- which.min(cl0)
if (up==1) d1 <- seq(min(d0[max(m0-1,1)],dobs),max(d0[min(m0+1,length(d0))],1),by=prec*10)
else d1 <- seq(min(d0[max(m0-1,1)],-1),max(d0[min(m0+1,length(d0))],dobs),by=prec*10)
cl1 <- numeric(length(d1))
j <- 1
for (d1i in d1){
cl1[j] <- prob_d(d1i,up)
j <- j+1
}
m1 <- which.min(cl1)
if (up==1) d2 <- seq(min(d1[max(m1-1,1)],dobs),max(d1[min(m1+1,length(d1))],1),by=prec)
else d2 <- seq(min(d1[max(m1-1,1)],-1),max(d1[min(m1+1,length(d1))],dobs),by=prec)
cl2 <- numeric(length(d2))
j <- 1
for (d2i in d2){
cl2[j] <- prob_d(d2i,up)
j <- j+1
}
cl3 <- d2[which.min(cl2)]
return(cl3)
}
# calculation of CI for delta
lower <- -1
if (dobs != -1) lower <- opt2(0)
upper <- 1
if (dobs != 1) upper <- opt2(1)
if (flg==1){
dum <- lower
lower <- -upper
upper <- -dum
dobs <- -dobs
}
dobs <- round(dobs, digits = dig)
lower <- round(lower, digits = dig)
upper <- round(upper, digits = dig)
return(c(lower,dobs,upper))
}
#-----------------------------------------------------------------------# ECOG data analysis based on Maruo and Kawai's methods
riskdif_ci_weight(14,11,2,1,0.1,1)#New likelihood weighting method
riskdif_ci_weight(14,11,2,1,0.1,2)#Beta weighting method
####################################################################################