# R/gsBinomialExact.R In gsDesign: Group Sequential Design

#### Documented in binomialSPRTgsBinomialExactplot.binomialSPRTplot.gsBinomialExact

```#####
# global variables used to eliminate warnings in R CMD check
#####
globalVariables(c("N","EN","Bound","rr","Percent","Outcome"))

gsBinomialExact <- function(k=2, theta=c(.1, .2), n.I=c(50, 100), a=c(3, 7), b=c(20,30))
{
checkScalar(k, "integer", c(2,Inf), inclusion=c(TRUE, FALSE))
checkVector(theta, "numeric", interval=0:1, inclusion=c(TRUE, TRUE))
checkVector(n.I, "integer", interval=c(1, Inf), inclusion=c(TRUE, FALSE))
checkVector(a, "integer", interval=c(-Inf, Inf), inclusion=c(FALSE, FALSE))
checkVector(b, "integer", interval=c(1, Inf), inclusion=c(FALSE, FALSE))
ntheta <- as.integer(length(theta))
theta <- as.double(theta)
if (k != length(n.I) || k!=length(a) || k != length(b))
stop("Lengths of n.I, a, and b must equal k on input")
m <- c(n.I[1], diff(n.I))
if (min(m) < 1) stop("n.I must must contain an increasing sequence of positive integers")
if (min(n.I-a) < 0) stop("Input a-vector must be less than n.I")
if (min(b-a) <= 0) stop("Input b-vector must be strictly greater than a")
if (min(diff(a)) < 0) stop("a must contain a non-decreasing sequence of integers")
if (min(diff(b)) < 0) stop("b must contain a non-decreasing sequence of integers")
if (min(diff(n.I-b)) < 0)  stop("n.I - b must be non-decreasing")

plo <- matrix(nrow=k, ncol = ntheta)
rownames(plo) <- paste(array("Analysis ", k), 1:k)
colnames(plo) <- theta
phi <- plo
en <- numeric(ntheta)

for(h in 1:length(theta))
{
p <- theta[h]

### c.mat is the recursive function defined in (12.5)
### plo and phi are the probabilities of crossing the lower and upper boundaries defined in (12.6)
c.mat <- matrix(0, ncol=k, nrow=n.I[k]+1)
for(i in 1:k)
{
if(i==1)
{
c.mat[,1] <- dbinom(0:n.I[k], m[1], p)
}
else
{
no.stop <- (a[i-1]+1):(b[i-1]-1)
no.stop.mat <- matrix(no.stop, byrow=T, nrow=n.I[k]+1, ncol=length(no.stop))
succ.mat <- matrix(0:n.I[k], byrow=F, ncol=length(no.stop), nrow=n.I[k]+1)
bin.mat <- matrix(dbinom(succ.mat-no.stop.mat, m[i], p),byrow=F,ncol=length(no.stop), nrow=n.I[k]+1)
c.mat[,i] <- bin.mat %*% c.mat[no.stop+1,(i-1)]
}
plo[i,h] <- sum(c.mat[(0:n.I[k]) <= a[i], i])
phi[i,h] <- sum(c.mat[(0:n.I[k]) >= b[i], i])
}
}

powr <-   array(1, k) %*% phi
futile <- array(1, k) %*% plo
en <- as.vector(n.I %*% (plo + phi) + n.I[k] * (t(array(1,ntheta)) - powr - futile))

x <- list(k = k, theta = theta, n.I = n.I,
lower = list(bound = a, prob = plo ),
upper = list(bound = b, prob = phi ), en=en)

class(x) <- c("gsBinomialExact", "gsProbability")
return(x)
}
# see http://theriac.org/DeskReference/viewDocument.php?id=65&SectionsList=3
binomialSPRT<-function(p0=.05,p1=.25,alpha=.1,beta=.15,minn=10,maxn=35){
lnA <- log((1-beta)/alpha)
lnB <- log(beta/(1-alpha))
a <- log((1-p1)/(1-p0))
b <- log(p1/p0)-a
slope <- -a / b
intercept <- c(lnA,lnB)/b
upper <- ceiling(slope*(minn:maxn)+intercept[1])
lower <- floor(slope*(minn:maxn)+intercept[2])
lower[lower< -1] <- -1
indx <- (minn:maxn >= upper)|(lower>=0)
# compute exact boundary crossing probabilities
y <- gsBinomialExact(k=sum(indx),n.I=(minn:maxn)[indx],
theta=c(p0,p1),a=lower[indx],b=upper[indx])
y\$intercept <- intercept
y\$slope <- slope
y\$alpha <- alpha
y\$beta <- beta
y\$p0 <- p0
y\$p1 <- p1
class(y) <- c("binomialSPRT","gsBinomialExact","gsProbability")
return(y)
}
plot.gsBinomialExact <- function(x,plottype=1,...){
if (plottype==6){
theta<-(max(x\$theta)-min(x\$theta))*(0:50)/50+min(x\$theta)
y <- gsBinomialExact(k=x\$k,theta=theta,n.I=x\$n.I,a=x\$lower\$bound,b=x\$upper\$bound)
xx <- data.frame(p=theta,EN=y\$en)
p<-ggplot(data=xx,aes(x=p,y=EN)) + geom_line() + ylab("Expected sample size")
}else if(plottype==3){
xx <- data.frame(N=x\$n.I,p=x\$upper\$bound/x\$n.I,Bound="Upper")
xx <- rbind(xx, data.frame(N=x\$n.I,p=x\$lower\$bound/x\$n.I,Bound="Lower"))
p<-ggplot(data=xx,aes(x=N,y=p,group=Bound))+
geom_point() +
ylab("Rate at bound")
}else if (plottype==2){
theta<-(max(x\$theta)-min(x\$theta))*(0:50)/50+min(x\$theta)
# compute exact boundary crossing probabilities
y <- gsBinomialExact(k=x\$k,n.I=x\$n.I,
theta=theta,a=x\$lower\$bound,b=x\$upper\$bound)
# compute probability of crossing upper bound for each theta
Power <- data.frame(rr=theta,
Percent=100*as.vector(matrix(1,ncol=length(y\$n.I),nrow=1)%*%
y\$upper\$prob),
Outcome="Reject H0")
# compute probability of crossing lower bound
futility <- data.frame(rr=theta,
Percent=100*as.vector(matrix(1,ncol=length(y\$n.I),nrow=1)%*%
y\$lower\$prob),
Outcome="Reject H1")
# probability of no boundary crossing
indeterminate <- data.frame(rr=theta,Percent=100-Power\$Percent-futility\$Percent,
Outcome="Indeterminate")
#combine and plot
outcome <- rbind(Power,futility,indeterminate)
p <- ggplot(data=outcome,aes(x=rr,y=Percent,lty=Outcome))+
geom_line()+
xlab("Underlying response rate")
}else{
xx <- data.frame(N=x\$n.I,x=x\$upper\$bound,Bound="Upper")
xx <- rbind(xx, data.frame(N=x\$n.I,x=x\$lower\$bound,Bound="Lower"))
p<-ggplot(data=xx,aes(x=N,y=x,group=Bound))+
geom_point() +
ylab("Number of responses")
}
return(p)
}
plot.binomialSPRT <- function(x,plottype=1,...){
p <- plot.gsBinomialExact(x,plottype=plottype,...)
if (plottype==1){
p <- p + geom_abline(intercept=x\$intercept[1],
slope=x\$slope) +
geom_abline(intercept=x\$intercept[2],
slope=x\$slope)
}
return(p)
}

# predictive probability bound
binomialPP <- function(a=.2, b=.8, theta=c(.2,.4), p1=.4, PP=c(.025,.95), nIA){
# initiate bounds outside of range of possibility
upper <- nIA + 1
lower <- array(-1,length(nIA))
j <- 1
# set bounds for each analysis sample size specified
for(i in nIA){
q <- 0:i
# compute posterior probability for value > p1
# for each possible outcome at analysis i
post <- pbeta(p1, a + q, b + i - q, lower.tail=F)
# set upper bound where posterior probability is > PP[2]
# that response rate is > p1
upper[j] <- sum(post < PP[2])
# set lower bound were posterior probability is <= PP[1]
# that response rate is < p1
lower[j] <- sum(post <= PP[1])
j <- j+1
}
# compute boundary crossing probabilities under
# response rates input in theta
y <- gsBinomialExact(k=length(nIA),n.I=nIA,
theta=theta,a=lower,b=upper)
# add beta prior parameters to return value
y\$abeta <- a
y\$bbeta <- b
# add input predictive probability bounds to return value
y\$PP <- PP
# define class for output
class(y) <- c("binomialPP","gsBinomialExact","gsProbability")
return(y)
}
```

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gsDesign documentation built on May 31, 2017, 2:15 a.m.