---
title: "Monitoring futility and efficacy in phase II trials with Bayesian posterior distributions -- a calibration approach"
author: "Annette Kopp-Schneider, Manuel Wiesenfarth, Ruth Witt, Dominic Edelmann, Olaf Witt and Ulrich Abel"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Monitoring futility and efficacy in phase II trials with Bayesian posterior distributions -- a calibration approach}
%\VignetteEncoding{UTF-8}
%\VignetteEngine{knitr::rmarkdown}
editor_options:
chunk_output_type: console
---
```{r setup, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>", fig.width=5, fig.height=5
)
```
# Introduction
This vignette reproduces results and figures in
Kopp-Schneider, A., Wiesenfarth, M., Witt, R., Edelmann, D., Witt, D. and Abel, U. (2018). \cr
Monitoring futility and efficacy in phase II trials with Bayesian
posterior distributions - a calibration approach.
*Biometrical Journal*, to appear.
Note that most figures are also shown in the interactive shiny app using `BDP2workflow()`
# Parameters
First we set the parameters as in the paper. Notation is identical as in the paper: p0, p1, pF, pE, cF, cE as in the paper,
shape1F, shape2F are the parameters for the prior for futility, and
shape1E, shape2E are the parameters for the prior for efficacy.
```{r, get parameters, message=F, warning=F}
library(BDP2)
p0=0.12
p1=0.3
pF=0.3
pE=0.12
shape1F=0.3
shape2F=0.7
shape1E=0.12
shape2E=0.88
cF=0.01
cE=0.9
```
# Figure 1
Figure 1: Posterior distribution of response probability used for futility stop decision: the trial will be stopped for futility if the coloured area is $< c_F$ (left panel). Posterior distribution of response probability used for efficacy calling: the trial will be called efficaceous if the coloured area is $\geq c_E$ (right panel). The Figure also shows the choice of $p_F=p_1$ and of $p_E=p_0$ as discussed in the actual design of the trial.
```{r, Figure 1, fig.show='hold',fig.width=6, fig.height=4}
##plot PF
plot(function(x) x,0,0.8,add=F,type="n",col="black",xlab="",ylab="",xaxt="n",yaxt="n",ylim=c(0,4),
cex.axis=1.5,cex.lab=1.5,xaxs='i',yaxs='i')
plot(function(x) dbeta(x,shape1F+2,shape2F+8),add=T,col="black",lwd=2,lty=1)
xy <- seq(0.3,1,length=1000)
fxy <- dbeta(xy,shape1F+2,shape2F+8)
xyx <- c(1,0.3,xy)
yyx <- c(0.0001,0.0001,fxy)
polygon(xyx,yyx,col='red')
lines(c(0.3,0.3),c(0,dbeta(0.3,shape1F+2,shape2F+8)),lwd=3)
axis(side=1,at=c(0.12,0.3),label=c(expression(p[E]),expression(p[F])),cex.axis=1.5)
axis(side=1,at=c(0.12,0.3),label=c(expression(paste("=",p[0])),expression(paste("=",p[1]))),cex.axis=1.5,line=1.5,lty=0)
text(0.65,0.7,labels=expression(paste("P(p > ",p[F],"| Data)")),cex=1.5)
lines(c(0.4,0.5),c(0.5,0.7),lwd=3)
##plot PE
plot(function(x) x,0,0.8,add=F,type="n",col="black",xlab="",ylab="",xaxt="n",yaxt="n",ylim=c(0,4),
cex.axis=1.5,cex.lab=1.5,xaxs='i',yaxs='i')
plot(function(x) dbeta(x,shape1E+2,shape2E+8),add=T,col="black",lwd=2,lty=1)
xy <- seq(0.12,1,length=1000)
fxy <- dbeta(xy,shape1E+2,shape2E+8)
xyx <- c(1,0.12,xy)
yyx <- c(0.0001,0.0001,fxy)
polygon(xyx,yyx,col='green')
lines(c(0.12,0.12),c(0,dbeta(0.12,shape1E+2,shape2E+8)),lwd=3)
axis(side=1,at=c(0.12,0.3),label=c(expression(p[E]),expression(p[F])),cex.axis=1.5)
axis(side=1,at=c(0.12,0.3),label=c(expression(paste("=",p[0])),expression(paste("=",p[1]))),cex.axis=1.5,line=1.5,lty=0)
text(0.65,0.7,labels=expression(paste("P(p > ",p[E],"| Data)")),cex=1.5)
lines(c(0.4,0.5),c(0.5,0.7),lwd=3)
```
# Figure 2
Upper tail function for $n=20$, for a design with $p_F=0.3$, $c_F = 0.01$ and prior distribution Be($p_F,1-p_F$) for futility (in red), and $p_E=0.12$, $c_E = 0.9$ and prior distribution Be($p_E,1-p_E$) for efficacy (in green). From the graphs, $b_F(20)=1$ and $b_E(20)=5$.
```{r, Figure 2}
n=20
plot(function(x) pbeta(pF,shape1=shape1F+x,shape2=shape2F+n-x,lower.tail = F)-cF,
0,n,ylim=c(-1,1),lwd=3,
xlab="x",ylab=expression(paste("P(p>",p[R],"|x,n = 20) - c")),col="red")
plot(function(x) pbeta(pE,shape1=shape1E+x,shape2=shape2E+n-x,lower.tail = F)-cE,
0,n,ylim=c(-1,1),lwd=3,col="green",add=TRUE)
abline(h=0)
legend("bottomright",legend=c(expression(paste("Futility decision: c=",c[F],", ",p[R],"=",p[F])),
expression(paste("Efficacy decision: c=",c[E],", ",p[R],"=",p[E]))),
lty=1,col = c("red", "green"),cex=1)
abline(v=c(0:20),lty=3,col="grey")
```
# Figure 3
Binomial density with zero responders,$(1-p)^n$, for varying $n$, evaluated for $p=p_0=0.12$ and for $p=p_1=0.3$ (left panel). Posterior probabilities $P(p \geq p_F | 0,n)$ and $P(p \geq p_F | 1,n)$ for varying $n$ for $p_F=0.3$ with prior distribution Beta($p_F,1-p_F$) (right panel).
```{r, Figure 3, fig.show='hold',fig.width=7}
nmin=4
nmax=15
plotBDP2(x="n",y="Prob0Successes",n=c(nmin,nmax),p0=p0,p1=p1)
plotBDP2(x="n",y="PostProb0or1Successes",n=c(nmin,nmax),pF=pF,shape1F=shape1F,shape2F=shape2F)
```
# Figure 4
Probability of calling efficacy at final analysis with $n_\text{final}=20$ patients as a function of $c_E$. Design parameters are $p_F=0.3, p_E=0.12$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$), $c_F=0.01$ and one interim at $10$ patients. For $p_{true}=p_0=0.12$ this corresponds to type I error, for $p_{true}=p_1=0.3$ this corresponds to power.
```{r, Figure 4,fig.width=6}
n=20
interim.at=10
cE=c(7500:9900)/10000
plotBDP2(x = "cE",
y = "PEcall",
n=n, interim.at=interim.at, p0=p0,p1=p1,
pF=pF,cF=cF,pE=pE,cE=cE,
shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,
col = c("green", "red"))
```
# Figure 5
Decision boundaries for futility (in red) and efficacy (in green) for a design with $c_F = 0.01$ and $c_E = 0.9$. Other design parameters are $p_F=0.3, p_E=0.12$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$).
```{r, Figure 5}
n=40
cE=0.9
cF=0.01
plotBDP2(x = "n",
y = "bFbE",
n=n,pF=pF,cF=cF,pE=pE,cE=cE,
shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,
col=c("red","green"))
```
# Figure 6
Type I error ($\text{CumP}_{\text{callE}}$) and probability of true stopping ($\text{CumP}_{\text{stopF}}$) for the uninteresting response rate $p=p_0=0.12$ (left panel). Power ($\text{CumP}_{\text{callE}}$) and probability of false stopping ($\text{CumP}_{\text{stopF}}$) for the target response rate $p=p_1=0.3$ (right panel). Other design parameters are $c_F = 0.01$ and $c_E = 0.9$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$), interim analyses every $10$ patients.
```{r, Figure 6,fig.width=7}
nvec=c(18:40)
interim.at=c(10,20,30)
ptrue=0.12
plotBDP2(x="n", y="PFstopEcall",
n =nvec, interim.at = interim.at,
pF=pF,cF=cF,pE=pE,cE=cE,ptrue=ptrue,shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E)
ptrue=0.3
plotBDP2(x="n", y="PFstopEcall",
n =nvec, interim.at = interim.at,
pF=pF,cF=cF,pE=pE,cE=cE,ptrue=ptrue,shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E)
```
# Figure 7
Power function ($\text{CumP}_{\text{callE}}$) at $n_\text{final}= 20$ (in green) and $n_\text{final}=30$ (in red) as a function of $p$. Other design parameters are $p_F=0.3, p_E=0.12$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$), $c_F=0.01$ and $c_E=0.3$, interim analyses every $10$ patients.
```{r, Figure 7,fig.width=6}
n=20
interim.at=10
ptrue=c(0:40)/100
plotBDP2(x = "ptrue", y = "PEcall",
n=n, interim.at=interim.at, ptrue=ptrue,
pF=pF, cF=cF, pE=pE, cE=cE, p0=p0, p1=p1,
shape1F=shape1F, shape2F=shape2F, shape1E=shape1E , shape2E=shape2E ,
col = "green")
n=30
interim.at=c(10,20)
plotBDP2(x = "ptrue", y = "PEcall",
n=n, interim.at=interim.at, ptrue=ptrue,
pF=pF, cF=cF, pE=pE, cE=cE, p0=p0, p1=p1,
shape1F=shape1F, shape2F=shape2F, shape1E=shape1E , shape2E=shape2E ,
col = "red",lty=2,add=TRUE)
abline(v=0.12,col="grey",lty=2)
abline(v=0.3,col="grey",lty=2)
```
# Figure 8
Expected number of patients in the trial for $n_\text{final}= 20$ (in green) and $n_\text{final}=30$ (in red) as a function of $p$. Other design parameters are $p_F=0.3, p_E=0.12$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$), $c_F=0.01$ and $c_E=0.3$, interim analyses every $10$ patients. Maximal numbers of patients ($n=20$ and $n=30$, resp.) are shown as dotted lines.
```{r, Figure 8,fig.width=6}
n=30
interim.at=c(10,20)
pvec=c(0:40)/100
interim.at=interim.at[interim.at<n]
plotBDP2(x="ptrue", y="ExpectedNumber",
n=n,interim.at=interim.at,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=pvec,
shape1F=shape1F,shape2F=shape2F,col="red",ylim=c(0,n),cex.lab=1.4)
n=20
interim.at=interim.at[interim.at<n]
plotBDP2(x="ptrue", y="ExpectedNumber",
n=n,interim.at=interim.at,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=pvec,
shape1F=shape1F,shape2F=shape2F,col="green",add=TRUE)
abline(h=20,col="grey",lty=2)
abline(h=30,col="grey",lty=2)
```
# Figure 9
Predictive power (including futility stop) as function of observed responders, evaluated at interim at $10$ patients (left panel) or $20$ patients (right panel). Final analysis at $n_\text{final}= 30$. Other design parameters are $p_F=0.3, p_E=0.12$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$), $c_F=0.01$ and $c_E=0.3$
```{r, Figure 9,fig.width=7}
interim.at=c(10,20)
nfinal=30
plotBDP2(x="k",y="PredictivePower",n=nfinal,interim.at = interim.at, cE=cE,cF=cF, pE=pE,pF=pF,
shape1E=shape1E,shape2E=shape2E,shape1F=shape1F,shape2F=shape2F)
```
# Figure 10
Decision boundaries for futility (in red) and efficacy (in green) for a design with $c_F = 0.01$ and $c_E = 0.9$ for up to $100$ enrolled patients. Other design parameters are $p_F=0.3, p_E=0.12$, prior distributions Beta($p_F,1-p_F$) and Beta($p_E,1-p_E$).
```{r, Figure 10}
n=100
plotBDP2(x = "n",
y = "bFbE",
n=n,pF=pF,cF=cF,pE=pE,cE=cE,
shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,
col=c("red","green"))
```
Session info
------------
```{r}
sessionInfo()
```