Introduction

In this vignette, the functionality of the brada package is outlined. The brada package provides access to functions which help to plan, analyze and conduct Bayesian response-adaptive (clinical) trial designs. In the current version, the brada package supports only phase IIA trials with a binary endpoint for response or success. Also, the scope of the package is on group-sequential designs, where based on interim analyses the trial can be stopped early for futility or efficacy. In the future, it is planned to include more Bayesian response-adaptive designs in the package.

The outline of the vignette is as follows:

This vignette builds the starting point, and the vignettes above provide further details.

The predictive probability design

The general setting considered in is a single-arm phase IIA trial with the goal to evaluate the response rate \(p\geq 0\) for a new drug or treatment. The null hypothesis \(H_0:p\leq p_0\) is tested against the alternative \(H_1:p>p_1\), where \(p_0,p_1\in [0,1]\), \(p_0 \leq p_1\) and \(p_0\) is a predefined threshold for determining the minimum clinically important effect (Kelter, 2021b). For simplicity, assume a Beta prior \(p\sim B(a_0,b_0)\) is selected for the response rate \(p\), which offers a broad range of flexibility in terms of modelling the prior beliefs about \(p\). For the Beta prior, \(a_0/(a_0+b_0)\) is the mean and \(a_0\) and \(b_0\) can be interpreted as the numbers of effective prior responses and non-responses. Thus, \(a_0+b_0\) gives a measure of how informative the prior is with larger values of \(a_0+b_0\) indicating a more informative prior distribution.

Let \(N_{\text{max}}\) be the maximum number of patients which is possibly recruited during the trial, and let \(X\) be the random variable which measures the number of responses in the current \(n\) enrolled patients, where \(n\leq N_{\text{max}}\). A reasonable assumption is that \(X\) follows a binomial distribution with parameters \(n\) and \(p\), \(X\sim \text{Bin}(n,p)\). The \(B(a_0,b_0)\) distribution is a conjugate prior for the binomial likelihood, and thus the posterior \(P_{p|X}\) is also Beta-distributed : \[ p|X=x\sim B(a_0+x,b_0+n-x) \] The idea of the predictive probability approach consists of analyzing the interim data to project whether the trial will result in a conclusion that the drug or treatment is effective or ineffective. Efficacy is declared when the posterior probability fulfills the constraint \[ P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T \] for some threshold \(\theta_T \in [0,1]\). That is, when \(n\) patients have been enrolled in the trial out of which \(X=x\) show a response, there remain \(m=N_{\text{max}}-n\) patients which can be enrolled in the trial. If out of these remaining \(m\) exactly \(i\) respond to the treatment, and the conditional probability \(P_{p|X,Y}(p>p_0|X=x,Y=i)\) is larger than a prespecified threshold \(\theta_T\), say, \(\theta_T=0.95\), this will be interpreted as the drug being effective. However, as the number \(Y\) of responses in the remaining \(m=N_{\text{max}}-n\) patients which can be enrolled in the trial is uncertain, this uncertainty must be modeled, too. Marginalizing out \(p\) of the binomial likelihood yields the prior predictive distribution which is Beta-Binomial \[ Y\sim \text{Beta-Binom}(m,a_0+x,b_0+n-x) \] Additionally, from the conjugacy of the beta prior we have the posterior \[ P_{p|X,Y}(X=x,Y=i)\sim B(a_0+x+i,b_0+N_{\text{max}}-x-i) \] and the expected predictive probability of trial success – henceforth abbreviated \(\text{PP}\) – can now be calculated by weighing the posterior probability of trial success \[ P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T \] when observing \(X=x\) and \(Y=i\) with the prior predictive probability \(P_{Y|X}(Y=i|X=x)\) of observing \(Y=i\) responses in the remaining \(m=N_{\text{max}}-n\) patients, after \(X=x\) responses have been observed among the current \(n\) patients: \[ \text{PP}=\mathbb{E}\left [ 1_{P_{p|X,Y}(p>p_0|X,Y)>\theta_T}|x\right ]=\int_{\mathcal{Y}}1_{P_{p|X,Y}(p>p_0|X,Y)>\theta_T}dP_{Y|X=x}=\sum_{i=0}^m P_{Y|X=x}(i)\cdot 1_{P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T} \] where \[ 1_{P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T}:=\begin{cases} 1, \text{ if } P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T\\ 0, \text{ if } P_{p|X,Y}(p>p_0|X=x,Y=i)\leq \theta_T \end{cases} \] is an indicator which measures whether the evidence against \(H_0:p\leq p_0\) is large enough – that is, \(P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T\) – conditional on \(X=x\) and \(Y=i\) or not. The predictive probability \(\text{PP}\) thus quantifies the expected predictive probability of trial success. To employ the approach in practice, futility and efficacy thresholds \(\theta_L\) and \(\theta_U\) out of \([0,1]\) must be fixed, so that the value of \(\text{PP}\) can be compared to these thresholds based on available interim data \(X=x\). Then, if \(\text{PP}<\theta_L\), respectively \(\text{PP}>\theta_U\), the trial can be stopped early for futility, respectively efficacy. Fig. 1 shows the PP group-sequential design, see also Berry (2011), Rosner (2020) and Rosner (2021). Note that in practice, \(\theta_U=1.0\) is often preferred because if the drug is effective one does not want to stop the trial. However, \(\theta_L>0\) is important to stop the trial in case the drug or treatment is not effective to avoid a waste of resources.

The predictive probability design for binary endpoints

Fig. 1 shows that based on the current data, the probability to obtain \(Y=i\) successes is weighted with the probability of success \(P_{p|X,Y}(p>p_0|X=x,Y=i)>\theta_T\) for each \(i=0,m\). This weighted sum is the predictive probability of trial success, should the trial be continued until the maximum trial size \(N_{max}\).

Getting started

To apply the package, first, load the package:

library(brada)