Efficacy and toxicity monitoring via Bayesian predictive probabilities in phase II clinical trials

Abstract

Bayesian monitoring strategies based on predictive probabilities are widely used in phase II clinical trials that involve a single efficacy binary variable. The essential idea is to control the predictive probability that the trial will show a conclusive result at the scheduled end of the study, given the information at the interim stage and the prior beliefs. In this paper, we present an extension of this approach to incorporate toxicity considerations in single-arm phase II trials. We consider two binary endpoints representing response and toxicity of the experimental treatment and define the result as successful at the conclusion of the study if the posterior probability of an high efficacy and that of a small toxicity are both sufficiently large. At any interim look, the Multinomial-Dirichlet distribution provides the predictive probability of each possible combination of future efficacy and toxicity outcomes. It is exploited to obtain the predictive probability that the trial will yield a positive outcome, if it continues to the planned end. Different possible interim situations are considered to investigate the behaviour of the proposed predictive rules and the differences with the monitoring strategies based on posterior probabilities are highlighted. Simulation studies are also performed to evaluate the frequentist operating characteristics of the proposed design and to calibrate the design parameters.

Appendices

Appendix 1: Detailed calculations to obtain the predictive probability of interest

It can be useful to illustrate the detailed calculations necessary to obtain the predictive probability in (5). We use very small and unrealistic sample sizes with the only purpose of exemplifying. More specifically, we set the total sample size equal to $N=10$ and assume that at the interim stage the data ${\varvec{x}}= (1,4,0,2)$ have been observed among $n=7$ current patients. The first column of Table 10 provides all the possible results ${\varvec{Y}}$ obtainable among the 3 future patients, that is all the possible entries of a $2 \times 2$ contingency table consistent with a total equal to 3. For each possible outcome, the second column provides its posterior predictive probability computed by using the Multinomial-Dirichlet distribution in (3). Then, the following two columns contain the posterior probabilities of an high efficacy (i.e. $Pr(p_E> p_E^* |{\varvec{\alpha }}, {\varvec{x}}, {\varvec{y}})$) and of a small toxicity (i.e. $Pr(p_T< p_T^* |{\varvec{\alpha }}, {\varvec{x}}, {\varvec{y}})$), based on the target values $p^*_E= 0.4$ and $p^*_T= 0.3$. The thresholds $\lambda _E$ and $\lambda _T$ are both set equal to 0.9 and in the table the color gray is used to highlight the cases in which the posterior quantities of interest are above 0.9. The final results are considered positive if they satisfy both the conditions on the posterior probabilities. In our case, we have three positive results and, simply by summing their predictive probabilities, we obtain the predictive probability of interest. Thus, in this fictitious example, the value of PP is equal to 0.5324 and, for reasonable choices of $\theta _L$ and $\theta _U$, the decision is to continue the trial to gather additional information before drawing a final conclusion.

Table 10 PP computation when ${\varvec{x}}= (1,4,0,2)$, $n=7$, $N=10$, $p^*_E= 0.4$, $p^*_T= 0.3$, $\lambda _E=\lambda _T=0.9$ and ${\varvec{\alpha }} = (0.25,0.25,0.25,0.25)$

Appendix 2: Description of the software tool to compute PP

We provide a very easy-to-use software tool to compute the predictive probability in (5) to implement the proposed decision rules. The program is coded in the R-programming language and uses the package gWidgets, which offers a relatively simple way of writing graphical user interfaces (GUIs). This package is therefore supposed to be correctly installed within R, as well as the packages gWidgetsRGtk2 and RGtk2. The file containing the R script is called “PP-BivEndpoints.R”.

The code can be run directly using R or its integrated development environment RStudio. When the script is contained in the current working directory, it is sufficient to let R run the source file by using the command line

> source("PP-BivEndpoints.R")

As a result a graphical interface is generated, where the user can insert the design parameters of interest. For instance, in Fig. 3 we show the GUI with the input parameters that correspond to the top-left cell of Table 3.

Fig. 3

Graphical user interface to compute the predictive probability in (5)

By clicking on the button “Compute PP”, the output produced on the R console summarizes the design parameters choices and provides the predictive probability of interest. For instance, using the input entries in Fig. 3, we obtain the following output:

Finally, by selecting “Show additional results”, more details about the calculations performed to obtain the value of PP are displayed. More specifically, the output provides the list of the future data that satisfy the conditions for a successful treatment, their posterior predictive probabilities and the corresponding posterior quantities of interest. In other words, results similar to those presented in Table 10 are displayed with reference only to the future outcomes that fulfill the conditions in (4)

Appendix 3: Numerical algorithm to evaluate the operating characteristics of the proposed design

First of all, we set suitable values for the minimum sample size $N_{min}$, the maximum planned sample size N, the target values $p_E^*$ and $p_T^*$, the vector of hyperparameters ${\varvec{\alpha }}$ and the probability thresholds $\lambda _E$, $\lambda _T$ and $\theta _L$. We also need to fix an assumed scenario by setting the true value of ${\varvec{p}}$, here denoted by ${\varvec{p}}^{\varvec{true}}$, and the number of simulated trials B.

Once these design parameters have been fixed, a single clinical trial with stopping rules based on the proposed approach can be simulated through the following steps:

• (a1) Set $n_{ASS}=0$ (the actually achieved sample size) and $j=1$ (an auxiliary index for indicating potential additional patients)

• (a2) Enrol the first $N_{min}$ patients into the trial by simulating the current data ${\varvec{x}}_{N_{min}}$ from a Multinomial distribution with index $N_{min}$ and parameter ${\varvec{p}}^{\varvec{true}}$.

  Compute PP based on ${\varvec{x}}_{N_{min}}$ by using formula (5) and

  • if $PP \le \theta _L$, the algorithm stops and set $n_{ASS}=N_{min}$;

  • if $PP > \theta _L$, the algorithm proceeds to the following step.

• (a3) Enrol the $(N_{min}+j)$th patient into the trial by simulating a value from a Multinomial distribution with index 1 and parameter ${\varvec{p}}^{\varvec{true}}$. Add the simulated value to the previous current data, obtaining the data ${\varvec{x}}_{N_{min}+j}$.

  Compute PP based on ${\varvec{x}}_{N_{min}+j}$ by using formula (5) and

  • if $PP \le \theta _L$, the algorithm stops and set $n_{ASS}=N_{min}+j$;

  • $PP > \theta _L$, set $j=j+1$ and, if $j \le N-N_{min}-1$, repeat the step (a3), otherwise proceeds to the following step.

• (a4) If $j = N-N_{min}$, the maximum sample size has been reached; therefore set $n_{ASS}=N$.

We simulate B clinical trials using the steps described above. Each simulated trial provides the value of the actually achieved sample size $n_{ASS}$. A value of $n_{ASS}$ less than N indicates that the trial has been terminated early because at a certain interim look the predictive probability of interest was below the desired threshold. In this case $H_0$ is not rejected and the experimental treatment is considered not sufficiently promising. When, instead, $n_{ASS}=N$ the null hypothesis is rejected and the new treatment is declared worthy of further investigation in phase III trials.

Therefore, the frequency of simulated trials where $n_{ASS}=N$ represents an empirical evaluation of the Type I error rate, under a scenario where $H_0$ is true, and of the statistical power when we assume that $H_0$ is not true.