For events of binary type, the total number of events observed in the trial follows \(B(n,p)\) distribution, where \(n\) is the number of the samples and \(p\) is an unknown parameter. From the Bayes point of view, the historical data can be used for the prior distribution of the unknown parameter \(p\). Since the total number of an event follows the binomial distribution, a beta distribution is used as the prior distribution of \(p\).

The prior information on \(p\) from \(_\) Bernoulli events can be denoted by a Beta distribution whose mean is \(_\), i.e., \(Beta(__,__))\). If \(_\) future Bernoulli events are observed and \(_\) of them are success events, letting \(_=(__+_)/_\) and \(_=_+_\), according to the likelihood of the observed data, the posterior distribution of p is also beta distribution [16, 17]: \(Beta(__,__))\).

Based on this, the control limits for a future sample of \(n\) Bernoulli events with \(T\) successes can be deduced. Given \(n\) and \(p\), the distribution of \(T\) is binomial, and the posterior predictive distribution of \(T\) can be derived as a beta-binomial distribution. That is, for future samples \(D=\left\_\right\},i=1,..,n\), the posterior predictive distribution of \(T (__)\) can be expressed as:

$$f\left( } \right) = ^ \left( p_ ,n - T + n_ \left( } \right)} \right)} \right)} \mathord^ \left( p_ ,n - T + n_ \left( } \right)} \right)} \right)} p_ ,n_ \left( } \right)} \right)} \right),\,0 \le T \le n}}} \right. \kern-0pt} p_ ,n_ \left( } \right)} \right)} \right),\,0 \le T \le n}}$$

(1)

The median of the posterior predictive distribution of \(T\) can be calculated easily through the derivation above. The median is used as a measurement to assess whether the current observation exceeds the QTL threshold, provided that the general performance of current data can be expressed by the median of the posterior predictive distribution.

Using an example of proportion of participants with protocol deviations (PDs) of special interest in a trial, historical data shows that the average proportion and 95th quantile of participants with PDs of interest are 17.95% and 27.17%, respectively. The prior distribution of \(p\) based on historical data was deduced as \(Beta\left(\mathrm\right)\).

Assuming that this QTL parameter was monitored in a trial with 300 enrolled participants and the expected proportion of participants with PDs is 17.95% (i.e., generating a sample of size 300 from the Bernoulli (0.1795)), based on the above derivation, the prior and posterior predictive probability density diagram of \(T\) (number of participants with PDs) in this process can be obtained, as shown in Fig. 7.

The quality tolerance limit uses the 95th quantile (27.17%) of historical data with the 80th quantile of the prior predictive distribution specified as the secondary limit (rendering sufficient time for actions) in the QTL monitoring process. The control chart using median of the posterior prediction distribution as the measurement to assess future data are shown in Fig. 8.

In-control process for proportion of participants with PDs of interest.

As previously mentioned, monitoring QTL parameters starts when the sample size is large enough, in this example, monitoring begins when the number of enrolled participants is 30, hence the above figure only shows the results when the number of participants ≥ 30. As can be seen from Fig. 8, the median of the posterior distribution doesn’t breach QTL threshold, which indicates an in-control process.

Figure 9 shows an out-of-control process on monitoring proportion of participants with interested PDs for the example of Bernoulli (0.30) event, the median of posterior predictive distribution of PD proportion breach both tolerance limits successively.

Out-of-control process for proportion of participants with PDs of interest.

This method is generally applicable to large sample size trials. Moreover, since the trial-specific assessment (median from the posterior predictive distribution) uses both historical and current data with thresholds established using historical data, it is important that the historical data are sufficiently homogeneous and similar in indication and ideally with a similar compound under investigation.

In practice, differences between trial data and historical data may be found as clinical trials are not like manufacturing. For example, there may be none, or insufficient historical data available for new clinical trials for new compounds or new indications or rare diseases. Should the historical data be inconsistent with current data, there is a potential for an inflated false alarm probability and may cause excursions on each run. One can use the minimal sufficient statistic method by Evans and Moshonov[18] to check for prior-data conflict.

When prior-data conflicts exist, one can consider adjusting the threshold or using power priors to update the posterior predictive distribution to prolong the run length of the process [19].