Vose SoftwareModelRisk example · Pharmaceuticals and biotech
Bayesian Estimate of a Trial's Success Probability
Halfway through a Phase II trial: how likely is it to succeed — and should it stop, or carry on?
From the model’s saved run
What the interim says
The drug's true response rate, after the earlier study and the interim data.
| Measure | Unit | Value |
|---|---|---|
| Response rate at the interim | probability | 35.0% |
| Posterior mean of the true rate | probability | 32.3% |
| …simulated mean | probability | 32.2% |
| 95% credible interval: from | probability | 21.3% |
| …to | probability | 44.3% |
| Chance the true rate beats the control rate | probability | 98.7% |
Histogram of True response rate from the saved simulation.
The business question
A biotech company is running a single-arm Phase II trial of a new oncology drug in 100 patients. The trial succeeds if at least 29 patients respond: with the historical response rate of 20%, so many responses would happen by chance only 2% of the time. An earlier Phase Ib study saw 5 responses in 20 patients.
At the planned interim analysis, after 40 patients, 14 have responded — 35%. The data monitoring committee asks the usual questions: how good does the drug look, how likely is the trial to reach its goal, and should it stop early — for futility if success is unlikely, for efficacy if it is all but certain?
The plug-in answer, assuming the remaining patients respond at 35%, says the trial is 96% sure to succeed. Is it?
- Who it is for
- Biostatisticians, clinical development teams and data monitoring committees; portfolio teams estimating the chance of success.
- Main outputs
- The response rate after the interim: estimate, 95% credible interval, chance of beating the control rate · the predictive probability of success, simulated and exact, against the plug-in conditional power · the decision · how the prior changes the answer · the interim results at which to stop.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
Patients in the trial, the control rate and the significance level that set the success threshold. Patients and responders at the interim; the earlier study that gives the prior; the futility and efficacy thresholds; three alternative priors.
Analysis sheet
The prior Beta(1 + responders, 1 + non-responders) from the earlier study, updated with the interim data, gives the posterior Beta. Each iteration draws a response rate from it (VoseBeta) and the remaining patients' responders (VoseBinomial), and checks whether the trial reaches its threshold. The exact predictive probability comes from the Beta-Binomial distribution (VoseBetaBinomialProb).
Results sheet
The posterior response rate and its credible interval. The predictive probability of success, simulated and exact, against the plug-in conditional power; the decision; four priors compared; the predictive probability for every possible interim result, with the stopping zones.
ModelRisk techniques
Find similar models
- Product
- Type
- Application
- Uses
The results in detail
Will the trial succeed?
Success: at least 29 responders of 100 patients.
Histogram of Responders at the final analysis from the saved simulation.
| Measure | Unit | Value |
|---|---|---|
| Predictive probability of success, simulated | probability | 82.8% |
| …exact (Beta-Binomial) | probability | 83.0% |
| Conditional power at the interim rate | probability | 96.4% |
| Final response rate, 10th percentile | probability | 27% |
| Final response rate, 90th percentile | probability | 40% |
| Decision | Continue the trial |
How much the prior matters
The same interim data on four priors.
Line chart: Predictive probability, Futility threshold, Efficacy threshold.
| Prior | α | β | Posterior mean | Beats control | Predictive probability |
|---|---|---|---|---|---|
| Phase Ib result | 6.0 | 16.0 | 32.3% | 98.7% | 83.0% |
| Flat | 1.0 | 1.0 | 35.7% | 99.0% | 88.9% |
| Sceptical | 4.0 | 16.0 | 30.0% | 96.3% | 75.0% |
| Enthusiastic | 8.0 | 12.0 | 36.7% | 99.8% | 92.8% |
| Measure | Unit | Value |
|---|---|---|
| Stop for futility with at most | responders | 8 |
| Stop for efficacy with at least | responders | 16 |
How to read these results
- Fourteen responders in 40 patients is 35%, but the drug's true response rate is still uncertain. Combined with the earlier study's 5 of 20, the best estimate is about 32%, with a 95% credible interval from 21% to 44%. The chance that it beats the control rate of 20% is about 99%.
- The trial needs at least 29 responders of 100, so 15 more among the remaining 60. The predictive probability of success is about 83% — the simulated value agrees with the exact Beta-Binomial one within sampling error — so the trial should continue: neither stopping rule applies.
- The conditional power, 96%, assumes the remaining patients respond at exactly the interim rate. It ignores that the rate itself is uncertain and overstates the chance of success — a common mistake in interim reports.
- The prior matters, but not decisively here: a flat prior gives 89%, a sceptical prior centred on the control rate 75%, an enthusiastic one 93% — all leading to the same decision. Where priors would disagree, the prior must be agreed before the interim, not after.
- The boundaries tell the committee in advance what to do: 8 responders or fewer at the interim would mean stopping for futility (a predictive probability below 10%), 16 or more stopping for efficacy (above 95%).
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
The trial 3 inputs
A single-arm trial against a historical control rate.
| Assumption | Unit | Value |
|---|---|---|
| Patients in the trial | patients | 100 |
| Control response rateResponse to standard treatment, from published studies. | probability | 20% |
| Significance level, one-sidedSets the success threshold. | probability | 2.5% |
Interim data 2 inputs
At the planned interim analysis.
| Assumption | Unit | Value |
|---|---|---|
| Patients evaluated | patients | 40 |
| Responders | patients | 14 |
Prior 2 inputs
The earlier study, on a uniform prior: Beta(1 + responders, 1 + non-responders).
| Assumption | Unit | Value |
|---|---|---|
| Earlier study: patientsThe Phase Ib dose-expansion cohort. | patients | 20 |
| Earlier study: responders | patients | 5 |
Stopping rules 2 inputs
Applied to the predictive probability of success.
| Assumption | Unit | Value |
|---|---|---|
| Stop for futility below | probability | 10% |
| Stop for efficacy above | probability | 95% |
Alternative priors 6 inputs
For the sensitivity table on the Results sheet.
| Assumption | Unit | Value |
|---|---|---|
| Flat prior: α | 1.0 | |
| Flat prior: βBeta(1, 1): every rate equally likely. | 1.0 | |
| Sceptical prior: α | 4.0 | |
| Sceptical prior: βCentred on the control rate, 20%. | 16.0 | |
| Enthusiastic prior: α | 8.0 | |
| Enthusiastic prior: βCentred on 40%. | 12.0 |
Units and conventions
- Response rate
- The drug's true chance that a patient responds — unknown, described by a Beta distribution.
- Success
- At least the threshold number of responders among all patients at the final analysis.
- Predictive probability
- The chance of success given the data so far, averaging over what the true rate might be.
- Conditional power
- The chance of success if the remaining patients respond exactly at the interim rate.
Key assumptions and limitations
- Patients
- Independent; every patient's response has the same chance.
- Prior
- The earlier study's patients are like this trial's: its result enters with full weight on a uniform prior.
- Threshold
- Set in advance from the control rate and the significance level (one-sided).
- Data
- The drug, the trial and all figures are invented for this example.
How to use it
- Enter your trial's size, control rate and significance level, and the interim data.
- Enter the evidence for the prior — an earlier study's responders and patients — and the stopping rules.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Read the decision; check how much it depends on the prior before acting on it.
ModelRisk functions used
- VoseBeta
- The response rate, drawn from its posterior distribution.
- VoseBinomial
- The responders among the remaining patients, given the drawn rate.
- VoseBetaBinomialProb
- The exact predictive probability: the Beta-Binomial cumulative probability.
- VoseInput, VoseOutput, VoseSimMean, VoseSimPercentile
- The simulated rate and outcomes, and their statistics.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Lee, J. J. & Liu, D. D. (2008). A predictive probability design for phase II cancer clinical trials. Clinical Trials 5(2), 93–106.
- Berry, S. M., Carlin, B. P., Lee, J. J. & Müller, P. (2010). Bayesian Adaptive Methods for Clinical Trials. CRC Press.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
Related example models
- ModelRisk★★★★★Drug Development Risk-Adjusted NPVWhat is a compound in Phase II worth — and should we develop it ourselves or license it out?Phases with uncertain cost, time and success, then launch and patent expiry: the textbook rNPV against the simulated NPV.
- ModelRisk★★★★★Launch Sales Forecast: Peak Sales and UptakeHow big will the new drug be?A patient funnel, an expert’s uptake curve as a correlated time series, and a competitor that may enter: peak sales against plan.
- ModelRisk★★★★★Patient Recruitment and Trial TimelinesWhen will the last patient be enrolled — and how many sites does it take to be confident of the deadline?Sixty sites, each with its own opening time and recruitment rate: a Poisson–Gamma model of enrolment, month by month.
An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.