What Is Being Predicted?
During a trial, investigators may ask whether the evidence collected so far is promising. A different question is whether the trial, after the planned remaining data are collected, will meet its final success criterion. The second question is forward-looking: it concerns a future decision based on both the data already observed and data that have not yet been collected.
Predictive probability of success (PPS) expresses that forward-looking question as a probability. It is conditional on the current data and on how the trial is planned to proceed. The success criterion must be specified for the final analysis, and the predictive calculation must represent the future data collection and analysis that will actually be used.
The event being predicted is not necessarily “the treatment works.” It is the operational event that the final analysis meets a specified rule. For example, a rule might require the final posterior probability that a treatment effect exceeds a clinically meaningful margin to be at least 0.95. PPS then asks how likely it is that the final posterior probability will reach that threshold, given the current data and planned future collection.
Specify the Rule and the Plan
A PPS value has no complete interpretation on its own. At minimum, it is tied to three things: the final success rule, the probability model, and the remaining-trial plan. The final rule states exactly what outcome counts as success. The model describes how current information and future observations are represented. The plan specifies features such as how many participants remain, their allocation, follow-up, and any planned adaptations.
For instance, suppose the final rule is \(P(\Delta>\delta\mid D_{\mathrm{final}})\geq 0.95\), where \(\Delta\) is the treatment effect and \(\delta\) is a clinically meaningful margin. The final decision is made by applying this rule to the posterior distribution after all planned data are available. PPS is not the current posterior probability \(P(\Delta>\delta\mid D_t)\). One is a probability about the parameter given current data; the other is a probability about a future final decision, conditional on current data and a plan.
The basic justification for the definition is direct. For any future-data outcome, the final rule either is or is not met. The indicator \(S\) records that outcome as 1 or 0. Its conditional expectation is therefore the probability of success:
This identity explains why PPS is a probability of a final event, rather than a summary of how favorable the current estimate looks. The future-data model and the final rule jointly determine which possible future outcomes count as successes.
Worked Examples
Worked Example: A Posterior Go Rule at the Final Analysis
Hypothetical setting. A randomized trial is evaluating whether a new treatment improves a binary clinical outcome. At the interim analysis, 80 participants have completed follow-up; 40 more are planned. The final success rule is that the posterior probability of exceeding the prespecified clinical margin must be at least 0.95. The current posterior probability of exceeding that margin is 0.90.
A predictive analysis under the trial’s stated model and remaining-participant plan gives the following distribution for the final posterior probability. These forecast probabilities are stipulated outputs for this example; the question here is how to interpret and use them.
| Final posterior probability of exceeding the margin | Predictive probability of this outcome |
|---|---|
| 0.90 | 0.15 |
| 0.94 | 0.25 |
| 0.96 | 0.35 |
| 0.98 | 0.25 |
The entries form a probability distribution because \(0.15+0.25+0.35+0.25=1.00\). Under the final rule, only outcomes of at least 0.95 count as successes. Thus the 0.96 and 0.98 outcomes count, while 0.90 and 0.94 do not.
The predictive probability of meeting the final rule is 0.60, or 60%, conditional on the current data, the stated predictive model, and the plan to collect data from 40 additional participants. The current posterior probability of exceeding the margin is 0.90; it is not the PPS. The difference arises because the final rule is a threshold on the final posterior probability, and future data may move that final quantity either above or below the threshold.
Worked Example: The Final Rule Changes the Forecast
Hypothetical setting. In a trial of a treatment for symptom relief, the final success rule requires the posterior probability that the treatment effect exceeds a clinically meaningful improvement of 2 points to be at least 0.90. Current data and a fixed remaining-data plan yield this stipulated predictive distribution for the final posterior probability:
| Final posterior probability of improvement greater than 2 points | Predictive probability |
|---|---|
| 0.82 | 0.20 |
| 0.89 | 0.30 |
| 0.93 | 0.35 |
| 0.97 | 0.15 |
For the prespecified threshold of 0.90, the outcomes 0.93 and 0.97 meet the final rule. Therefore,
Suppose, solely to examine the role of the rule, that the threshold were instead 0.85, with the same data model and plan. The outcomes 0.89, 0.93, and 0.97 would then count as successes:
The change from 0.50 to 0.80 does not reflect new evidence. It follows from changing the definition of final success. In an actual trial, the final rule should be prespecified; changing it after viewing interim data would change the decision problem and can undermine the intended interpretation of the analysis.
Worked Example: The Remaining-Data Plan Matters
Hypothetical setting. Investigators are studying whether a new intervention reduces a clinical score. The final rule is fixed: the final posterior probability of a reduction greater than 1 point must be at least 0.95. A predictive analysis reports the following results for two possible plans, conditional on the same current data. Each row lists predictive probabilities for the final posterior probability of exceeding 1 point.
| Final posterior probability | Plan A: 20 additional participants | Plan B: 60 additional participants |
|---|---|---|
| 0.90 | 0.30 | 0.10 |
| 0.94 | 0.30 | 0.20 |
| 0.96 | 0.25 | 0.35 |
| 0.98 | 0.15 | 0.35 |
Under the fixed 0.95 final threshold, the successful outcomes are 0.96 and 0.98. For Plan A,
For Plan B,
The forecast is 40% under Plan A and 70% under Plan B. These are plan-conditional forecasts, not guarantees that enrolling more participants will improve the observed treatment effect or ensure success. The numerical distributions are hypothetical outputs from the model; their purpose is to show that a PPS must be reported together with the future-data plan it conditions on.
Assumptions, Diagnostics, and Limitations
A predictive success calculation is meaningful only if its assumptions reasonably represent the trial. The current data must be correctly represented in the analysis, and the model must provide a defensible probability distribution for future outcomes. Planned enrollment, treatment allocation, outcome definitions, follow-up, and analysis rules must be represented accurately. If the design includes adaptations, the predictive plan must account for how those adaptations will operate; assuming a fixed plan when actual data collection will respond to interim results can produce a forecast for the wrong trial.
Useful checks are practical as well as statistical. Confirm that the predictive probabilities over possible final outcomes sum to 1, that the final success rule is applied consistently, and that the remaining sample size and follow-up assumptions match the operational plan. Check whether the forecast is driven by a small number of implausible future outcomes or by assumptions that are weakly supported by current information. If the model is poorly calibrated for the endpoint or population, a precise-looking PPS may still be misleading.
PPS is conditional, not unconditional. It is not the probability that the treatment is truly effective, the probability that the trial will be completed as planned, or a promise about what will happen. It also does not automatically include operational risks such as delayed enrollment, loss to follow-up, or changes in standard care. Such features matter only if they are represented in the predictive model and plan.
The principal alternatives answer different questions. A current posterior go rule applies a threshold to present posterior evidence and makes an action decision now. PPS forecasts whether the final analysis will satisfy a specified rule. A frequentist conditional-power calculation commonly evaluates the chance of a final test crossing its boundary under a selected fixed effect assumption; PPS instead uses a Bayesian predictive distribution and can target a final Bayesian rule. These quantities may be informative alongside one another, but they are not interchangeable: they can use different assumptions and define success differently.
Common Pitfalls
- Calling current evidence the predictive probability. A current posterior probability of 0.90 does not mean there is a 90% chance of meeting the final rule. PPS requires a distribution for future data and the final decision event.
- Reporting a number without its conditions. “PPS is 0.60” is incomplete. State the current analysis, final success rule, model, and remaining-data plan that define the forecast.
- Changing the rule without recognizing the consequence. A different posterior threshold defines a different success event and can change PPS even when the data and plan are unchanged.
- Interpreting a larger planned sample as guaranteed success. More planned observations can change the predictive distribution, but they cannot guarantee a favorable final result.
- Using a model for a trial that will not be conducted. If actual allocation, follow-up, or adaptive decisions differ from the plan used in the prediction, the reported PPS may no longer describe the trial in progress.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A final posterior go rule requires \(P(\Delta>\delta\mid D_{\mathrm{final}})\geq 0.95\). Explain in one sentence what PPS refers to in this setting.
- A predictive distribution assigns probabilities 0.10, 0.25, 0.40, and 0.25 to final posterior probabilities 0.88, 0.93, 0.96, and 0.99, respectively. What is PPS for a final threshold of 0.95?
- For the distribution in Question 2, what is PPS if the final threshold is 0.90? Why does the answer change?
- True or false: If the current posterior probability that the treatment effect exceeds the clinical margin is 0.95, the PPS must also be 0.95. Explain.
- Name two parts of the remaining-trial plan or model that should be checked before interpreting a PPS estimate.
Solutions. (1) It is the conditional chance, given current data and the stated remaining-data plan and model, that the final posterior probability will be at least 0.95. (2) The successful final values are 0.96 and 0.99, so \(\mathrm{PPS}=0.40+0.25=0.65\). (3) Values 0.93, 0.96, and 0.99 meet the 0.90 threshold, so \(\mathrm{PPS}=0.25+0.40+0.25=0.90\). The final event has changed because the threshold is lower. (4) False. The current posterior probability concerns the treatment effect given current data; PPS concerns whether a final rule will be met after future data are collected. (5) Valid answers include remaining enrollment, allocation, follow-up, the future-outcome model, and any planned adaptations. The key is that the prediction must correspond to how the trial will actually proceed.