Why an Event Projection Needs More Than One Date
A projected date for reaching an event target is a forecast, not a promise. Even if a model gives one central date, the actual target may be reached earlier or later because future events vary and because the assumptions used for the projection may not hold. Communicating only a single date can therefore suggest more certainty than the data support.
A useful projection states what date or time is being forecast, gives a suitable measure of uncertainty, and explains the main assumptions behind that uncertainty. The measure might be a prediction interval under a specified model, or a set of dates under alternative plausible assumptions. These are different summaries and should not be presented as interchangeable.
A Predictive Range Under a Simple Event Model
To see what a probability-based range means, suppose that, from the forecast origin onward, events follow a Poisson process with constant rate \(\lambda\) events per month. This model assumes independent event increments, a constant average rate over the forecast period, and no dependence between future increments. For the calculation below, treat \(\lambda\) as known. These are simplifying assumptions, not facts guaranteed by the observed data.
Let \(r\) be the number of additional events needed to reach the target, and let \(T\) be the waiting time in months until those \(r\) events occur. The count by time \(t\), \(N(t)\), has a Poisson distribution with mean \(\lambda t\). Therefore, the event target has been reached by \(t\) exactly when \(N(t)\ge r\).
Equivalently, \(T\) has a gamma distribution with shape \(r\) and rate \(\lambda\). Its mean is \(r/\lambda\), and its variance is \(r/\lambda^2\). A central \(100(1-\alpha)\%\) predictive interval is formed from the \(\alpha/2\) and \(1-\alpha/2\) quantiles of this distribution. Its probability statement is conditional on the model and the specified rate: if the assumptions and rate were correct over repeated comparable forecasting situations, the interval procedure would contain the event time at the stated frequency.
Worked Example: A Predictive Interval for the Target Date
Suppose a hypothetical trial needs 10 more adjudicated events, and the forecast model assumes a constant rate of 2 events per month from the forecast origin. The origin is 1 August 2026. Calculate the expected waiting time and an approximate central 90% predictive interval.
Under the stated Poisson-process model, \(r=10\) and \(\lambda=2\) events per month. The expected waiting time is
For the interval, \(2\lambda T\) has a chi-square distribution with \(2r=20\) degrees of freedom. The 5th and 95th percentiles of that chi-square distribution are approximately 10.85 and 31.41. Thus,
A clear summary is: “Under a constant rate of 2 events per month, the expected time to the remaining 10 events is 5.00 months; the model-based central 90% predictive interval is approximately 2.71 to 7.85 months from 1 August 2026.” The expected time corresponds roughly to early January 2027, while the interval spans roughly late October 2026 to late March 2027. These calendar conversions are approximate because months have different lengths.
The interval is not a guarantee that the target will be reached within that period. Nor does it include uncertainty about whether 2 events per month is a suitable rate: in this example, the rate was treated as known. The model and rate assumption must be communicated with the numbers.
From Model-Based Uncertainty to Scenario Ranges
A predictive interval describes random variation under a probability model. In practice, uncertainty about future recruitment, event rates, dropout, or operational processing may also matter. One way to show the effect of such assumptions is to calculate projections under several plausible scenarios. A scenario range is useful for sensitivity analysis, but it does not have a probability interpretation merely because it has lower and upper dates.
Worked Example: Three Accrual Scenarios and an Operational Delay
A hypothetical study needs 8 more events. Consider three constant-rate planning scenarios: 2.5 events per month in an optimistic scenario, 2.0 in a central scenario, and 1.5 in a conservative scenario. Suppose the corresponding assumed delays from reaching the adjudicated-event target to having the analysis ready are 0.50, 0.75, and 1.00 months.
For each scenario, calculate the mean time to the event target as \(r/\lambda\), then add that scenario’s operational delay. The resulting forecast is:
| Scenario | Rate (events/month) | Mean time to target (months) | Assumed delay (months) | Analysis-ready time (months) |
|---|---|---|---|---|
| Optimistic | 2.5 | 8 / 2.5 = 3.20 | 0.50 | 3.70 |
| Central | 2.0 | 8 / 2.0 = 4.00 | 0.75 | 4.75 |
| Conservative | 1.5 | 8 / 1.5 = 5.33 | 1.00 | 6.33 |
The scenario-based analysis-ready times range from 3.70 to 6.33 months after the forecast origin. The correct interpretation is that these are results under three specified combinations of rate and delay assumptions. It is not correct to call 3.70 to 6.33 months a 90% interval, or to say that the target has a particular probability of being reached within that range, because no probabilities were assigned to the scenarios.
The event-target time and the analysis-ready time are separate quantities. The delay is added here as a simplifying fixed scenario value. If the delay itself varies, a probability-based forecast for the analysis-ready date would need to account for that variation as well.
Updating a Projection Without Overstating Precision
A projection can be updated when the observed event count or the remaining target changes. The updated forecast should make clear that it is conditional on information available at its new forecast origin. It should not be described as though the earlier forecast had been a fixed commitment. The same principles apply whether the projection is based on a constant rate, a changing recruitment pattern, or a more detailed accrual model.
Worked Example: A Revised Forecast for an Analysis-Ready Date
At a new forecast origin, a hypothetical trial needs 6 more adjudicated events. Assume a constant rate of 1.8 events per month and, for simplicity, a fixed operational delay of 0.50 months. Calculate the expected analysis-ready time and an approximate central 80% predictive interval under the event model.
The expected time to the target is \(6/1.8=3.333\) months. Adding the fixed delay gives an expected analysis-ready time of \(3.333+0.50=3.833\) months. For an 80% interval for the event time, use the 10th and 90th percentiles of a gamma distribution with shape 6 and rate 1.8. Since \(2\lambda T\) has a chi-square distribution with 12 degrees of freedom, the relevant chi-square quantiles are approximately 6.304 and 18.549:
Adding the fixed 0.50-month delay shifts both endpoints, giving an approximate analysis-ready interval of 2.251 to 5.653 months after the new forecast origin. A suitable report would identify this as an 80% model-based predictive interval under the stated rate and fixed-delay assumptions. If the delay is not actually fixed, this calculation does not include its uncertainty.
How to Present the Forecast
A concise presentation can still distinguish the forecast target, central estimate, uncertainty measure, and assumptions. State the forecast origin and the units, and identify whether the reported range is probabilistic or scenario-based. If converting times to calendar dates, specify that the dates are approximate when the calculation uses month-based time.
Distinguish reaching the event target from completing adjudication, database preparation, or another operational step before an analysis can occur.
Label a model-based interval with its coverage level, or label an assumption-based range as a set of scenarios. Do not imply that a scenario range has a probability it was not designed to provide.
Briefly identify the assumptions that materially affect the range, such as the event rate or operational delay, and say whether the displayed uncertainty includes uncertainty in those quantities.
Diagnostics, Alternatives, and Limitations
A narrow interval is not automatically a reliable interval. Check whether the recent accrual pattern is reasonably compatible with the assumed future pattern, whether the event rate has changed, and whether the remaining events are expected to accrue under conditions comparable to those used for the forecast. Earlier material on monitoring and updating event accrual, and on diagnosing miscalibrated projections, provides relevant checks. A marked change in recruitment, event ascertainment, or processing is a reason to reconsider a constant-rate projection rather than to report its range without qualification.
Three common communication choices have different meanings:
| Approach | What it communicates | Main limitation |
|---|---|---|
| One expected date | A central model-based time, such as \(r/\lambda\). | Hides the spread of possible event times and can sound like a commitment. |
| Predictive interval | A probability-based range under a stated model. | May omit uncertainty in model parameters, structural changes, and operational delays. |
| Scenario range | How the forecast changes under selected assumptions. | Has no probability interpretation unless scenario probabilities are justified and used. |
A predictive interval based on a known rate is especially limited when the rate was estimated from sparse data or when future conditions may differ from the past. Treating an estimated rate as known can make the interval too narrow. A scenario analysis can expose sensitivity to assumptions, but it does not by itself quantify the likelihood of each scenario. Conversely, a probability-based interval can look precise while being conditional on a model that does not capture important changes in accrual.
Common mistake: “The expected target date is 1 January, so the analysis will occur on 1 January.” This treats an average forecast as a guaranteed date and may also confuse the event date with the analysis-ready date. The correction is to call 1 January the expected date under stated assumptions, provide an uncertainty summary where possible, and explain what additional delay may occur before analysis.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A model gives an expected time of 4 months to reach a target and a central 90% predictive interval of 2 to 7 months. Does this mean there is a 90% probability that the mean waiting time is between 2 and 7 months?
- A scenario analysis gives projected times of 3, 4, and 6 months under three selected event-rate assumptions. Can these be reported as a 90% predictive interval?
- A study needs 12 more events and assumes a constant rate of 3 events per month. What is the expected time to the event target, in months?
- A forecast concerns the event-target date, but the final analysis requires an additional variable processing delay. What should be clarified when presenting an analysis-ready date?
Solutions. (1) No. The interval is intended to contain the future event time with 90% probability under the stated predictive model; it is not an interval for the mean waiting time. (2) No. Those are scenario results unless probabilities are assigned and used to form a probabilistic range. (3) The expected time is \(12/3=4\) months. (4) State that the forecast is for the analysis-ready date, explain how the processing delay was handled, and indicate whether its variability is included. If it is omitted or treated as fixed, say so.