Reporting Event Projection Assumptions

Why Assumptions Belong With the Projection

An event projection is not a forecast that stands on its own. Its meaning depends on who is expected to enroll, when they are expected to enroll, how event and censoring rates are modeled, and which events count by the specified cutoff. A projected count without these details cannot be reproduced, and a projected date can be misread if it does not distinguish the event cutoff from the later date when data are ready for analysis.

The purpose of reporting assumptions is not to imply that future recruitment or event times are known. It is to make clear which inputs produced the projection, so readers can judge its relevance and update it when observed trial operations differ. The introductory material on communicating event projection uncertainty distinguishes a central projection from uncertainty around it; here the emphasis is on stating the assumptions behind the projection itself.

A Clear Model for Event Accrual

Let \(C\) be the calendar date through which events are to be counted, and let \(s\) denote a participant's enrollment date. The participant's potential follow-up by that cutoff is \(C-s\), if \(s\le C\). A projection must specify the planned or assumed enrollment pattern, because participants enrolled earlier generally have more opportunity to experience an event before \(C\).

Definition: An event projection is the expected number of qualifying events observed by a stated event-data cutoff, under specified assumptions about enrollment, event occurrence, censoring, and event ascertainment. The projected analysis-ready date may be later than the event-data cutoff because of reporting, cleaning, or adjudication delays.

For a simple model, suppose participant \(i\) has event hazard \(\lambda_i(u)\) at time \(u\) since enrollment and independent loss-to-follow-up censoring hazard \(\gamma_i(u)\). Conditional on the stated model and any covariates used, the probability of observing the event by follow-up time \(\tau\) is

p_i(\tau)=\int_0^\tau S_i(u)G_i(u)\lambda_i(u)\,du, \qquad S_i(u)=\exp\left\{-\int_0^u\lambda_i(v)\,dv\right\}, \qquad G_i(u)=\exp\left\{-\int_0^u\gamma_i(v)\,dv\right\}.

Here \(S_i(u)\) is the probability of remaining event-free through time \(u\), and \(G_i(u)\) is the probability of remaining uncensored through that time. The formula assumes censoring is independent of the event time conditional on the modeled information. It does not treat a genuine competing event as ordinary censoring; a competing event requires its own event-process specification. If censoring depends on prognosis in a way not handled by the model, this calculation may not represent the observed event probability.

If the event and censoring hazards are constant, \(\lambda_i(u)=\lambda\) and \(\gamma_i(u)=\gamma\), then

p_i(\tau)=\frac{\lambda}{\lambda+\gamma} \left\{1-\exp[-(\lambda+\gamma)\tau]\right\}.

With no loss to follow-up, set \(\gamma=0\), giving \(p_i(\tau)=1-\exp(-\lambda\tau)\). These expressions assume the hazard and follow-up time use compatible units, such as hazards per year and follow-up measured in years.

For a fixed set of planned enrollment dates, the expected number of observed events by cutoff \(C\) is the sum of the participant-specific probabilities:

E(D_C)=\sum_{i:s_i\le C}p_i(C-s_i).

If enrollment is represented by an intensity \(R(s)\), the corresponding expected count is \(\int_{s\le C}R(s)p(C-s)\,ds\). In either representation, the result is an expectation, not a promise that the realized event count will equal it.

What to State in a Report

A reproducible report should identify the assumptions that materially determine the projected count or date. State the following in words or a compact table, and give enough detail that another analyst could reconstruct the calculation.

Enrollment: the planned number of participants, start and end dates or calendar-time enrollment rates, any pauses or ramp-up assumptions, and whether the projection uses a fixed schedule or a changing recruitment pattern.
Event process: the endpoint counted, the time origin, the event-rate scale and units, and whether rates are constant, piecewise constant, or otherwise time-varying. If arms or subgroups have different assumptions, report them separately.
Censoring: the assumed loss-to-follow-up rate or distribution, any administrative censoring at the cutoff, and the condition under which censoring is treated as independent of event occurrence.
Cutoff and target: the calendar date or event-count target being projected, the rule for counting an event at the boundary, and whether the target concerns events occurring by that date or events confirmed in the data by that date.
Operational timing: the expected time from event occurrence or cutoff to reporting, data cleaning, adjudication, and a usable analysis data set. Distinguish fixed delays from variable or incomplete ascertainment.
Uncertainty and updates: which inputs are estimated or uncertain, the scenarios or ranges considered, and what observed information would trigger an update.

For time-to-event endpoints, specifying the time scale matters. A hazard described as “0.4” is incomplete unless its unit and interval are given, for example, a constant hazard of \(0.4\) per participant-year since randomization. Under non-proportional hazards, a single average rate can conceal changes over time or differences between treatment groups. Report the time pattern and group-specific assumptions used rather than implying a common constant hazard where that is not intended.

Worked Projections

Worked Example: A Simple Enrollment and Cutoff Assumption

Suppose a hypothetical trial plans to have 200 participants enrolled by the event cutoff. Assume 150 participants (75%) enroll at least one year before the cutoff, while the remaining 50 enroll on the cutoff date and contribute no follow-up by that date. For illustration, assume a constant event hazard of \(0.4\) per year, no loss to follow-up, and complete event ascertainment through the cutoff. Calculate the expected number of events.

The 150 earlier enrollees each have one year of follow-up, so their event probability is \(1-\exp(-0.4)\). The 50 participants enrolled on the cutoff have zero follow-up, so their event probability by the cutoff is zero. Equivalently, the expected count is

E(D_C)=200\left[0.75\{1-\exp(-0.4)\}+0.25(0)\right] =200[0.75(0.329680)] \approx 49.45\text{ events}.

The assumptions are internally consistent with the calculation: the hazard is in inverse years, the nonzero follow-up is one year, and censoring and ascertainment delays are assumed absent before the stated cutoff. The report should not describe 49.45 as a guaranteed count; it is the model-based expected count. The enrollment timing is essential: 200 participants with a different distribution of follow-up would generally yield a different expectation.

Worked Example: Uniform Recruitment Over a Defined Window

In a separate hypothetical projection, 120 participants are assumed to enroll uniformly over the first six months of a one-year accrual period. The event-data cutoff is at the end of the year. Assume a constant event hazard of \(0.5\) per year, no loss to follow-up, and complete ascertainment through the cutoff. Find the expected event count.

Let \(s\) be enrollment time in years after accrual begins. Under uniform recruitment, \(s\) is distributed uniformly from 0 to 0.5, so follow-up by the cutoff ranges from one year down to half a year. For a participant enrolled at \(s\), the event probability is \(1-\exp[-0.5(1-s)]\). Averaging this probability over the enrollment window gives

\begin{aligned} \bar p &=1-\frac{1}{0.5}\int_0^{0.5}\exp[-0.5(1-s)]\,ds\\ &=1-\frac{\exp(-0.25)-\exp(-0.5)}{0.25}\\ &=1-\frac{0.778801-0.606531}{0.25}\\ &\approx 0.31092. \end{aligned}

Therefore,

E(D_C)=120(0.31092)\approx 37.31\text{ events}.

The constant hazard and no-censoring assumptions make the exponential event-probability formula applicable, while the uniform-enrollment assumption justifies averaging over the six-month window. A report using this projection should state that recruitment is assumed uniform over those six months; “120 participants by one year” alone is not enough to reproduce the result.

Worked Example: Censoring and the Analysis-Ready Date

Suppose 100 hypothetical participants each have one year of potential follow-up before the event-data cutoff. Assume a constant event hazard of \(0.4\) per year and an independent constant dropout-censoring hazard of \(0.1\) per year. The data are expected to be ready for analysis 28 days after the cutoff. Calculate the expected observed event count and state how the delay should be reported.

The probability of observing an event before either the event or dropout process removes the participant from event-free follow-up is

p(1)=\frac{0.4}{0.4+0.1}\{1-\exp[-(0.4+0.1)(1)]\} =0.8(1-\exp(-0.5)) \approx 0.314775.

Thus the expected count is

E(D_C)=100(0.314775)\approx 31.48\text{ events}.

The operational assumption is that the analysis-ready data set is expected 28 days after the event-data cutoff, subject to the stated process for reporting and cleaning. Those 28 days should not be added to each participant’s follow-up in this calculation: the target is events by the original cutoff, not events occurring during the later processing period. If the operational process also causes events before the cutoff to be reported late or missed, that ascertainment assumption must be specified separately.

Justification and Practical Checks

The expected-count formula follows by writing the total count as a sum of event indicators. Let \(I_i=1\) when participant \(i\) has an observed qualifying event by cutoff \(C\), and \(I_i=0\) otherwise. Then \(D_C=\sum_i I_i\). By linearity of expectation, which does not require the indicators to be independent,

E(D_C)=E\left(\sum_i I_i\right)=\sum_i E(I_i) =\sum_i P(I_i=1)=\sum_i p_i(C-s_i).

The formula therefore requires the individual event probabilities to match the enrollment and follow-up assumptions, but not independence among participants for the expected count itself. Dependence can still matter for uncertainty around that count. The continuous-enrollment version follows by replacing the sum over planned enrollees with an integral over the assumed enrollment intensity.

Before sharing a projection, check that the dates, time scales, and definitions agree. In particular, verify that the enrollment window reaches the stated total, that each participant’s follow-up is measured to the specified cutoff, and that rates expressed per month or year are paired with durations in the same units. Compare assumed recruitment and event rates with observed accrual when available, and state how far the projection depends on extrapolation beyond current data.

Fix the estimand for the projection.
Define the qualifying event and whether the target is events occurring by a calendar cutoff or a planned event count.

Describe enrollment and follow-up.
Give the calendar-time enrollment pattern and calculate each participant’s potential follow-up to the event-data cutoff.

Specify event, censoring, and ascertainment assumptions.
Give rate models, units, censoring conditions, and any assumptions about delayed or incomplete event capture.

Report the result with operational timing and checks.
Label the expected count or date as a projection, report when data are expected to be analysis-ready, and identify key diagnostics or sensitivity scenarios.

Alternatives, Diagnostics, and Limitations

The reporting method should match the complexity of the assumed process. A single average event rate may be a useful concise summary when hazards and recruitment are reasonably stable, but it can obscure follow-up differences or time-varying hazards. A participant-level sum or a piecewise integration over enrollment and follow-up is more transparent when recruitment changes or event rates vary over time. Simulation can represent more complicated combinations of enrollment, censoring, event-time patterns, and operational delays, but it does not remove the need to report the inputs and rules used.

Projection approach	Useful when	Reporting caution
Single-rate summary	Recruitment and event rates are stable enough for a concise approximation.	State the follow-up and rate assumptions; an average can hide changing patterns.
Participant-level or piecewise calculation	Enrollment dates, follow-up, or event rates vary over time.	Describe the schedule or rate segments so the calculation can be reconstructed.
Simulation	Several interacting processes or variable operational delays need representation.	Report the simulated assumptions and distinguish model variation from uncertainty in the inputs.

Warning signs include recruitment that is slower or more uneven than assumed, observed event accrual that persistently differs from the modeled pattern, and an unexplained gap between events occurring and events available in the analysis data. These do not automatically prove that the original projection was calculated incorrectly; they may indicate that inputs or operational processes have changed and should be reviewed. Important limitations also remain even when assumptions are clearly reported: event-rate estimates may be imprecise, future recruitment may depart from plan, and a simple independent-censoring model may be unsuitable when dropout is informative.

Common mistake: reporting “the target will be reached in October” without saying which recruitment pattern, event rate, censoring rule, or data cutoff supports that date. The statement can be repeated but not meaningfully evaluated. Instead, give the assumptions and label October as conditional on them; then state any separate expected delay from the event cutoff to an analysis-ready data set.

Key takeaway: A useful event projection reports not only a count or date, but also the enrollment pattern, event-rate model and units, censoring and ascertainment rules, event-data cutoff, and operational delay. These details determine what the projection means and whether it can be reproduced.

Check Your Understanding

Answer each question using the definitions and assumptions stated above. Brief solutions follow the questions.

With constant event hazard \(0.3\) per year, no censoring, and two years of follow-up, what is the event probability for one participant?
Why is a planned total enrollment number insufficient to reproduce an event projection?
With constant event and independent censoring hazards of \(0.2\) and \(0.1\) per year, respectively, what is the probability of observing an event by one year?
If the event-data cutoff is June 30 and adjudication takes an expected 20 days, what two dates or timing statements should a report distinguish?

Solutions. (1) \(1-\exp(-0.3\times2)=1-\exp(-0.6)\approx0.4512\), or about a 45.12% event probability by two years under the stated model. (2) Participants enrolled at different times have different follow-up by the cutoff, so the enrollment pattern is needed to calculate their event probabilities. (3) \(0.2/(0.2+0.1)\{1-\exp[-(0.2+0.1)]\}=(2/3)(1-\exp(-0.3))\approx0.1728\), or about 17.28%. (4) State that events are counted through June 30, and separately state that the analysis-ready data are expected about 20 days later, subject to the adjudication process.