Scenario Analysis For Event Timing

Why Use Event-Timing Scenarios?

A single event-accrual projection can look precise while depending on uncertain inputs: how quickly participants enroll, how often the event occurs, and how often participants drop out before it. Scenario analysis makes that uncertainty visible by calculating event accrual under a small set of clearly specified, plausible combinations of assumptions.

An optimistic scenario represents conditions expected to produce faster event accrual; a central scenario represents the planning assumptions judged most representative; and a conservative scenario represents conditions expected to produce slower accrual. These labels describe the projected timing of events, not the treatment effect, trial quality, or probability that a scenario will occur. A scenario is useful only when its assumptions are explicit and internally coherent.

Definition: An event-timing scenario is a specified set of assumptions about enrollment, event occurrence, and loss to follow-up used to project event accrual over calendar time. A central scenario uses the primary planning assumptions; optimistic and conservative scenarios vary one or more uncertain inputs in directions that respectively accelerate or delay expected accrual.

A Basic Projection Model

First consider a simple model. Assume participants enroll at a constant rate \(r\) participants per month from calendar month 0 through month \(T\). After enrollment, each participant is followed until the first of the event, dropout, or the calendar cutoff. Suppose the event and dropout have constant cause-specific hazards \(\lambda\) and \(\delta\), respectively, measured per person-month. Assume event and dropout times are independent, and administrative censoring occurs at the cutoff without otherwise changing the event or dropout process.

Under these assumptions, a participant with \(x\) months of potential follow-up has probability \(\lambda/(\lambda+\delta)\{1-\exp[-(\lambda+\delta)x]\}\) of having the event before dropout and before that follow-up ends. This expression accounts for the fact that dropout can prevent the event from being observed.

p_{\text{event}}(x) = \frac{\lambda}{\lambda+\delta} \left[1-\exp\{-(\lambda+\delta)x\}\right]

At calendar time \(T\), a participant enrolled at time \(u\) has \(T-u\) months of potential follow-up. With constant enrollment, expected events are obtained by adding these event probabilities across enrollment times. In the continuous-rate approximation, this is an integral over the enrollment period.

E\{D(T)\} = r\int_0^T p_{\text{event}}(T-u)\,du = r\frac{\lambda}{\lambda+\delta} \left[ T-\frac{1-\exp\{-(\lambda+\delta)T\}}{\lambda+\delta} \right]

Here \(D(T)\) is the number of observed events by month \(T\). The result is an expected count, which can be fractional; it is not a claim that the realized count will equal that value. The formula assumes enrollment continues throughout the full interval at rate \(r\). For a specified enrollment pattern that is not constant, calculate or approximate the contribution of each enrollment cohort using its own follow-up time.

Worked Scenarios Under Constant Enrollment

Worked Example: Calculating the Central Projection

Suppose a hypothetical trial plans to enroll 20 participants per month for 12 months. The assumed event hazard is \(0.08\) per person-month and the dropout hazard is \(0.02\) per person-month. Calculate the expected number of observed events by month 12 under the model above.

The total rate in the exponential term is \(\lambda+\delta=0.08+0.02=0.10\) per person-month. The central projection is

E\{D(12)\} = 20\left(\frac{0.08}{0.10}\right) \left[ 12-\frac{1-e^{-0.10(12)}}{0.10} \right]

Since \(e^{-1.2}=0.3011942\), the bracket is \(12-(1-0.3011942)/0.10=5.011943\) months. Therefore, \(E\{D(12)\}=20(0.8)(5.011943)=80.1911\), or about 80.19 events.

The units are consistent: participants per month multiplied by months gives participants, while the hazard ratio is dimensionless. The calculation assumes constant enrollment and hazards, independent event and dropout times, and complete event ascertainment up to dropout or the administrative cutoff. It is a planning expectation, not a guaranteed event count.

A useful scenario set changes inputs in a way that has a clear operational interpretation. For example, faster enrollment increases the number of participants with follow-up by a given cutoff; a higher event hazard increases the probability of observing an event; and a higher dropout hazard decreases it. The following scenarios change all three inputs, while holding the combined event-plus-dropout hazard constant to make the calculations easy to compare.

Worked Example: Comparing Three Explicit Scenarios

For a hypothetical 12-month projection, consider a central enrollment rate of 20 participants per month, event hazard \(0.08\), and dropout hazard \(0.02\). Define the optimistic scenario as 22 participants per month, event hazard \(0.09\), and dropout hazard \(0.01\). Define the conservative scenario as 18 participants per month, event hazard \(0.07\), and dropout hazard \(0.03\). All rates are per person-month where applicable.

In each scenario, \(\lambda+\delta=0.10\), so the bracketed time quantity is the same: \(12-(1-e^{-1.2})/0.10=5.011943\) months.

\begin{aligned} E\{D_{\text{optimistic}}(12)\} &=22\left(\frac{0.09}{0.10}\right)(5.011943) =99.2365 \approx 99.24,\\ E\{D_{\text{central}}(12)\} &=20\left(\frac{0.08}{0.10}\right)(5.011943) =80.1911 \approx 80.19,\\ E\{D_{\text{conservative}}(12)\} &=18\left(\frac{0.07}{0.10}\right)(5.011943) =63.1505 \approx 63.15. \end{aligned}

The optimistic assumptions produce the largest expected count by month 12, and the conservative assumptions the smallest. The labels are justified here because enrollment and event rates move in the faster-accrual direction in the optimistic case, while dropout is lower. The scenarios are hypothetical planning cases, not estimates from an actual trial.

This comparison does not establish that any one scenario is likely, nor does it quantify the probability that the count will fall within the range from 63.15 to 99.24. To make a probability statement, additional probabilistic modeling would be needed. Even the central expected count is not necessarily the most probable realized count.

Accounting for Different Enrollment Times

The constant-rate formula is convenient, but scenario analysis also works with staggered enrollment. For each group of participants, use the follow-up available by the cutoff and calculate that group’s expected events. Adding these expectations gives the total expected count. This calculation assumes that participants within a group share the same event and dropout model.

Worked Example: Two Enrollment Cohorts and a Cutoff

Suppose a hypothetical trial has 60 participants enrolled at month 0 and another 60 enrolled at month 6. The data cutoff is month 12. Assume the event hazard is \(0.08\) and the dropout hazard is \(0.02\) per person-month, with independent constant hazards as above. The first group has 12 months of potential follow-up; the second has 6 months.

For 12 months, the event probability before dropout is \(0.8(1-e^{-1.2})=0.5590446\), approximately 0.5590. For 6 months, it is \(0.8(1-e^{-0.6})=0.3609507\), approximately 0.3610. Thus the expected count is

60(0.5590446)+60(0.3609507) = 33.5427+21.6570 = 55.1997 \approx 55.20\text{ events}.

For comparison, if all 120 participants enrolled at month 6, each would have 6 months of potential follow-up. The expected count would be \(120(0.3609507)=43.3141\), or about 43.31 events. Both calculations use the same hazards and total sample size; the difference arises from the enrollment timing and the follow-up available by month 12.

Constructing and Checking Scenarios

A scenario table makes assumptions auditable. Record the enrollment pattern, event-rate assumptions, dropout assumptions, projection cutoff, and resulting expected counts. Where useful, state the source or rationale for each input, such as the recruitment plan or a range of plausible event rates. Do not adjust several inputs without documenting them: otherwise a difference in projected accrual cannot be attributed to a particular assumption.

The monotone ordering in the constant-hazard examples has a direct explanation. Holding other inputs fixed, a higher event hazard raises the chance of an event before a fixed follow-up time; a higher dropout hazard lowers it; and adding participants earlier gives them at least as much follow-up by the cutoff. These statements depend on the model and the compared inputs. In more complex models, interactions between time-varying hazards, enrollment patterns, and follow-up can make scenario ordering less obvious, so calculate and verify it rather than relying on the labels.

Set the projection basis.
Specify the cutoff, target population, enrollment schedule, event definition, and follow-up rules consistently across scenarios.

Choose interpretable inputs.
State the central assumptions and specify which enrollment, event-rate, or dropout assumptions change in the optimistic and conservative cases.

Calculate accrual under each case.
Use the constant-hazard formula when its assumptions fit, or sum cohort-specific expected counts under the chosen survival and dropout assumptions.

Check the results.
Verify units and arithmetic, confirm that labels match the computed ordering, and assess whether each scenario remains plausible.

Alternatives, Diagnostics, and Limitations

A single central projection is simpler to communicate, but it can conceal how strongly the result depends on uncertain inputs. Scenario analysis shows sensitivity to selected assumptions, but it does not automatically account for all possible parameter values. Monte Carlo projection, covered in the earlier material on Monte Carlo projection of event accrual, can represent variability in simulated event counts under a specified model. It answers a different question from a small scenario set: scenarios compare chosen assumption sets, whereas simulation characterizes outcomes under a probabilistic model. Neither method repairs unrealistic input assumptions.

Check a scenario analysis for warning signs before using it for planning. Examples include an “optimistic” case with fewer expected events than the central case, a conservative case that assumes implausible recruitment failure, or projected counts that exceed the number of enrolled participants. Also check that dropout is not being treated as an observed event, that follow-up ends at the correct cutoff, and that event and dropout rates use the same time units. If enrollment or event rates change over time, a constant-rate projection may systematically misstate accrual; use a model that reflects the planned pattern and examine the fit of its assumptions.

Scenario ranges are not confidence intervals, prediction intervals, or guarantees. Their endpoints are determined by the analyst’s chosen assumptions, and values outside them remain possible. The simple model also assumes constant hazards and independent event and dropout times. Informative dropout, changing event hazards, delayed effects, or incomplete event ascertainment may require more appropriate survival and censoring models. Scenario analysis is most informative when its assumptions are plausible, its purpose is clear, and its limitations are reported alongside the projected counts.

Common pitfall: Calling the optimistic and conservative forecasts “upper and lower confidence limits” implies a probability-based uncertainty calculation that scenario analysis has not supplied. Instead, describe them as expected counts under named assumptions, state those assumptions, and avoid assigning probabilities to the scenarios unless a separate model justifies them.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a constant-hazard model, what does the ratio \(\lambda/(\lambda+\delta)\) represent?
A central case projects 50 expected events. An alternative raises dropout while leaving enrollment and event hazard unchanged. Under the stated model, should its expected count be higher or lower? Why?
Why is a scenario range not automatically a prediction interval?
At a cutoff of month 10, 40 participants enrolled at month 0 and 40 enrolled at month 5. If each group has the same event probability for its respective follow-up, how should the expected total be calculated?
Give one reason a constant-hazard projection might be misleading and one diagnostic check that could reveal a problem.

Solutions. (1) It is the probability that an event, rather than dropout, occurs first when the two independent competing times have constant hazards \(\lambda\) and \(\delta\); the follow-up limit is then accounted for by the exponential term. (2) Lower: greater dropout reduces the probability of observing the event before dropout, holding the other inputs fixed. (3) Scenario endpoints are selected assumption sets, not quantiles derived from a probability model, so they do not provide a stated coverage probability. (4) Multiply the 40-person cohort counts by their respective event probabilities and add: \(40p_{\text{event}}(10)+40p_{\text{event}}(5)\). (5) For example, a changing event hazard can make a constant rate misleading; compare observed accrual over time with projections or check whether the assumed rate remains plausible across follow-up.