Sensitivity To Dropout And Event-Rate Assumptions

Why Assumptions Matter for an Event Date

A projected event-target date is conditional on assumptions about how quickly events occur and how often participants drop out before an event can be observed. Even if the enrollment pattern is held fixed, changing either assumption can shift the forecast materially. Sensitivity analysis makes that dependence explicit: it recalculates the date under alternative plausible inputs rather than presenting a single date as if it were assumption-free.

The earlier material on sensitivity to the enrollment pattern considers changes in when participants enter. Here, hold that pattern fixed and vary the event-rate and dropout assumptions. The aim is not to decide which assumptions are correct from the projection alone. It is to show what the forecast would imply if those assumptions were different.

A Competing-Hazard Model for Event Accrual

For a transparent calculation, suppose that after enrollment each participant is followed until the event of interest or dropout, whichever occurs first. Assume a constant event hazard $\lambda$ per month and a constant dropout hazard $\delta$ per month. The hazards are independent in the competing-risks sense: conditional on remaining event-free and not yet dropped out, the instantaneous event and dropout rates are $\lambda$ and $\delta$. Assume complete ascertainment of the event up to dropout, no other event that prevents observing the event of interest, and no change in either hazard over follow-up.

Definition: Under constant competing hazards $\lambda$ and $\delta$, the probability that an enrolled participant has the event of interest by follow-up time $v$, before dropping out, is $$ F_E(v;\lambda,\delta) = \frac{\lambda}{\lambda+\delta} \left[1-\exp\{-(\lambda+\delta)v\}\right], \qquad v\geq 0. $$ The corresponding long-run event probability is $\lambda/(\lambda+\delta)$. It is less than one when $\delta>0$, because some participants drop out before having the event.

For a fixed enrollment schedule, the expected cumulative number of events by calendar time $T$ is the sum, or integral, of the event probabilities for the participants enrolled by then. For example, if $N$ participants all enroll at calendar time zero, then

D(T;\lambda,\delta)=N F_E(T;\lambda,\delta).

With staggered enrollment, a participant enrolled at calendar time $u$ has follow-up $T-u$, so their contribution is $F_E(T-u;\lambda,\delta)$ when $u\leq T$. The enrollment schedule is kept the same across sensitivity scenarios; only the specified hazard assumptions change.

A projected target date is the calendar time $T^*$ solving $D(T^*;\lambda,\delta)=d$, where $d$ is the planned event target. This is a date at which the expected count reaches the target under the model. It is not a guarantee that the observed count will equal $d$ on that date, and it is not generally the expected value of the random date on which the actual count first reaches $d$.

Worked Example: Deriving a Baseline Target Date

Suppose, hypothetically, that 100 participants enroll at month 0. The target is 60 events. Use a constant event hazard of $\lambda=0.050$ per month and a constant dropout hazard of $\delta=0.010$ per month. Under the model just stated, the event probability by month $T$ is $F_E(T)$, and the expected count is $100F_E(T)$. The assumptions make the hazards constant over follow-up and treat dropout as preventing later event observation.

Set the expected count equal to the target and solve:

100\frac{0.050}{0.050+0.010} \left[1-\exp\{-(0.050+0.010)T\}\right]=60.

Dividing by 100 gives an event probability of $0.600$. Since $0.050/0.060=0.8333$,

0.8333[1-\exp(-0.060T)]=0.600, \qquad \exp(-0.060T)=0.2800,

and therefore

T=\frac{-\log(0.2800)}{0.060}=21.22\text{ months}.

Thus, under these hypothetical assumptions, the expected event count reaches 60 about 21.22 months after enrollment. The long-run expected count is $100(0.050/0.060)=83.33$, so the target is attainable under this model.

One-Way Sensitivity Analysis

A one-way sensitivity analysis changes one input at a time and holds the other inputs at their baseline values. Here, one can vary $\delta$ while fixing $\lambda$, then vary $\lambda$ while fixing $\delta$. The resulting date differences isolate the model’s response to each input, but they are not a decomposition of all uncertainty: assumptions may vary together, and the impact of one input can depend on the value of the other.

For the all-enrolled-at-time-zero model, a finite date exists only if the target fraction $q=d/N$ is below the long-run event probability $\lambda/(\lambda+\delta)$. When that condition holds, solving the target equation gives

T^*(\lambda,\delta) = \frac{-\log\left[1-q(\lambda+\delta)/\lambda\right]} {\lambda+\delta}.

This expression makes the roles of the assumptions visible. Increasing dropout generally reduces the fraction who can eventually experience an observed event and slows expected event accumulation. Increasing the event hazard generally raises the event probability and brings the target forward. The exact date response is nonlinear, so a percentage change in a hazard does not imply the same percentage change in the date.

Worked Example: Varying Dropout and Event Hazards Separately

Continue with the hypothetical 100 participants and target of 60 events, so $q=0.600$. The baseline assumptions are $\lambda=0.050$ and $\delta=0.010$ per month, giving $T^*=21.22$ months. Check each alternative against the same model and enrollment schedule.

Higher dropout only. Set $\delta=0.020$ and keep $\lambda=0.050$. The target fraction is below the long-run event probability $0.050/0.070=0.7143$, so a finite solution exists:

T^*= \frac{-\log[1-0.600(0.070/0.050)]}{0.070} = \frac{-\log(0.1600)}{0.070} = 26.18\text{ months}.

The projected date is 4.96 months later than baseline. The change reflects both the greater chance of dropout before an event and the resulting lower event probability over a given follow-up period.

Lower event hazard only. Set $\lambda=0.040$ and keep $\delta=0.010$. The long-run event probability is $0.040/0.050=0.8000$, still above 0.600:

T^*= \frac{-\log[1-0.600(0.050/0.040)]}{0.050} = \frac{-\log(0.2500)}{0.050} = 27.73\text{ months}.

This is 6.51 months later than baseline. In this example, a 20% decrease in the event hazard shifts the forecast by more than 20% of the baseline date; the nonlinear formula is why date changes should be calculated rather than inferred by scaling.

Joint Sensitivity and Attainability

One-way analyses are useful for identifying influential assumptions, but they can miss combinations that matter in practice. A joint sensitivity analysis changes event and dropout assumptions together. Choose combinations that are scientifically or operationally plausible, keep the enrollment schedule and target fixed, and recalculate the target date for each combination. A scenario grid is often clearer than a single “best” alternative because it shows how the inputs interact.

A crucial diagnostic is whether the target can be reached at all under a scenario. In the all-enrolled model, as follow-up grows without bound, the expected event count approaches $N\lambda/(\lambda+\delta)$. If $d$ exceeds this limit, the equation $D(T)=d$ has no finite solution. If $d$ equals the limit, the expected count approaches the target asymptotically but does not reach it at any finite time. Numerical root-finding cannot fix this structural limitation.

Worked Example: A Joint Scenario and an Unattainable Target

Suppose the hypothetical event hazard is lower and dropout is higher together: $\lambda=0.040$ and $\delta=0.020$ per month. With $N=100$ and $d=60$, the long-run expected count is $100(0.040/0.060)=66.67$. The target remains attainable, although the margin above 60 is limited. The target date is

T^*= \frac{-\log[1-0.600(0.060/0.040)]}{0.060} = \frac{-\log(0.1000)}{0.060} = 38.38\text{ months}.

Compared with the 21.22-month baseline, this joint scenario moves the projected date 17.16 months later. That shift is not obtained by adding the two one-way date shifts: the target-date equation depends jointly on both hazards.

Now consider a more severe hypothetical combination, $\lambda=0.030$ and $\delta=0.020$. The long-run expected event count is

100\frac{0.030}{0.030+0.020}=60.00.

Because 60 is exactly the asymptote, the expected count approaches 60 only as follow-up tends to infinity. There is no finite target date. If the event hazard were $0.025$ per month with the same dropout hazard, the long-run expected count would be $100(0.025/0.045)=55.56$, below the target; under that scenario the target is unattainable even with unlimited follow-up.

Why the Date Moves: A Local Justification

For a more general projection, let $D(T;\theta)$ be the expected event count by date $T$, where $\theta$ is one model input such as the dropout hazard. Suppose a finite target solution satisfies $D(T^*;\theta)=d$, and that $D$ is differentiable with a positive rate of expected event accumulation at the target date, so $D_T(T^*;\theta)>0$. A small change in $\theta$ produces a corresponding date change. Differentiating the target equation gives

D_T\,dT^*+D_\theta\,d\theta=0, \qquad \frac{dT^*}{d\theta} = -\frac{D_\theta}{D_T}.

This shows why a change that reduces expected events at a fixed date ($D_\theta<0$) moves the target date later when $D_T>0$. The derivative is a local approximation near the baseline; for substantial changes, recalculate the target date under each scenario rather than relying on a linear extrapolation. If $D_T$ is near zero because event accrual is approaching a plateau, the date can be extremely sensitive to small input changes.

Diagnostics, Alternatives, and Limitations

A sensitivity analysis is useful only if its assumptions and calculations remain credible. Check the following before interpreting the scenario dates:

Check the target against the plateau. Calculate $N\lambda/(\lambda+\delta)$ in the simple model. A target at or above the plateau has no finite solution.
Check units and follow-up scale. Hazards must use units compatible with the date, such as per month when $T$ is in months. A mismatch can produce a numerically plausible but meaningless date.
Check the enrollment schedule. Hold it fixed when attributing a date shift to dropout or event rates. Otherwise, the change combines multiple sources.
Check model shape against observed accrual. A constant hazard may be inappropriate if hazards vary with time since enrollment, calendar time, or participant characteristics. A simple scenario grid cannot repair a misspecified hazard model.
Check whether the changed assumptions are plausible together. Event and dropout assumptions may be related, so independent extremes can create unrealistic joint scenarios.

One-way analysis is easiest to explain and helps identify which individual input has the largest effect around a baseline. Joint scenario analysis is more informative when assumptions plausibly change together, but its conclusions depend on the selected combinations. A probabilistic projection can represent distributions for uncertain inputs and report a distribution of projected counts or dates; it requires defensible probability models and should not be replaced by an arbitrary list of scenarios.

All of these methods remain conditional on the event and dropout model, the enrollment projection, and the endpoint’s ascertainment process. A sensitivity table does not establish that any scenario is likely, and a single projected date does not describe the full uncertainty in when the actual target will be observed. Where hazards are strongly time-varying or dropout is informative, the constant independent-hazard calculation should be treated as an illustration, not as an adequate primary model.

Common Pitfall

Common mistake: “The baseline date is 21.22 months, so increasing dropout by a given percentage must delay the date by that same percentage.” This confuses a model input with its nonlinear effect on event accumulation. Dropout changes both the rate of competing exits and the long-run fraction who experience the event. Correction: state the alternative hazards, check target attainability, and solve the target equation again. Report the date difference as conditional on that scenario, not as a universal percentage rule.

Key takeaway: A projected event date is conditional on assumptions. Keep the enrollment pattern fixed when studying dropout and event-rate sensitivity, recalculate dates for one-way and joint scenarios, and verify that each scenario can reach the target under its long-run event probability.

Check Your Understanding

Answer each question using the competing-hazard model in this tutorial. The solutions follow the questions.

With $\lambda=0.040$ and $\delta=0.010$ per month, what is the long-run probability of an observed event before dropout?
For 100 participants and a target of 70 events, can a finite target date exist under $\lambda=0.040$ and $\delta=0.020$? Explain.
In a one-way sensitivity analysis of dropout, which assumption is held fixed, and why?
Why does a target date that solves $D(T)=d$ not guarantee that the observed event count will equal $d$ on that date?
In the local relation $dT^*/d\theta=-D_\theta/D_T$, what happens to the target date when an input reduces expected events at each fixed date, assuming $D_T>0$?

Solutions. (1) The probability is $\lambda/(\lambda+\delta)=0.040/0.050=0.8000$, or 80.00%. (2) The long-run expected count is $100(0.040/0.060)=66.67$, below 70, so no finite date exists under this model. (3) Hold the event hazard fixed, along with the enrollment schedule and other model inputs, so the date shift isolates the dropout change. (4) $D(T)$ is an expected count; actual events vary randomly, so the observed count may be above or below that expectation. (5) Then $D_\theta<0$, and with $D_T>0$, the derivative is positive: increasing that input locally delays the target date.