Why Vary Assumptions About Censored Outcomes?
When a participant is censored before the end of follow-up, the competing event that would have occurred later is unknown. An analysis may estimate cumulative incidence under assumptions about censoring, or use imputation to assign plausible outcomes beyond censoring. But these approaches cannot reveal what actually happened after follow-up ended. A sensitivity analysis addresses that limitation directly: it asks how the estimate would change under a set of defensible alternative assumptions about those unobserved outcomes.
The aim is not to choose a scenario that makes a preferred result appear. It is to expose the dependence of the conclusion on assumptions that the observed data cannot fully verify. This tutorial uses a simple scenario-based calculation. The introductory material on imputing event times beyond censoring gives context for the broader problem; here the emphasis is on systematically varying assumptions and summarizing their consequences.
A Scenario-Based Estimator
Suppose a cohort has \(N\) individuals. By time \(\tau\), \(D_k\) have an observed event of type \(k\), and \(C\) have been censored before \(\tau\) while still event-free. The other individuals are known to have remained free of all events through \(\tau\). For the moment, suppose censored individuals can be divided into strata \(s=1,\ldots,S\), with \(C_s\) censored individuals in stratum \(s\). In a specified scenario, let \(q_{sk}\) be the assumed probability that a censored individual in stratum \(s\) would experience event type \(k\) by \(\tau\), before any other event.
The probabilities for the event types in each stratum must be coherent: \(q_{sk}\geq 0\), and \(\sum_k q_{sk}\leq 1\). The remaining probability represents no event by \(\tau\). These are assumptions about outcomes that were not observed; they are not probabilities estimated from the known outcomes of the censored individuals.
The denominator remains the original cohort size. Each censored person contributes their scenario-specific probability of event \(k\), rather than being removed from the cohort or counted as if they definitely had that event. If information supports different assumptions by censoring reason, baseline risk, or follow-up time, those features can define the strata. The assumptions should be stated clearly enough that each scenario has a substantive interpretation.
Why the Calculation Works
The formula follows from the expected number of type-\(k\) events by \(\tau\) under a given scenario. The \(D_k\) observed events contribute exactly \(D_k\). For each censored individual \(i\) in stratum \(s\), define \(Y_{ik}=1\) if their unobserved outcome would be event \(k\) by \(\tau\), and \(Y_{ik}=0\) otherwise. Under the scenario, \(E(Y_{ik})=q_{sk}\). Thus the expected total count is the observed count plus the sum of these assumed probabilities. Dividing by \(N\) gives the scenario-specific estimated risk.
This is an expected-count calculation, not a claim that a fractional number of people experienced an event. The estimator summarizes a hypothetical completed cohort under the chosen assumptions. If the analysis instead uses an event-time model, the scenario must specify assumptions compatible with that model and the time horizon. Whatever the method, competing outcomes must remain mutually exclusive: assigning more probability to one cause can leave less probability for another cause or for remaining event-free.
Worked Example: Two Censoring Strata and Two Causes
Consider a hypothetical cohort of \(N=100\) people followed to \(\tau\). Before \(\tau\), 16 have an observed event of cause 1, 8 have an observed event of cause 2, 20 are censored while event-free, and 56 are known to remain event-free. The 20 censored people comprise two strata of 10 each. In a reference scenario, assume the probabilities of an unobserved event by \(\tau\) are \(q_{A1}=0.08,\ q_{A2}=0.04\) in stratum A and \(q_{B1}=0.24,\ q_{B2}=0.12\) in stratum B. These are valid probabilities because their sums are 0.12 and 0.36, respectively, both no greater than 1.
For cause 1, the assumed contribution from censored people is \(10(0.08)+10(0.24)=0.8+2.4=3.2\) people in expected-count terms. Therefore,
The estimated cause-1 risk in this scenario is 19.2% by \(\tau\). For cause 2, the corresponding contribution is \(10(0.04)+10(0.12)=0.4+1.2=1.6\), so
Now consider a scenario in which the cause-1 probabilities rise to 0.16 in stratum A and 0.40 in stratum B, while the cause-2 probabilities remain 0.04 and 0.12. The probability sums are now 0.20 and 0.52, still valid. Cause 1 becomes \(\{16+10(0.16)+10(0.40)\}/100=21.6/100=0.216\). Cause 2 remains \(\{8+10(0.04)+10(0.12)\}/100=9.6/100=0.096\). The cause-1 risk increases by 2.4 percentage points, while the cause-2 risk is unchanged because its assumed probabilities did not change. This is a comparison of assumptions, not evidence that the increased cause-1 risk is more likely to be true.
Designing a Useful Sensitivity Grid
A useful analysis starts with the estimand and the assumptions that matter for it. State the time horizon, the event type or contrast of interest, the population denominator, and which people are considered censored before that horizon. Then identify the unobserved quantities that drive the result. Scenarios can vary probabilities by stratum, or vary parameters in an event-time model used to obtain those probabilities. The selected values should be justified using subject-matter knowledge, observed predictors, external information, or a transparent range of plausible departures from the reference assumptions.
A grid is often more informative than a single “best guess.” It may include the reference scenario, progressively higher and lower risks after censoring, and scenarios in which the competing-event probabilities change too. When there are two treatment groups, allow assumptions to differ between groups if that is scientifically plausible; assuming identical post-censoring outcomes in both groups is itself an assumption. For each row, report the scenario parameters, the cause-specific risks, and the contrast in its original units, such as a risk difference in percentage points by \(\tau\).
Worked Example: A Group Contrast and a Tipping Scenario
Suppose two hypothetical groups each contain 100 people. By \(\tau\), group A has 12 observed cause-1 events and 40 censored individuals; group B has 15 observed cause-1 events and 40 censored individuals. Everyone else is known to be event-free through \(\tau\). In a reference scenario, assume that for censored people in both groups, the probability of cause 1 by \(\tau\) is 0.15 and the probability of cause 2 is 0.05. The remaining probability, 0.80, is for no event by \(\tau\), so the assumptions are coherent.
The cause-1 risks are \(\widehat F_{A1}=(12+40(0.15))/100=18/100=0.18\) and \(\widehat F_{B1}=(15+40(0.15))/100=21/100=0.21\). The group B minus group A risk difference is \(0.21-0.18=0.03\), or 3 percentage points by \(\tau\).
Next consider a departure in which the assumed cause-1 probability among censored people is 0.20 in group A and 0.10 in group B. Keep the cause-2 probability at 0.05 in each group; the remaining probabilities are 0.75 and 0.85, respectively. Both sets still sum to 1. The risks are now \(\widehat F_{A1}=(12+40(0.20))/100=20/100=0.20\) and \(\widehat F_{B1}=(15+40(0.10))/100=19/100=0.19\). The risk difference is \(0.19-0.20=-0.01\), or −1 percentage point by \(\tau\). This scenario reverses the direction of the contrast. It demonstrates that the conclusion is sensitive to differential post-censoring assumptions; it does not show that these assumptions are correct.
Assumptions, Diagnostics, and Alternatives
A scenario-based calculation is only as interpretable as its assumptions. At minimum, the analysis assumes that observed event types and censoring times are correctly classified, that the target population and time horizon are defined consistently, and that the specified probabilities describe plausible outcomes for the censored individuals to whom they are applied. If probabilities are assigned within strata, the strata must be defined using information available for those individuals, and the assumptions within each stratum must be defensible. The calculation alone does not correct bias from informative censoring; it shows what follows under the stated alternative assumptions.
Useful diagnostics are primarily checks on coherence and sensitivity. Verify for every stratum and scenario that probabilities are between 0 and 1 and sum to no more than 1; confirm that each risk is between 0 and 1; and check that the assumed contributions do not exceed the number censored. If cause-specific risks are reported together, their combined event probability must be compatible with the competing-risk structure. Inspect whether conclusions change gradually or abruptly across the grid, and identify the scenario at which a contrast changes sign or crosses a clinically meaningful threshold.
A common mistake is to treat every censored person as if they had the event of interest, or to remove them from the denominator and call the result a cohort risk. The first assigns probability 1 to an unobserved outcome without justification; the second changes the target quantity by conditioning on not being censored. The correction is to keep the original cohort denominator and add a contribution based on explicitly stated scenario probabilities. Another mistake is to call a range of scenario estimates a confidence interval. It is not: it describes variation across assumptions, not sampling uncertainty. Uncertainty intervals, if needed, require a separate variance procedure that accounts for estimation and any model-fitting or imputation steps.
This approach complements, rather than replaces, methods based on conditional independent censoring, such as censoring-weighted estimation, and model-based imputation. Those methods rely on their own assumptions and may be useful reference analyses. Sensitivity scenarios make departures from assumptions visible, but they do not identify the unobserved outcomes from the data alone. They also depend on the chosen range: scenarios that are too narrow can understate sensitivity, while arbitrary extreme values may be uninformative. Later approaches formalize particular model structures for informative censoring; here the essential practice is to define plausible departures and report their impact plainly.
Name the cause-specific cumulative incidence, time horizon, cohort denominator, and any group contrast.
Describe how censored individuals' unobserved outcomes are represented, including any strata or model parameters.
Choose a transparent grid, checking that competing-event probabilities remain coherent in every scenario.
Report risks and contrasts for each scenario, then identify whether the substantive conclusion changes.
Reporting the Result
A clear report separates the reference estimate, the sensitivity range, and statistical uncertainty. Give enough detail for another analyst to reproduce the calculations: the number censored, the probability assumptions by cause and stratum, the time horizon, and the formula or model used. State whether the direction or practical interpretation of the contrast is robust across the examined scenarios. If it changes, report where and explain why the assumptions producing that change are plausible or implausible in context. Do not describe a scenario-specific estimate as an observed fact about what happened after censoring.
Check Your Understanding
Answer each question using the definitions and calculations above.
- A cohort has \(N=80\), with 10 observed cause-1 events and 20 individuals censored before \(\tau\). Under a scenario assigning \(q_1=0.25\) to each censored individual, what is the estimated cause-1 risk?
- In one stratum, assumed probabilities are \(q_1=0.65\) and \(q_2=0.40\). Is this a coherent competing-risk scenario? Explain.
- Two groups have the same number censored, but a sensitivity scenario assumes a higher cause-1 probability after censoring in group B. What should be examined when interpreting the group contrast?
- Why is the range of estimates across sensitivity scenarios not a confidence interval?
Solutions. (1) \(\{10+20(0.25)\}/80=15/80=0.1875\), or 18.75% by \(\tau\). (2) No. The probabilities sum to \(0.65+0.40=1.05>1\), so they cannot represent mutually exclusive competing outcomes and no event. (3) Recalculate both group risks and their contrast under the scenario, then check whether its direction or practical interpretation changes; equal censoring counts do not imply equal unobserved outcome distributions. (4) The range records variation across chosen assumptions. A confidence interval quantifies sampling uncertainty under a specified analysis and requires a separate uncertainty calculation.