Why Review Interim Survival Data Carefully?
At an interim analysis, survival data can suggest that a treatment effect changes over time: curves may separate only after a delay, approach one another, or reverse ordering. These patterns matter when interpreting non-proportional hazards (NPH), because a single hazard ratio can obscure how effects evolve. But interim displays are incomplete and often based on few participants still under observation at later times. Their role is to describe and diagnose the available data, not to turn an unplanned visual impression into a confirmatory conclusion.
A sound review reads Kaplan–Meier (KM) estimates alongside the number at risk, the timing of events and censoring, and suitably cautious time-varying summaries. It asks whether a pattern is plausible and adequately supported, not whether the interim data have proved that hazards are non-proportional.
Reading Kaplan–Meier Curves and Risk Tables
Suppose \(t_j\) are the distinct observed event times, \(n_j\) is the number at risk just before \(t_j\), and \(d_j\) is the number of events at \(t_j\). The KM estimate is the product of the estimated conditional probabilities of surviving each event time:
This estimator assumes that censoring is non-informative for the survival distribution being estimated: conditional on the information used in the analysis, participants censored at a time have the same subsequent event prospects as those who remain under observation. It also requires correctly recorded event times and event status. In a randomized comparison, the estimate is calculated separately within each assigned group; randomization does not remove the need to consider whether censoring is informative.
A KM curve steps down at event times and stays flat between events. Censoring removes a participant from later risk sets but does not itself cause a downward step. The risk table reports how many participants remain under observation and event-free just before selected times. It therefore helps reveal how much information supports the tail of a curve. A visually large late separation when only a few participants remain at risk is much less stable than a similar separation supported by many participants.
Sampling uncertainty also grows as data become sparse. One standard estimate of the variance of \(\hat{S}(t)\) is Greenwood’s formula:
This formula illustrates why a curve should not be read as exact, particularly late in follow-up when risk sets are small. Confidence intervals for KM estimates can add context, but overlapping or non-overlapping pointwise intervals are not, by themselves, a formal test of a treatment effect or of proportional hazards.
Worked Example: Locating an Ordering Reversal
Consider hypothetical interim KM estimates at selected months. At month 6, both groups have an estimated survival of 0.450. The curves remain equal between month 6 and month 8. Just before the month-8 events, each group has 9 participants at risk. At month 8, two control participants and one treatment participant experience the event. What does the display show?
| Time | Control KM estimate | Treatment KM estimate |
|---|---|---|
| Month 6 | 0.450 | 0.450 |
| Between months 6 and 8 | 0.450 | 0.450 |
| Month 8, after events | 0.350 | 0.400 |
The month-8 updates follow directly from the KM factor. For control, the factor is \(1-2/9=7/9\), so the estimate becomes \(0.450(7/9)=0.350\). For treatment, the factor is \(1-1/9=8/9\), so the estimate becomes \(0.450(8/9)=0.400\). The estimates are equal from month 6 until month 8; their ordering reverses at the month-8 event time. The display is compatible with different event patterns over time, but this small hypothetical calculation does not establish that the underlying hazards differ or that the reversal is reliable.
Use Risk Tables to Judge How Much the Curves Can Tell You
At an interim cut, groups can have different numbers at risk because of event experience, censoring, and the timing of enrollment. A risk table does not explain those differences by itself. Review it with the censoring marks, follow-up duration, and event counts. In particular, do not interpret a thin tail as if it were supported by the same amount of information as the earlier part of the curve.
Worked Example: A Late Separation With Few Participants at Risk
In a hypothetical trial, the risk table shows 42 control and 39 treatment participants at risk at month 6, but only 15 control and 9 treatment participants at risk at month 12. Suppose the curves look similar through month 6 and are visibly separated at month 12.
The separation is a feature of the observed estimates and merits review, but the late risk sets are small, especially in the treatment group. A small number of events or changes in who remains under observation could materially alter the tail estimates. The next diagnostic questions are whether censoring differs by group, whether the groups had comparable follow-up opportunities, how many late events occurred, and whether the apparent separation is consistent across nearby times rather than driven by one step.
The risk table does not justify concluding that the treatment benefit begins at month 6, nor does it indicate the probability that such a benefit is real. It describes how many participants contribute to estimation near each displayed time. For a firmer assessment, the analyst should examine uncertainty and the prespecified inferential summaries, while retaining the interim pattern as exploratory.
Time-Varying Effect Summaries
A KM display describes survival probabilities, while a hazard ratio describes a relative instantaneous event rate among participants still event-free. Under proportional hazards, the treatment-to-control hazard ratio is constant over time. Under NPH, it can vary. Time-specific summaries can help describe such variation, but their interpretation depends on the summary’s assumptions and on how many events support each time period.
One simple diagnostic divides follow-up into intervals and compares event rates within each interval. This is only a rough summary: a rate ratio is not automatically a hazard ratio, and an interval-specific estimate can be noisy. Under a piecewise-constant hazard model, with comparable definitions of person-time and adequately independent event counting, the event rate in a group is estimated by events divided by person-time. The approximate standard error of a log rate ratio based on independent Poisson event counts is the square root of the sum of the reciprocals of the two event counts. Such a calculation can describe a pattern; choosing intervals after inspecting the data and then treating their estimates as definitive is not confirmatory.
Worked Example: Describing a Delayed Rate Pattern
Suppose, hypothetically, a trial has the following events and person-time. Assume for this descriptive calculation that hazards are approximately constant within each interval, person-time is measured consistently, and the event counts are treated as independent Poisson counts. These assumptions are simplifying; a full trial analysis may require covariate adjustment, censoring review, and a prespecified model.
| Interval | Control events | Control person-time | Treatment events | Treatment person-time |
|---|---|---|---|---|
| 0–6 months | 30 | 300 person-months | 36 | 360 person-months |
| 6–12 months | 20 | 180 person-months | 10 | 180 person-months |
In months 0–6, the estimated rates are \(30/300=0.1000\) and \(36/360=0.1000\) events per person-month, giving a treatment-to-control rate ratio of \(1.000\). In months 6–12, the rates are \(10/180=0.05556\) and \(20/180=0.1111\) events per person-month, giving a ratio of \(0.5000\). The later interval therefore shows a lower estimated treatment rate in this hypothetical summary, while the earlier interval shows equal estimated rates.
For the later interval, the approximate standard error of the log rate ratio is \(\sqrt{1/10+1/20}=0.3873\). A nominal 95% interval on the log scale is \(\log(0.5000)\mathbin{\pm}1.96(0.3873)=-0.6931\mathbin{\pm}0.7591\), or \((-1.452, 0.0660)\). Exponentiating gives an approximate interval of \((0.234, 1.068)\) for the rate ratio. It includes 1.000. This interval is a rough, interval-specific summary under the stated assumptions, not a group-sequentially adjusted confirmatory interval. The estimates suggest a possible delayed pattern, but they do not establish one.
A Disciplined Diagnostic Review
A useful review proceeds from data quality and follow-up to the visual pattern, then to time-specific summaries and their uncertainty. If the pattern motivates a different analysis, distinguish that scientific motivation from the confirmatory status of the analysis. Repeatedly inspecting interim data and selecting the most favorable interval or summary can inflate false-positive risk if the selection is then treated as though it had been prespecified.
Confirm the event definition, analysis groups, data cutoff, and censoring conventions. Look for discrepancies in follow-up or data completeness that could distort the curves.
Describe when estimates separate, remain equal, converge, or reverse. Identify the event times at which changes occur and note how many participants remain at risk in each group.
Compare the visual impression with event counts and appropriately cautious time-varying summaries. Treat sparse intervals and intervals chosen after looking at the data as exploratory.
Report a possible pattern and its limitations. Do not present an interim visual feature as proof, a treatment effect estimate with guaranteed validity, or a reason to alter confirmatory inference without a prespecified method.
Several diagnostics deserve particular attention. Unequal censoring patterns can make the tails less comparable. A reversal based on one or two events may be unstable. A time-specific estimate with few events can have wide uncertainty even if its point estimate appears dramatic. A visible pattern may also result from chance, especially when many cutoffs, time intervals, or summaries have been inspected. Reviewing model-based time-varying estimates, such as treatment-by-time terms or trends in Schoenfeld residuals, may add information, but those diagnostics also depend on model specification and do not convert an exploratory interim look into a confirmatory test.
What the Displays Can and Cannot Establish
KM curves are direct, descriptive estimates of survival over time under the censoring assumptions. They can reveal features that a single constant hazard ratio may conceal. A log-rank test is a global comparison with its own assumptions and may be less informative about when effects emerge; weighted log-rank methods emphasize different portions of follow-up, but require prespecification and appropriate sequential calibration when used for monitoring. RMST contrasts summarize differences in average event-free time up to a chosen horizon and can be easier to interpret when hazards are non-proportional, but the horizon and interim inference must also be handled appropriately.
These approaches answer related but different questions. None should be selected after seeing which one gives the most favorable interim result and then reported as if it had been the sole planned analysis. Diagnostic review is valuable because it can flag whether a constant-effect interpretation seems inadequate and inform scientific discussion. It is limited by incomplete follow-up, censoring assumptions, sparse late risk sets, and the multiplicity created by looking at many possible patterns. Any confirmatory claim must rest on the trial’s prespecified analysis strategy and valid control of the sequential error rate.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why does censoring not create a downward step in a Kaplan–Meier curve, and what does it do to later risk sets?
- Two groups have equal KM estimates at month 6 and stay equal until an event at month 8 changes their ordering. At what time does the ordering reverse?
- A late curve separation is accompanied by risk sets of 7 and 5 participants. Name two reasons to interpret that tail cautiously.
- An analyst examines many time intervals and reports only the interval with the smallest treatment-to-control rate ratio as confirmatory. What is the problem?
Solutions. (1) A censored participant has no observed event at the censoring time, so contributes no event factor to the KM product; that person is removed from subsequent risk sets. (2) The ordering reverses at the month-8 event time, not at an unspecified point between months 6 and 8. (3) With so few participants at risk, a small number of events can change the estimates substantially, and the tail estimates have greater sampling uncertainty; censoring differences may also matter. (4) Selecting the most favorable interval after inspecting many intervals creates selection and multiplicity concerns. The selected result is exploratory unless the procedure and its error control were prespecified and appropriately calibrated.