Choosing a Time-to-Event Estimand
A time-to-event outcome records how long it takes for a prespecified event to occur—for example, time from randomization to disease progression or death. A treatment comparison depends not only on how event times are analyzed, but also on which feature of their distributions is the target. Survival probability at a chosen time, an event-time quantile, and restricted mean survival time (RMST) answer different clinical questions; they are not interchangeable summaries.
The time origin, event definition, target population, treatment strategies, and handling of intercurrent events must be specified. In particular, the chosen intercurrent-event strategy determines what the potential event time means after an event such as treatment discontinuation or rescue medication. The estimand is the target quantity; censoring assumptions and the statistical model are needed to estimate it, but do not themselves define the clinical question.
A Bayesian analysis represents uncertainty about the estimand with its posterior distribution. As in the earlier material on Bayesian estimands as posterior distributions, a practical way to obtain that distribution is to calculate the target contrast for every posterior draw of the model parameters. The resulting draws describe uncertainty about the estimand, conditional on the model, data, and prior—not variability among individual patients alone.
Survival Probability at a Prespecified Time
For a prespecified time \(t^*\), the arm-specific survival estimand is \(S_a(t^*)\). It answers a direct question: what proportion of the target population would be expected to remain event-free beyond \(t^*\) under strategy \(a\)? A common contrast is the survival-probability difference, \(S_1(t^*)-S_0(t^*)\), measured in probability points. A ratio, \(S_1(t^*)/S_0(t^*)\), is another possibility when the denominator is positive. The protocol should identify the time and contrast scale in advance; choosing a favorable time after viewing results can distort interpretation.
Worked Example: Survival Difference at 12 Months
Suppose a hypothetical Bayesian analysis of a randomized trial produces the following four illustrative posterior draws of the 12-month survival probabilities. These few draws are shown only to demonstrate the calculation; an actual analysis would use many more draws.
| Draw | \(S_1(12)\) | \(S_0(12)\) | Difference |
|---|---|---|---|
| 1 | 0.90 | 0.82 | 0.08 |
| 2 | 0.88 | 0.80 | 0.08 |
| 3 | 0.92 | 0.84 | 0.08 |
| 4 | 0.86 | 0.81 | 0.05 |
For each draw, subtract the control survival probability from the experimental survival probability. For draw 1, for example, \(0.90-0.82=0.08\), or 8 percentage points. The mean of these illustrative contrast draws is
Thus, the mean across these four draws is a 7.25-percentage-point higher 12-month survival probability under the experimental strategy. All four draws have a positive difference, so the fraction of these draws above zero is \(4/4=1\). This fraction is only an illustration from a deliberately tiny set of draws; it is not a reliable posterior probability for a real analysis. With adequate posterior simulation, the corresponding fraction estimates the posterior probability that the difference is positive.
A positive difference means a greater probability of being event-free beyond the chosen time under the experimental strategy. It does not say how event times differ among people who experience the event, or how much time is gained on average. Those questions motivate quantiles and RMST.
Event-Time Quantiles
An event-time quantile describes a point in the distribution of time to event. For \(0<p<1\), define the \(p\)-quantile in arm \(a\) by \(q_a(p)=\inf\{t:F_a(t)\ge p\}\), where \(F_a(t)=P(T^a\le t)=1-S_a(t)\). Equivalently, it is the earliest time at which the cumulative event probability reaches \(p\). The median event time is \(q_a(0.5)\). A common treatment contrast is \(q_1(p)-q_0(p)\), in units of time; a positive value means the specified event quantile occurs later under the experimental strategy.
Quantiles require adequate support in the time range being analyzed. If the event probability never reaches \(p\), the \(p\)-quantile is not defined from that distribution. For instance, if fewer than half the target population has experienced the event by the end of follow-up, the median cannot be read directly from the observed survival curve. A model may extrapolate to a median beyond follow-up, but that estimate relies on extrapolation assumptions rather than direct information at that time.
Worked Example: Bayesian Median-Time Difference
Consider a hypothetical exponential event-time model, under which \(S_a(t)=\exp(-\lambda_a t)\) with a constant event rate \(\lambda_a\) per month. This is a strong distributional assumption, not a generic property of time-to-event data. Suppose the displayed posterior draws of the rates are:
| Draw | \(\lambda_1\) (per month) | \(\lambda_0\) (per month) |
|---|---|---|
| 1 | 0.040 | 0.050 |
| 2 | 0.045 | 0.055 |
| 3 | 0.050 | 0.060 |
The median solves \(S_a(t)=0.5\), so \(\exp(-\lambda_a t)=0.5\), giving \(q_a(0.5)=\log(2)/\lambda_a\). In draw 1, the experimental median is \(0.6931/0.040=17.33\) months and the control median is \(0.6931/0.050=13.86\) months. The contrast is \(17.33-13.86=3.47\) months.
Applying the same calculation to all three draws gives contrasts of \(2.80\) months for draw 2 and \(2.31\) months for draw 3. Their mean is \((3.47+2.80+2.31)/3=2.86\) months, and all three are positive. This example supports a positive model-based median contrast in the displayed draws. It does not establish that an exponential model fits a real trial, and three draws cannot provide a meaningful posterior probability or uncertainty summary.
Restricted Mean Survival Time
Restricted mean survival time summarizes event-free time up to a fixed horizon \(\tau\). Define \(\mu_a(\tau)=E[\min(T^a,\tau)]\). A participant with an event before \(\tau\) contributes their event time; a participant still event-free at \(\tau\) contributes \(\tau\). The treatment contrast is \(\mu_1(\tau)-\mu_0(\tau)\), measured in time units. For example, a difference of 1.2 months means 1.2 additional expected event-free months accumulated within the first \(\tau\) months under the experimental strategy. It does not imply that every participant gains 1.2 months.
For a nonnegative event time, RMST also equals the area under the survival curve up to \(\tau\):
To justify the identity, for each realized \(T^a\), the value \(\min(T^a,\tau)\) is the length of the interval from 0 to \(\tau\) for which \(T^a>t\). Thus, apart from an endpoint that does not affect the integral, \(\min(T^a,\tau)=\int_0^\tau \mathbf{1}(T^a>t)\,dt\). Taking expectations and interchanging expectation and integration for this bounded, nonnegative quantity gives
The finite, prespecified horizon makes the summary interpretable even when the full mean event time is not estimable—for example, when follow-up is limited or the event-time distribution has a long tail. However, the result is horizon-specific. Changing \(\tau\) changes the estimand and can change the treatment contrast.
Worked Example: RMST Difference Through 12 Months
Suppose, for illustration, posterior survival curves are represented by straight-line segments between months 0, 6, and 12. In one posterior draw, the experimental curve has survival values \(1.00, 0.82, 0.62\), and the control curve has values \(1.00, 0.72, 0.42\), at those times. The trapezoidal area on each six-month interval gives the exact integral for these piecewise-linear curves.
For the experimental strategy, the areas are \(6(1.00+0.82)/2=5.46\) months and \(6(0.82+0.62)/2=4.32\) months. Therefore, its RMST through month 12 is \(5.46+4.32=9.78\) months. For control, the areas are \(6(1.00+0.72)/2=5.16\) months and \(6(0.72+0.42)/2=3.42\) months, for an RMST of \(8.58\) months. The draw-specific contrast is \(9.78-8.58=1.20\) months.
Repeat this calculation for two further illustrative posterior draws:
| Draw | Experimental survival at months 6, 12 | Control survival at months 6, 12 | RMST difference through month 12 |
|---|---|---|---|
| 1 | 0.82, 0.62 | 0.72, 0.42 | 1.20 months |
| 2 | 0.80, 0.58 | 0.74, 0.46 | 0.72 months |
| 3 | 0.84, 0.64 | 0.70, 0.40 | 1.56 months |
For draw 2, the arm-specific areas are \(5.40+4.14=9.54\) months and \(5.22+3.60=8.82\) months, giving \(0.72\) months. For draw 3, they are \(5.52+4.44=9.96\) months and \(5.10+3.30=8.40\) months, giving \(1.56\) months. The mean of the three displayed contrasts is \((1.20+0.72+1.56)/3=1.16\) months; all three are positive. As before, this tiny illustrative sample of draws is not sufficient for real posterior inference.
Estimation Assumptions, Diagnostics, and Alternatives
A survival estimand is defined by the target population and treatment strategies, but estimating it requires assumptions about the trial and the observed data. In a randomized trial, random assignment supports comparisons of the assigned strategies, subject to appropriate handling of intercurrent events and missingness. A survival analysis must also account for right censoring: a censored participant is known to have remained event-free up to a last observed time, but their later event time is unknown. The likelihood or other model must use that partial information rather than treating censoring time as an event time.
A common identifying condition is that censoring is independent of the event time, at least conditional on variables included in the model. Informally, among participants with the same modeled information, those censored at a time should not have systematically different unobserved future event times from those remaining under observation. This condition can fail if, for example, worsening health affects both dropout and event risk and is not adequately accounted for. Adequate follow-up and positivity are also important: the data must contain information about survival over the times and groups being compared. Model fit matters, especially when a parametric model extrapolates beyond observed follow-up.
Diagnostics should therefore examine the number still at risk over time, the amount and timing of censoring, and whether observed survival patterns are compatible with the model. Check whether predictions track the observed survival information over the supported range, and investigate sensitivity to plausible modeling choices when conclusions depend on extrapolation. For quantiles, check explicitly that the relevant event-probability threshold is reached within the informative range. For RMST, check that the prespecified horizon is supported by follow-up in both arms. A smooth posterior curve is not proof that the data identify its distant tail.
These estimands emphasize different features. Survival probability is a point-in-time comparison; a quantile compares a particular location in the event-time distributions; RMST summarizes the entire survival curve up to \(\tau\). A hazard ratio, by contrast, compares instantaneous event rates among those still event-free, and its interpretation and assumptions differ from these marginal summaries. No one scale is universally best. Choose the scale to answer the clinical question, and report the horizon or quantile level so the contrast is interpretable.
Bayesian analysis does not remove these limitations. Posterior probabilities and intervals are conditional on the likelihood, model, prior, and estimand definition. A precise posterior under a misspecified survival model, informative censoring, or an unsupported time horizon can still be misleading. The estimand and horizon should be chosen for clinical relevance before examining treatment results, and the modeling assumptions should be made explicit.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In words, interpret a survival-probability difference of 0.06 at 18 months.
- A model gives median event times of 10 and 12 months for experimental and control strategies. What is the experimental-minus-control median contrast, and how should its sign be interpreted?
- Explain why RMST through 24 months is not generally the same estimand as RMST through 36 months.
- If fewer than half of participants have experienced the event by the informative end of follow-up, what is the problem with estimating the median directly from the observed survival curve?
- Give one censoring assumption needed for valid estimation and one diagnostic that could reveal a problem.
Solutions. (1) The experimental strategy has a 6-percentage-point higher probability of remaining event-free beyond 18 months in the target population. (2) The contrast is \(10-12=-2\) months. The experimental median is two months earlier, not later, than the control median. (3) The integrals cover different portions of the survival curves, so the accumulated event-free time and treatment contrast can change with the horizon. (4) The observed curve has not reached survival probability 0.5, so the median is not directly identified within the informative follow-up range; a model-based value beyond that range would rely on extrapolation. (5) One condition is censoring independent of event time conditional on modeled information. A diagnostic is to inspect censoring patterns and numbers at risk over time, alongside checks of model predictions against observed survival information.