Time-to-Event

Time-to-Event Outcomes in Trials and Economic Models

A time-to-event outcome is the length of time each patient goes, counted from a fixed starting point, before a named event such as death or disease progression happens. Overall survival and progression-free survival in cancer trials are the best-known examples, and in health economic evaluation outcomes of this kind decide how long patients spend in each health state, and so how many costs and quality-adjusted life years a model accrues. What sets these data apart is that many patients have not had the event when the analysis is run, so their times are only partly known. The statistical tools belong to survival analysis. The economic questions are which summary of the event times a model needs and how one fitted curve becomes the input each model type uses.

What an event time records

An event time has meaning only once its start and its end are fixed. Clark and colleagues, introducing survival analysis in the British Journal of Cancer, stress the need to state what the event is and when observation starts and finishes, and advise choosing an end-point that cannot be misclassified, such as death from any cause rather than from one named cause. The FDA guidance on cancer trial endpoints, issued in December 2018, defines the common oncology outcomes in these terms.

The guidance prefers progression-free survival to time to progression, noting that because it includes deaths it can be a better correlate of overall survival.

For each patient the data hold two items, the time observed and whether the event was seen at that time. In the usual notation for right-censored data

$$Y_i = \min(T_i, C_i), \qquad \delta_i = I(T_i \le C_i)$$

where $T_i$ is the true event time for patient $i$, $C_i$ is the patient's censoring time, when follow-up would end without an event, $Y_i$ is the time recorded, $I(\cdot)$ is the indicator function and $\delta_i$ is 1 for an observed event and 0 for a censored record. Clark and colleagues note that censored times understate the true time to event, and the censoring page explains how such records enter an analysis.

Why the mean event time matters most

Event times are rarely symmetric. Clark and colleagues describe survival data as skewed, with many early events and relatively few late ones, so the median and the mean can sit far apart. The distribution is described by the survival function, the hazard rate and the cumulative hazard, and the mean is the whole area under the survival curve.

$$E[T] = \int_{0}^{\infty} S(t)\, dt, \qquad S(t) = \Pr(T > t) = e^{-H(t)}$$

where $T$ is the event time, $S(t)$ is the probability of remaining event-free beyond time $t$, $H(t)$ is the cumulative hazard up to $t$ and $E[T]$ is the mean event time.

NICE Decision Support Unit Technical Support Document (TSD) 14 states that it is mean rather than median effects that matter for economic evaluation, so entire survival distributions are needed. Because many patients are still event-free when trials end, TSD 14 adds that extrapolation is required unless the data are complete or very close to complete. Within follow-up, the area under the Kaplan-Meier estimator gives the restricted mean survival time. Beyond it, the mean depends on a parametric survival model or a more flexible curve. The NICE manual (PMG36, section 4.2.24) notes that a lifetime time horizon often requires extrapolation beyond the available studies, with the uncertainty that goes with it.

How event times become model inputs

The same fitted curve can feed three common model structures, and each uses it in a different form. All three forms below come from the fitted survival function or its parameters.

Cohort state transition models

A Markov model moves its cohort in fixed cycles, so it needs a transition probability for each cycle and not a continuous curve. Each probability applies only to those still event-free when the cycle starts. From a survival curve, the probability of the event during the cycle ending at time $t$ is

$$p(t) = 1 - \frac{S(t)}{S(t-u)}$$

where $p(t)$ is the per-cycle transition probability, $u$ is the cycle length and $S(\cdot)$ is the fitted survival function. The result is constant only when the hazard is constant, and TSD 14 observed that exponential models were often used where Markov transition probabilities were not time dependent. Any other shape needs a time-inhomogeneous model or, where the clock is time in a state, a chain of tunnel states, and TSD 15 notes that a cohort model can hold only so many such states before it becomes unwieldy.

Partitioned survival models

A partitioned survival model uses the curves directly. TSD 19 describes state membership as derived from a set of non-mutually exclusive survival curves, so the proportion in the progressed state at each time is overall survival minus progression-free survival. Because each endpoint is modelled separately, its extrapolation reflects within-trial trends in that endpoint alone unless external evidence on later hazards is added.

Discrete event simulation

In a discrete event simulation, TSD 15 describes the distributions of time to each event type as the main inputs, with each patient's times sampled from them. A changing risk is handled by sampling one time from a non-exponential distribution, which TSD 15 calls easier than computing time-dependent transition probabilities for every cycle. For a Weibull distribution it gives

$$t_0 = \beta\,[-\ln U]^{1/\alpha}, \qquad t_1 = \beta\left[\frac{-\ln U}{HR}\right]^{1/\alpha}, \qquad \gamma = e^{-\ln(HR)/\alpha}$$

where $U$ is a uniform random number between 0 and 1, $\alpha$ is the shape and $\beta$ the scale of the comparator curve, $t_0$ is the sampled comparator time, $HR$ is a constant hazard ratio, $t_1$ is the sampled time on treatment and $\gamma$ is the equivalent acceleration factor, the effect in accelerated failure time form. The conversion holds only when both arms follow a Weibull with a common shape. Otherwise TSD 15 suggests fitting separate curves to each arm.

Worked example with one Weibull curve in three model forms

All numbers here are illustrative. Progression-free survival on standard care follows a Weibull curve with shape 1.2 and scale 10 months, so the hazard rises over time. A new treatment has a hazard ratio of 0.70 with the same shape.

Step 1. Landmark survival. At 12 months, 12/10 raised to the power 1.2 is 1.2446, so survival is $e^{-1.2446} = 0.288$. About 28.8% of patients on standard care are progression-free at one year.

Step 2. Median and mean. The median solves $S(t) = 0.5$ and the mean uses the gamma function.

$$\text{Median} = \beta\,(\ln 2)^{1/\alpha} = 10 \times 0.7368 = 7.37, \qquad \text{Mean} = \beta\,\Gamma\!\left(1 + \frac{1}{\alpha}\right) = 10 \times 0.9407 = 9.41$$

where $\Gamma(\cdot)$ is the gamma function, evaluated at 1.8333 to give 0.9407, and both results are in months. The mean exceeds the median by about two months because of the long right tail.

Step 3. Monthly probabilities for a cohort model. With a one-month cycle, the probability for month 1 is one minus the survival ratio 0.9389 over 1.0000, or 0.061. Month 12 uses 0.2881 over 0.3259, giving 0.116, and month 24 uses 0.05731 over 0.06608, giving 0.133. No single constant probability can match this rising hazard.

Step 4. Sampled times for a simulation. For a patient drawn with $U = 0.30$, minus the log of 0.30 is 1.2040. On standard care the sampled time is 10 × 1.2040 to the power 1/1.2, or 11.67 months. On the new treatment it is 10 × (1.2040/0.70) to the power 1/1.2, which is 10 × 1.7200 to the power 0.8333, or 15.71 months. The acceleration factor is 0.70 to the power minus 1/1.2, or 1.346, and 11.67 × 1.346 = 15.71, so both routes agree.

Step 5. Treatment gain. Under a common shape the new treatment's scale is 10 × 1.346 = 13.46 months, so its median is 7.37 × 1.346 = 9.92 months and its mean is 9.407 × 1.346 = 12.66 months. Progression-free survival at 12 months rises to 0.288 to the power 0.70, or 0.418. The median gain is 2.55 months (9.92 less 7.37), while the mean gain is 3.26 months (12.662 less 9.407).

Interpretation. The model should value the mean gain of 3.26 progression-free months per patient, not the median gain, which understates it by 0.71 months. A cohort model built from Step 3, a partitioned survival model using the area under the same curve and a simulation sampling as in Step 4 should give approximately the same means over a long enough horizon, within cycle-length and simulation error. In a real appraisal part of that area may lie beyond trial follow-up, so the means are only as credible as the curve's tail.

Choices that change the result

Several analysis decisions move the time gained and the QALYs attached to it. The main ones in appraisal work follow.

• Endpoint definition. Counting or censoring deaths changes the event count. The FDA notes that in time-to-progression analyses deaths are censored, either at death or at an earlier visit, and describes the latter as informative censoring.
• Assessment schedule. Progression is found only at scheduled visits, and the FDA lists balanced timing of assessments between arms as critical for progression-based endpoints.
• Choice of curve. PMG36 (section 4.6.25) lists standard parametric and more flexible approaches, such as spline-based and cure models, asks for the clinical plausibility of their hazard functions to be assessed, and warns that functions with stable or decreasing variance over time are likely to understate uncertainty in the extrapolation.
• Proportional hazards. PMG36 (section 4.6.22) says the proportional hazards assumption should always be assessed, preferably with log-cumulative hazard plots, and section 4.6.24 points to other methods when it fails.
• Treatment switching. The FDA notes that overall survival may be affected when control patients switch to the new treatment. PMG36 (section 4.6.26) advises against censoring or excluding patients who cross over, because such simple methods are very susceptible to selection bias.
• Evidence synthesis. PMG36 (section 4.6.21) says synthesis of survival outcomes needs individual patient data, and that methods such as that of Guyot and colleagues can otherwise reconstruct Kaplan-Meier data.

Misreadings and limits

Five misreadings of time-to-event outcomes can distort an economic model. Four concern how event times or treatment effects are summarised, and the fifth concerns reading one endpoint across to another.

1. A median used as a mean. In TSD 14's review of 45 NICE technology appraisals, mean estimates were used in 36 but median estimates in 16, sometimes directly as model parameters. Medians cannot be added up into life years or QALYs.
2. An average of observed times. Averaging recorded times treats censored records as events and understates the mean.
3. One minus Kaplan-Meier for a cause-specific event. Austin, Lee and Fine show that when other events can occur first, this overstates the incidence of the event of interest, whether or not the events are independent. Competing risks methods give the cumulative incidence instead.
4. A hazard ratio read as time gained. The months gained depend on the baseline curve and its shape, as Step 5 shows, and a constant ratio may not hold over the extrapolated period.
5. Progression-free gains assumed to carry over to survival. The FDA notes that progression-based endpoints may not always correlate with survival and that data are usually insufficient to evaluate the relationship between effects on the two.

Sources

• Austin PC, Lee DS, Fine JP. Introduction to the analysis of survival data in the presence of competing risks. Circulation. 2016;133(6):601-609. https://doi.org/10.1161/CIRCULATIONAHA.115.017719
• Clark TG, Bradburn MJ, Love SB, Altman DG. Survival analysis part I: basic concepts and first analyses. British Journal of Cancer. 2003;89(2):232-238. https://pmc.ncbi.nlm.nih.gov/articles/PMC2394262/
• Davis S, Stevenson M, Tappenden P, Wailoo A. NICE DSU Technical Support Document 15: cost-effectiveness modelling using patient-level simulation. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; April 2014. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD15_Patient-level_simulation.pdf
• Food and Drug Administration. Clinical Trial Endpoints for the Approval of Cancer Drugs and Biologics. Guidance for Industry. December 2018. https://www.fda.gov/media/71195/download
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR; June 2011, last updated March 2013. https://doi.org/10.15131/shef.data.33745843
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 4.2.24, 4.6.21, 4.6.22, 4.6.24, 4.6.25 and 4.6.26. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: Partitioned survival analysis for decision modelling in health care: a critical review. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; June 2017. https://orda.shef.ac.uk/articles/report/TSD_19_Partitioned_survival_analysis_as_a_decision_modelling_tool/33848653