Time-to-Event Questions: First Events, Competing Events and Recurrences Need Different Maps

อ่านฉบับภาษาไทย (Thai version)

Abstract

Time-to-event analysis is used when the outcome asks not only whether an event happens but when it happens. Before choosing a model, three questions decide the map: is only the first event of interest, can the same event happen again in one patient, and can another event stop the event of interest from ever happening? The answers lead to three families. A first event, such as death or a first recurrence, is described with the Kaplan-Meier survival curve and modelled with the Cox model, which estimates a hazard ratio adjusted for covariates. Recurrent events, such as repeated asthma exacerbations, use the Andersen-Gill model, which puts all events on one clock, or the Prentice-Williams-Peterson model, in which the previous event matters and, in gap time, the clock restarts. Competing risks, such as death before cancer recurrence, use the cause-specific hazard model or the Fine-Gray subdistribution hazard model. This article walks through each family, the decision path between them and the phrases that point to each method.

Visual summary: three questions before a time-to-event model lead to three families: first event with Kaplan-Meier and Cox, recurrent events with Andersen-Gill and Prentice-Williams-Peterson, and competing risks with the cause-specific hazard and Fine-Gray models; an sHR of 0.70 is not a 30% lower risk at every time. — Visual summary.

One cohort, three kinds of outcome

A research team follows patients after surgery for colorectal cancer. Some have a recurrence. Some are admitted to hospital again and again. Some die before the cancer ever returns.

For every outcome the team wants to know not only whether it happens but when. Before anyone fits a model, the team has to decide what shape each question has, because the shape picks the model.

Big picture: whether, and when

Time-to-event analysis is used when the outcome asks not only whether the event happens but also when it happens [1]. Each patient contributes a length of follow-up and the way it ended: with the event, or without it.

When follow-up ends without the event, because the study closed or contact was lost, the patient is censored: known to be event-free up to that time, with the event still possible later [1].

Three questions before choosing a model

Three questions come before any model:

Is only the first event of interest?
Can the event happen repeatedly in the same patient?
Is there another event that prevents the event of interest from ever happening?

The first points to first-event analysis, the second to recurrent-event analysis and the third to competing-risk analysis. The questions are not exclusive: a study of the first recurrence of cancer asks the first and the third at once, because death can come before recurrence.

The master map

The three answers give three families of methods, each with two tools. Kaplan-Meier and Cox belong to first-event analysis; Andersen-Gill (AG) and Prentice-Williams-Peterson (PWP) to recurrent-event analysis; the cause-specific hazard model and the Fine-Gray model to competing-risk analysis.

Inside each family the two tools do different jobs, and the sections below take them one family at a time.

⤢ click to enlarge

The master map of time-to-event analysis: pick the family by the question, then the tool by the job.

First-event analysis

First-event analysis is used when the interest is the time until the first event. The analysis stops at that event; whatever happens to the patient afterwards does not enter it.

Examples:

Time to first recurrence
Time to death
Time to first hospitalisation
Time to treatment failure

The two main tools

Kaplan-Meier (KM): the descriptive survival curve. At each time it estimates the proportion of patients still free of the event, using every censored patient for as long as that patient was followed [2].
Cox proportional hazards model: estimates the hazard ratio while adjusting for covariates [3]. The hazard is the event rate at a given moment among patients still event-free and still followed; the hazard ratio compares that rate between groups, and the model assumes the ratio stays constant over follow-up (proportional hazards).

Kaplan-Meier describes; Cox compares groups after adjustment. Part 2 of this series takes both apart.

Recurrent-event analysis

Recurrent-event analysis is used when the same patient can have the event repeatedly. Stopping at the first event would throw the later ones away.

Examples:

Repeated asthma exacerbations
Recurrent infection
Recurrent hospitalisation
Recurrent cancer relapse

Two models are worth knowing [4].

Andersen-Gill (AG)

The Andersen-Gill model treats recurrent events as one continuous process on total time: the clock starts once, at the start of follow-up, and keeps running through every event [4, 5].

Each patient's follow-up is split into stretches between events, a layout called a counting process. Every event counts in the same way, whether it is the patient's first or a later one, and after each event the patient stays at risk of the next.

Prentice-Williams-Peterson (PWP)

The PWP model gives weight to event order [4, 6]. It analyses the first, second and later events in separate strata, and a patient is at risk of a second event only after a first: each event is analysed conditional on the previous one. The effect of treatment or a covariate can be allowed to differ by event number or be shared across them.

PWP is most often met in its gap-time form: after each event the clock is reset to zero, so time is counted from the previous event rather than from the start of follow-up [4]. A total-time form also exists, in which the clock keeps running while the strata still follow event order.

Part 6 fits both models.

Competing-risk analysis

Competing-risk analysis is used when another event, once it happens, makes the event of interest impossible [7].

Example:

Outcome: cancer recurrence
Competing event: death before recurrence

Death is not ordinary censoring, because after death the recurrence can never happen. A censored patient could still have the event later; a patient who has died cannot. Treating death as censoring describes a population in which nobody dies first, and so it overstates the share of patients who will actually have a recurrence [7, 8].

Two main approaches

Cause-specific hazard model: models the rate of the event of interest among patients still alive and still free of the event, usually with a Cox model in which a patient who has the competing event leaves the risk set (the patients counted as at risk) at that time [7]. Its hazard ratio speaks to whether a factor changes that rate.
Fine-Gray subdistribution hazard model: models the subdistribution hazard, whose risk set keeps patients who have already had the competing event [9]. Its ratio, the subdistribution hazard ratio (sHR), is tied to the cumulative incidence: the probability of having had the event by a given time when death can come first [7, 8].

The two models estimate different quantities, different estimands, so they are two answers to two questions rather than two steps, and a study that asks both questions reports both [8]. Part 4 builds the cumulative incidence, and part 5 fits the two models side by side.

Two misreadings to avoid

"Death is just another censored observation."

Censoring assumes the event could still happen after follow-up ends. After death a recurrence cannot happen, so censoring at death describes a population in which nobody dies first, and the share with a recurrence comes out too high.

Fix: Treat death as a competing event. Removing patients from the risk set at death is right for the cause-specific hazard; for the risk a patient actually faces, use the cumulative incidence.
"sHR = 0.70 means the risk is 30% lower at every time."

The subdistribution hazard ratio compares subdistribution hazards, not risks. When the subdistribution hazards are proportional, an sHR of 0.70 implies a lower cumulative incidence at every time, but the size of that difference changes over follow-up and is not a fixed 30%.

Fix: Report the sHR as a ratio of subdistribution hazards, and read the absolute difference in risk at a stated time from the cumulative incidence curves.

The decision path

Asked in order, the three questions become a decision path. The first two choose between first-event and recurrent-event analysis. The third is asked whatever the answers to the first two: a first recurrence of cancer is a first event, and death before recurrence still competes with it.