Choosing Survival Estimands For Omics Studies

Start With the Question, Not the Model

An omics feature can be associated with survival in several different senses. It might distinguish patients with different probabilities of being alive at two years, different expected time alive during the first three years, or different rates of entering an event state among those who remain event-free. These are related questions, but they are not interchangeable. A model cannot repair an outcome definition that does not match the scientific question.

For a survival analysis, first state what counts as the event, when the clock starts, and which population the conclusion concerns. Then specify the quantity to be compared and the scale of the comparison. In an omics study, also make clear whether the feature is a measured baseline characteristic, a prespecified score, or a group defined by a threshold. This tutorial focuses on defining that target, not on fitting a particular survival model.

Four Decisions That Define a Survival Estimand

1. Define the event. “Survival” does not by itself identify an outcome. The event could be death from any cause, death attributed to a specified cause, disease recurrence, or a composite such as recurrence or death. State the event operationally, including any relevant adjudication rule. Different event definitions answer different questions: a composite counts any component as an event, while a recurrence-only outcome does not count death without recurrence as the same event.

2. Define time zero. The time origin is the point from which event time is measured. It might be diagnosis, surgery, treatment initiation, or biospecimen collection. The choice should correspond to the clinical question and be consistently observable. For example, measuring from diagnosis addresses time since diagnosis; measuring from a later blood draw addresses time from that draw among people who reached it. Those populations and interpretations need not coincide.

3. Define the target population. Specify who the result is intended to describe: for example, adults with a particular disease stage who provided a pretreatment sample at diagnosis. A cohort selected because participants survived until sample collection may not represent all patients at diagnosis. Likewise, an estimate in one hospital cohort does not automatically describe a different population.

4. Define the contrast and summary scale. State the omics comparison and the outcome summary. A comparison might be between two prespecified expression groups or between values one standard deviation apart. The summary might be risk by a specified horizon, survival probability, restricted mean survival time, or an event rate. If the intended quantity is adjusted for clinical covariates, identify the adjustment set and the kind of adjusted comparison; “adjusted association” alone does not fully specify a target.

Common Prognostic Targets

Let \(T\) denote time from the chosen origin to the specified event, and let \(Z\) denote an omics feature or group. For a target population and group \(z\), the survival function is \(S_z(t)=P(T>t\mid Z=z)\): the probability of remaining free of the event beyond time \(t\), under the specified event definition and population. The corresponding cumulative risk by time \(t\) is \(F_z(t)=P(T\leq t\mid Z=z)=1-S_z(t)\), when the event is defined as the event of interest and the survival probability is defined accordingly.

A fixed-horizon prognostic contrast might be \(F_1(\tau)-F_0(\tau)\), the difference in event risk by horizon \(\tau\), or \(S_1(\tau)-S_0(\tau)\), the difference in event-free probability at that horizon. These contrasts are on an absolute scale and have a direct interpretation at the chosen horizon. A risk ratio \(F_1(\tau)/F_0(\tau)\) is a relative-scale alternative when the denominator is nonzero. State which scale is used: the same two groups can have the same risk ratio but a different risk difference at another baseline risk.

Another prognostic target is the restricted mean survival time through horizon \(\tau\):

This is the expected event-free time accumulated from time zero through \(\tau\), with times beyond \(\tau\) counted as \(\tau\). Its between-group difference is expressed in units of time. It does not require summarizing group differences by one constant relative event rate over all follow-up.

Event-Process Targets and Their Interpretation

For an event time \(T\), the hazard at time \(t\), when defined, is the instantaneous event rate at \(t\) among members of the target group who have remained event-free just before \(t\). In continuous time, it can be expressed as the limiting conditional probability of an event in a short interval after \(t\), divided by the interval length. Thus, a hazard concerns the evolving event process and its current risk set, not the probability that a randomly selected member of the original group experiences the event by \(t\).

An event-process question could ask whether the conditional event rate at each time differs by omics group, or whether that difference changes over time. A fixed-horizon risk question instead asks about the accumulated probability by a selected time. The two can move differently: groups may have different early event rates but similar later rates, or different rates that produce a particular cumulative risk contrast. A single rate ratio should not be described as a risk ratio.

A related, simpler summary is a crude incidence rate: number of first events divided by total observed person-time at risk. It is an average rate over the observed follow-up, not generally the same quantity as a time-specific hazard. If the scientific question concerns repeated events, the event count and time-at-risk definition must say whether each occurrence is counted; a first-event survival estimand does not answer a recurrent-event question.

Why the RMST Formula Is an Area

The identity defining RMST follows from a basic representation of a nonnegative random variable. For any event time \(T\), the truncated time \(\min(T,\tau)\) equals the length of time in \([0,\tau]\) for which the person remains event-free. In indicator form,

Taking expectations and interchanging expectation and integration for this bounded nonnegative quantity gives

The integrand is the probability of remaining event-free at each time, so its area accumulates expected event-free time. This also explains why the horizon \(\tau\) is part of the estimand: changing it changes the area and potentially the comparison.

Checks, Alternatives, and Limitations

Before selecting an analysis, write a one-sentence estimand such as: “Among adults with stage II disease who provided a pretreatment sample at diagnosis, compare the two-year risk of all-cause death from diagnosis between prespecified high and low expression groups.” That sentence identifies the target population, contrast, event, origin, horizon, and scale. If the scientific aim is to characterize event timing among those still event-free, state that event-process target instead.

The target must be distinguished from the conditions needed to estimate it. For example, incomplete follow-up can prevent direct observation of every participant's status at a horizon; an appropriate estimation strategy then requires assumptions about the observation process. Those assumptions affect whether the target can be identified from the data, but they do not change what the target was defined to be. The method for handling right censoring is a separate modeling question.

Several checks help reveal a mismatched estimand. Confirm that the omics measurement precedes the defined follow-up origin when it is intended as a baseline predictor; otherwise, the question may concern a different population or involve information measured after follow-up began. Check whether the specified event can be applied consistently across participants and cohorts. Confirm that the selected horizon is supported by the observed follow-up and is clinically meaningful. Finally, ensure that the interpretation uses the same scale as the estimand: percentage-point differences, time differences, and rate ratios are not synonyms.

A frequent mistake is to say, “The high-expression group has twice the risk because its event rate is twice as large.” The calculation in the event-rate example supports a twofold ratio of crude events per person-year, not a twofold risk by a fixed time and not necessarily a twofold hazard at every time. The correction is to report the quantity actually calculated and, if the scientific question concerns risk by a horizon, define and estimate that horizon-specific risk contrast.

Another limitation is that an omics association is not automatically causal. Even a carefully defined prognostic contrast describes outcomes across groups as observed; differences in clinical characteristics, treatment, or selection into the study may contribute. If the aim is a causal effect of an intervention that changes an omics feature or acts through it, that requires a different question and additional causal assumptions.