Two Different Averages
At each follow-up time, an average biomarker is calculated from the participants whose biomarker values are available. If some participants have experienced an event, however, that average may describe a changing subset of the original cohort. It answers a different question from asking how the full baseline cohort’s biomarker trajectory would have evolved.
The distinction is not automatically a matter of one average being correct and the other biased. The right target depends on the scientific question. The mean among people who are still event-free can be meaningful in its own right. A full-cohort trajectory can also be a meaningful target, but after a terminal event it may refer to an unobserved, hypothetical biomarker outcome rather than an ordinary measurement.
The notation \(Y_i^0(t)\) is a conceptual target, not a value automatically recoverable from the data. It might be scientifically interpretable if “prevent the event” describes a meaningful scenario and the biomarker could in principle be measured under that scenario. Otherwise, a full-cohort post-event average may not be a well-defined or useful target. This distinction matters especially for terminal events such as death: there is no ordinary post-death biomarker measurement to enter an average.
Why the Survivor Mean Can Move
The participants contributing to the survivor mean can change over time. If event-free status is associated with biomarker level or biomarker progression, the composition of that group changes too. A shift in its average can therefore reflect both within-person change and a change in who remains in the group.
Here is a precise decomposition when the outcome is defined for everyone at time \(t\). Write \(S_i(t)=I(T_i\geq t)\), the indicator that participant \(i\) is event-free at \(t\), and let \(p(t)=P\{S_i(t)=1\}\). The cohort mean is a mixture of the means among event-free and event-experienced participants:
Subtracting the survivor mean gives the difference between the two averages:
Thus, if participants with an event would have had higher biomarker values than event-free participants, the survivor mean is below the cohort mean, provided some participants have experienced an event. If the two group means are equal, selection on event-free status does not separate these averages at that time. For a terminal event, the mixture identity applies only to an outcome defined for both groups—for example, a hypothetical \(Y_i^0(t)\). It does not make an undefined post-event biomarker observable.
Worked Example: A Survivor Mean Below the Cohort Mean
Consider a hypothetical cohort of 100 participants. At a follow-up time, 80 remain event-free and have a mean biomarker value of 13. The other 20 have experienced an event. Suppose the biomarker is still defined after their event, and their mean is 16. Calculate the survivor mean, the full-cohort mean, and their difference.
The survivor mean is the mean for the 80 event-free participants: 13 units. The full-cohort mean includes both groups:
The survivor mean is \(13-13.6=-0.6\) units below the cohort mean. The mixture identity gives the same result: \(p=0.8\), so \( (1-p)(13-16)=0.2(-3)=-0.6\) units. In this example the two means are both defined and directly comparable. If the event were terminal and the post-event biomarker undefined, the value 16 could not be treated as an observed post-event mean; it would need to refer to a specified hypothetical outcome.
A Survivor Trend Need Not Be the Cohort Trend
Comparing means at two times can be particularly misleading if the membership of the event-free group changes substantially. Even when each participant’s biomarker would increase over time, the observed survivor mean can decrease if participants with high values are more likely to experience the event and leave the event-free group.
Worked Example: The Observed Mean Falls While Individual Values Rise
Suppose a hypothetical cohort has 100 participants at baseline. Fifty have biomarker values of 18 units and fifty have values of 2 units, so the baseline mean is:
At the next time, suppose every participant’s value has increased by 1 unit: the first group would have values of 19, and the second group values of 3. But only one participant from the first group remains event-free; all 50 from the second group remain event-free. The observed survivor mean is:
The survivor mean has fallen from the baseline mean of 10 units to about 3.314 units, even though every participant’s value in the hypothetical calculation increased. This comparison does not mean the same 100 people were averaged at both times: baseline includes everyone, while the follow-up survivor mean includes only 51 participants. If the hypothetical follow-up values are defined for all 100, their full-cohort mean is \( \{50(19)+50(3)\}/100=11\) units. The falling survivor mean and rising full-cohort mean answer different questions.
This example assumes the stated follow-up values are meaningful for every participant. If the event prevents the biomarker from being defined, the value for someone who experienced that event cannot simply be supplied by carrying forward an earlier value. A full-cohort trend then requires an explicit hypothetical definition and additional assumptions; the observed survivor trend remains a description of those who are event-free.
When Is a Full-Cohort Trajectory Well Defined?
First ask what happens to the biomarker after the event. If it remains meaningful and can be measured, a mean over all baseline participants may be a direct descriptive target, subject to having or appropriately handling the necessary measurements. If the event makes the biomarker undefined, as death generally does, “the cohort’s mean biomarker after the event” has no automatic observed-data interpretation.
One possible target is the mean trajectory \(E\{Y_i^0(t)\}\), where \(Y_i^0(t)\) is the value participant \(i\) would have at \(t\) under a specified scenario preventing the terminal event. This is a hypothetical estimand: its definition depends on the scenario, and its estimation needs assumptions about outcomes that are not observed after the event. It is not the same as the survivor mean. Nor should the survivor mean be described as the hypothetical outcome for the entire cohort; it is conditional on actually being event-free.
A useful first step is therefore to state the target in words before calculating an average. For example: “mean biomarker among participants alive and event-free at 12 months” defines a survivor average. “Mean 12-month biomarker that the baseline cohort would have had if the terminal event had been prevented” describes a hypothetical target that needs further specification. The second statement cannot be estimated merely by averaging the observed 12-month measurements among survivors.
Worked Example: Separating an Observed Mean From a Hypothetical Target
In a hypothetical cohort of 50 participants, 30 are event-free at a follow-up visit and have an observed mean biomarker of 8 units. The 20 others experienced a terminal event before that visit. The survivor mean is directly calculated as 8 units. What can be said about the full-cohort mean at that time?
There is no observed biomarker for the 20 participants after the terminal event, so their post-event values cannot be added to the calculation. The survivor mean is not the full-cohort mean: it is conditional on being event-free. If the question instead concerns the mean under a scenario preventing the event, denote the hypothetical values for the 20 participants by \(Y_i^0(t)\). Suppose, solely for illustration, that a model or assumption supplied a mean of 11 units for those 20 hypothetical values. The implied hypothetical cohort mean would be:
The 9.2-unit result is conditional on the assumed or estimated mean of 11 for the unobserved hypothetical values; it is not calculated from the survivor measurements alone. Without a stated assumption or model for those values, the data described here identify the survivor mean of 8 units, but do not determine that hypothetical cohort mean.
Diagnostics, Alternatives, and Limitations
A survivor average is easiest to interpret when it is reported with information about who contributes to it. Useful descriptive checks include the number and proportion remaining event-free at each time, the distribution of baseline characteristics among those still event-free, and the history of biomarker values before participants experience events. If these summaries change markedly, a trend in the survivor mean may reflect changing composition as well as individual progression. Such checks can reveal when selection is plausible; they cannot reveal the unobserved post-event biomarker values.
The main alternatives answer different questions:
| Approach | Question answered | Key limitation |
|---|---|---|
| Report the survivor mean | What is the mean among participants who are event-free at this time? | It does not by itself describe the baseline cohort’s trajectory. |
| Average a biomarker defined for everyone | What is the mean in the original cohort at this time? | Requires the outcome to be meaningful for participants who experienced an event, as well as suitable data. |
| Specify a hypothetical no-event trajectory | What would the baseline-cohort mean be under a stated scenario preventing the terminal event? | Post-event hypothetical values are unobserved, so results depend on assumptions and a well-defined scenario. |
The survivor mean is often the most transparent descriptive summary when the scientific question is explicitly about people who remain event-free. It should not be called a biased estimate without first naming a different target. Conversely, a hypothetical full-cohort trajectory should not be presented as though it were observed simply because a model returns a number. The target, assumptions, and sensitivity of the result to those assumptions matter.
These distinctions concern event-free status. Participants can also miss visits or leave follow-up for other reasons, so the people with observed measurements may be a further selected subset of those who remain event-free. Describing the survivor mean does not, by itself, resolve that additional selection. A useful report states which participants contribute to each average and avoids treating all absent measurements as if they had the same meaning.
Common Mistake
Wrong reasoning: “The mean among participants still event-free at each visit is the cohort’s mean trajectory, because each visit includes the people who can be measured then.”
Correction: Each visit may include a different group. The mean among people still event-free is conditional on event-free status at that time. A cohort-wide mean is a different target when the outcome is defined for everyone; after a terminal event it may instead require a clearly defined hypothetical outcome. Always state which average is being reported before interpreting its trend.
Check Your Understanding
Answer each question, then check the concise solutions below.
- At a given time, 75% of a cohort is event-free. The mean biomarker is 6 units among event-free participants and 10 units among participants who experienced an event. If the biomarker is defined for both groups, calculate the cohort mean.
- Why can the observed survivor mean fall even if every participant’s biomarker value increases?
- After a terminal event makes the biomarker undefined, does the survivor mean equal the mean that the full baseline cohort would have had if the event had been prevented? Explain.
- Name two summaries that help describe how the event-free group changes over follow-up, and state one thing these summaries cannot establish.
Solutions. 1. The cohort mean is \(0.75(6)+0.25(10)=4.5+2.5=7.0\) units. 2. Participants with higher values may be more likely to experience an event, changing the composition of the event-free group; the resulting survivor average can fall despite increases in individual values. 3. No. The survivor mean is conditional on being event-free, while the no-event mean concerns hypothetical outcomes for everyone under a specified scenario. Those unobserved outcomes require assumptions and are not supplied by the survivor mean. 4. Report the number or proportion event-free at each time and summarize baseline characteristics or prior biomarker values in that group. These descriptions can show changes in composition, but cannot establish the biomarker values participants would have had after a terminal event.