Why the Landmark Cohort Can Look Different
A landmark prediction concerns patients who are still event-free at a specified time. That condition is part of the question being answered, but it also changes who is in the analysis. If a biomarker predicts earlier events, patients with high values may be less common among those who reach the landmark. If both the biomarker and other patient characteristics affect survival to the landmark, the survivors can also differ in those characteristics in ways that alter associations measured after the landmark.
This is not automatically a flaw in landmark analysis. A prediction for patients who are event-free at time \(t_L\) should describe that conditional population. The interpretive problem arises when an association estimated in that selected group is treated as if it described the original cohort, or as if it were a causal effect without additional assumptions.
For a longitudinal biomarker, the value used at \(t_L\) may be a last observed value or a summary of measurements before the landmark. Such a value may only be defined or available for patients who survive to \(t_L\). The conditioning event and the rule for obtaining the predictor therefore both help determine the analyzed population. Here the central issue is selection by event-free survival, not the separate problem of how missing or irregularly timed measurements are handled.
How Conditioning Changes a Marker Distribution
First consider a baseline marker \(X\) measured for everyone. Let \(S(t_L\mid X=x)=P(T>t_L\mid X=x)\) be the probability of surviving event-free to the landmark for patients with marker value \(x\). If \(X\) has density \(f_X(x)\), the distribution among landmark survivors is
This follows from Bayes’ rule: the joint density of having marker value \(x\) and surviving is \(f_X(x)S(t_L\mid X=x)\); dividing by the overall survival probability \(P(T>t_L)\) gives the conditional density. Thus, marker values associated with higher landmark survival receive greater representation among survivors. The analogous formula for a discrete marker replaces the density and integral with probabilities and a sum.
The formula describes a change in composition, not necessarily an error. It also shows why one cannot infer the survivor distribution from the original marker distribution alone: the survival probabilities matter. For a marker measured only at or near the landmark, the same selection logic applies to the patients for whom a value is defined and observed, but the distribution also reflects the measurement and availability process.
Worked Example: A High-Risk Marker Becomes Less Common
Suppose, hypothetically, that 40% of an original cohort has a high baseline biomarker and 60% has a low value. By \(t_L\), 50% of the high-marker group and 80% of the low-marker group remain event-free. Find the high-marker proportion among survivors.
The proportion of the original cohort that is both high-marker and event-free is \(0.40(0.50)=0.20\). The proportion that is low-marker and event-free is \(0.60(0.80)=0.48\). Therefore the total event-free proportion is \(0.20+0.48=0.68\), and
So about 29.4% of landmark survivors have a high marker, compared with 40.0% of the original cohort. This difference is consistent with the stipulated lower landmark survival in the high-marker group. It does not, by itself, establish why the groups differ or what the marker’s causal effect might be.
Worked Example: A Biomarker Mean Shifts Among Survivors
Consider a hypothetical baseline biomarker \(X\), measured in units, with values 0, 2, and 4. Their proportions in the original cohort are 0.25, 0.50, and 0.25, respectively. Their event-free probabilities at \(t_L\) are 0.90, 0.60, and 0.30. Calculate the original and survivor means.
The original mean is
The unnormalized survivor weights for the three values are \(0.25(0.90)=0.225\), \(0.50(0.60)=0.300\), and \(0.25(0.30)=0.075\). Their sum, the overall probability of reaching the landmark, is \(0.600\). Dividing each weight by this total gives survivor proportions \(0.375\), \(0.500\), and \(0.125\). Hence
The mean is 0.500 units lower among survivors. The calculation is valid because the example specifies the marker distribution and survival probability for every marker value, and the probabilities sum to a positive landmark survival probability. It illustrates selection changing the marker distribution, not a change in any individual’s biomarker.
Selection Can Also Change Associations
A shift in the distribution of one marker is only part of the issue. Suppose a marker \(B\) and another characteristic \(U\) are independent in the original cohort, but both affect survival to \(t_L\). Among survivors, knowing \(B\) can provide information about \(U\), because survival depends on both. In causal diagrams this is often described as conditioning on a collider: survival is a common consequence of the two characteristics. The induced relationship is a feature of the selected group, even though the characteristics were independent before selection.
This can alter an association between \(B\) and an event after the landmark. For example, if \(U\) is protective, survivors with a high-risk value of \(B\) may be unusually enriched for protective \(U\). A crude comparison of future outcomes by \(B\) can then be weaker than the comparison within levels of \(U\), or even point in the opposite direction. Such a pattern does not show that \(B\) has become protective; it can reflect the changed mixture of patients.
Worked Example: A Survivor Association Can Be Misleading
In a hypothetical cohort, binary marker \(B\) and protective characteristic \(U\) are independent at baseline, each occurring in half the patients. The probabilities of surviving to \(t_L\) are stipulated as follows. Rows give \(B\), and columns give \(U\).
| \(U=0\) | \(U=1\) | |
|---|---|---|
| \(B=0\) | 0.80 | 0.90 |
| \(B=1\) | 0.30 | 0.70 |
Because the four baseline combinations are equally common, within the \(B=0\) group the survivor weights are 0.80 and 0.90, giving
Within the \(B=1\) group, the corresponding calculation is
Thus \(B\) and \(U\), initially independent, are associated among survivors: the \(B=1\) survivors are more likely to have the protective characteristic. Now suppose, also hypothetically, that the event risk over a fixed post-landmark horizon is 0.40 for \(U=0\) and 0.10 for \(U=1\), with no difference by \(B\) within levels of \(U\). The marginal horizon risk for \(B=0\) survivors is
whereas for \(B=1\) survivors it is
The crude survivor comparison shows lower risk for \(B=1\), although the stipulated post-landmark risk is identical by \(B\) within each \(U\) level. The arithmetic is internally consistent with the specified survivor compositions and conditional risks. It demonstrates how selection and mixing can produce an association; it is not evidence that \(B\) causes lower risk.
What a Landmark Association Does and Does Not Say
A landmark model targets risk or association after \(t_L\), conditional on being event-free at \(t_L\) and on the predictor information used in that model. This is a meaningful target for decisions made at the landmark: the relevant patients are those who have reached it. A coefficient can describe how post-landmark outcomes differ across predictor values in that survivor population, subject to the model and censoring assumptions.
It does not automatically describe the association in the original cohort. Nor does it establish a causal effect of changing a biomarker. A causal interpretation would require a well-defined intervention and assumptions addressing confounding, selection, and the measurement process, among other matters. In particular, adjusting for measured covariates may help explain or standardize composition, but it does not guarantee removal of selection due to unmeasured characteristics.
The distinction is especially important when comparing estimates from different landmark times. Later landmarks condition on surviving longer and can represent increasingly different populations. A change in an estimated coefficient across landmarks may reflect a genuine time-varying association, a change in the survivor composition, or both. It should not be interpreted as a time-varying biological effect without further justification.
Assumptions, Diagnostics, and Alternatives
The selection calculations require a clearly defined time origin, event, and landmark, plus a consistent rule for determining who is event-free at \(t_L\). The event-free probabilities used in a calculation must correspond to the same population and event definition as the target landmark cohort. In an analysis, right censoring before \(t_L\) is not the same as confirmed event-free survival: whether patients with incomplete follow-up can be classified as eligible requires appropriate follow-up or censoring methods. The simple examples above assume survival status is known for all patients through \(t_L\).
Useful checks include comparing measured baseline characteristics and biomarker distributions in the original cohort and landmark survivors, and plotting or tabulating landmark survival across predictor groups. For longitudinal predictors, also report how many eligible survivors lack a usable value and whether that availability varies across patient groups. These checks reveal observed composition changes and data gaps; they cannot establish that selection on unmeasured factors is absent. Marked changes in composition, sparse predictor groups among survivors, or very few patients at later landmarks warn that estimates may be unstable or difficult to generalize.
The main alternatives answer different questions rather than simply removing selection:
| Approach | What it can address | Key limitation for this issue |
|---|---|---|
| Landmark analysis | Prediction or association among patients event-free at a specified time. | Results are conditional on reaching that time; the composition may differ from the original cohort. |
| Standardization or weighting | Can describe results under a specified measured-covariate distribution when the target is defined and required probabilities are estimable. | Cannot recover outcomes after the landmark for patients who had an event before it; weights also rely on adequate overlap and correctly modeled selection factors. |
| Joint modeling | Can model longitudinal measurements and event times together under its specified shared-model assumptions. | It does not make survivor conditioning disappear; the target and assumptions still need to be stated. |
Weighting is not a universal correction. For example, inverse survival weighting may reweight observed survivors toward a baseline covariate distribution under suitable assumptions and known or estimable survival probabilities. But it cannot observe a post-landmark biomarker value or post-landmark outcome for someone who experienced the event before \(t_L\). A joint model may use a broader model for the longitudinal and event processes, but its estimates depend on that model and are not interchangeable with the conditional landmark estimand.
Common Pitfalls
- Calling all survivor differences bias. A different marker distribution among survivors is expected when survival varies with the marker. It is a problem only relative to an incorrectly stated target or an unsupported interpretation.
- Assuming baseline independence persists. Characteristics independent in the original cohort can be associated after conditioning on survival if both affect survival.
- Reading a survivor coefficient as causal. A conditional association does not by itself identify the effect of intervening on the predictor.
- Treating a late landmark as representative of the starting cohort. Later landmarks may select a smaller and differently composed subset, limiting transport of their estimates.
Check Your Understanding
Use the survivor-conditioning calculations and interpret each answer in context.
- A marker is present in 30% of an original cohort. Landmark survival is 0.60 for marker-positive and 0.90 for marker-negative patients. What proportion of survivors are marker-positive?
- Can survival selection create an association among two characteristics that were independent at baseline? Briefly explain the condition under which this can occur.
- In the worked example with three biomarker values, why is the survivor mean not the original mean?
- A landmark coefficient is estimated among patients event-free at \(t_L\). What population does it describe, and what additional claim does it not establish by itself?
- Would weighting survivors allow direct observation of post-landmark outcomes for patients who experienced the event before \(t_L\)? Explain.
Solutions. (1) The positive survivor weight is \(0.30(0.60)=0.18\); the negative survivor weight is \(0.70(0.90)=0.63\). The positive proportion is \(0.18/(0.18+0.63)=0.2222\), or 22.2%. (2) Yes. If both characteristics affect survival, conditioning on survival can make them associated even when independent before selection. (3) The three biomarker values have different landmark survival probabilities, so their survivor proportions are reweighted; the survivor distribution is not the original distribution. (4) It describes the modeled association among patients event-free at \(t_L\) with the specified predictor information. It does not, without further assumptions, establish a causal effect or describe the original cohort. (5) No. Weighting can change the representation of observed survivors under assumptions, but it cannot supply unobserved post-landmark outcomes for patients who had the event before \(t_L\).