Landmark Analysis in Survival Studies

Introduction

Survival studies often begin with a clearly defined time zero, such as randomization, treatment initiation, diagnosis, or surgery. The difficulty arises when an important exposure is not known at that initial time.

Examples include whether a patient eventually achieves a tumor response, whether a patient receives a transplant, whether a biomarker becomes positive, whether a treatment modification occurs, or whether a patient remains progression-free long enough to satisfy a clinically defined condition.

A tempting approach is to classify patients according to what happens during follow-up and then compare survival between the resulting groups. This can create serious bias because membership in one group may require the patient to remain alive and event-free long enough to enter that group.

The Problem: Exposure Is Determined After Baseline

Suppose investigators want to know whether achieving a clinical response is associated with longer overall survival.

Patients enter the study at time zero. Some eventually respond, while others do not. A response may occur weeks or months after treatment begins.

If investigators simply label a patient as a "responder" because that patient eventually responds, the responder group has a hidden requirement: the patient must survive long enough to respond.

This creates a period during which the patient cannot yet be classified as a responder but must remain alive and event-free in order eventually to enter the responder group.

A Simple Example of Immortal Time Bias

Consider a study in which patients are classified according to whether they eventually receive a treatment intensification.

Patient D, for example, had to survive at least 60 days to be classified in the intensification group.

If all 500 days of Patient D's survival are subsequently attributed to the "intensification" group, the first 60 days are being credited to an exposure that had not yet occurred.

The same issue occurs for every patient whose exposure is determined after baseline.

What Is Landmark Analysis?

Landmark analysis solves this problem by selecting a fixed time point \(L\), called the landmark time.

Only patients who are alive and otherwise eligible at \(L\) enter the landmark analysis.

Exposure status is then determined using information available at or before \(L\). Survival is subsequently analyzed from \(L\) forward.

The Landmark Time

The landmark time is central to the validity and interpretation of the analysis. Let:

For example, investigators might select 30 days, 90 days, 6 months, or 1 year after treatment initiation.

The landmark should ideally have a strong clinical rationale rather than being selected after inspecting the survival results.

Formal Definition of the Landmark Population

Let \(T\) denote the original event time and \(C\) the censoring time. At landmark \(L\), a patient is eligible if the patient has survived beyond the landmark and remains observable according to the analysis definition.

The landmark population can therefore be represented conceptually as:

The exact eligibility definition depends on the study and the endpoint. The essential principle is that the analysis begins with patients who have reached the landmark.

Resetting the Survival Clock

A critical feature of landmark analysis is that follow-up is measured from the landmark rather than from the original study entry.

For patient \(i\), define the landmark survival time as:

when the patient experiences the event after the landmark.

If the patient is censored after the landmark, the corresponding landmark censoring time is:

The analysis therefore asks a different question from the original survival analysis: among patients who have reached the landmark, how does subsequent survival differ according to their status at the landmark?

Landmark Analysis vs. Baseline Analysis

Kaplan–Meier Landmark Analysis

Suppose patients are classified at landmark \(L\) into two groups:

• Landmark responders
• Landmark nonresponders

The Kaplan–Meier estimator can then be calculated using survival times measured from \(L\).

For group \(g\), the landmark Kaplan–Meier estimator is:

where \(n_{gj}\) is the number at risk immediately before landmark-follow-up time \(t_j\), and \(d_{gj}\) is the number of events at that time.

The resulting curves compare post-landmark survival, not survival from original treatment initiation.

Landmark Cox Regression

Landmark analysis can also be incorporated into a Cox proportional hazards model. Suppose \(Z_i(L)\) represents the exposure status measured at the landmark. The landmark Cox model is:

Here:

• \(h_i^{(L)}(t)\) is the hazard after the landmark;
• \(h_0^{(L)}(t)\) is the landmark baseline hazard;
• \(Z_i(L)\) is the landmark exposure;
• \(\mathbf{X}_i(L)\) contains baseline or landmark covariates;
• \(\beta\) is the log hazard ratio associated with landmark exposure.

The corresponding hazard ratio is:

A Complete Worked Example

Consider a hypothetical oncology study evaluating whether achieving disease control by 12 weeks is associated with subsequent overall survival.

Patients enter the study at treatment initiation. Disease-control status is evaluated during the first 12 weeks. The investigators specify:

Suppose 120 patients enter the study. By Week 12:

The landmark analysis therefore contains:

patients. The 13 patients who did not reach the landmark in an analyzable state are not included in the post-landmark comparison.

Step 1: Define the Landmark

The landmark is:

The landmark should be specified before examining the final survival comparison whenever possible.

Step 2: Exclude Patients Who Do Not Reach the Landmark

Patients who die before Week 12 cannot be assigned to a post-Week-12 disease control group. They therefore do not contribute to the landmark survival comparison.

This is an important limitation of the method: the analysis conditions on surviving to the landmark.

Step 3: Assign Landmark Exposure

Each patient who reaches the landmark is classified according to disease-control status known by Week 12.

The classification does not use information that occurs after the landmark.

Step 4: Reset Follow-Up Time

Suppose a patient dies at Week 40. The post-landmark survival time is:

If another patient is censored at Week 50, that patient's landmark follow-up is:

This time reset is essential. The Kaplan–Meier and Cox analyses should use follow-up measured from the landmark.

Step 5: Estimate Landmark Survival

Suppose the resulting 24-month post-landmark survival estimates are:

A Kaplan–Meier plot would display survival beginning at the Week-12 landmark.

The curves should therefore be interpreted as conditional survival curves: survival among patients who reached Week 12.

Step 6: Fit a Landmark Cox Model

Suppose the Cox model produces:

The corresponding hazard ratio is:

Thus, in this hypothetical example, patients with disease control by Week 12 have an estimated post-landmark hazard of death approximately 51% lower than patients without disease control.

Adjusting for Prognostic Covariates

A landmark analysis can include covariates that are known by the landmark. For example:

• age;
• disease stage;
• baseline performance status;
• baseline biomarker status;
• prior treatment;
• tumor burden;
• other clinically important prognostic factors.

A multivariable landmark Cox model could be written as:

This allows the association between landmark exposure and subsequent survival to be estimated after adjustment for selected covariates.

Choosing the Landmark Time

The landmark time involves a trade-off.

A clinically motivated landmark is generally preferable to selecting the landmark solely because it produces the most favorable statistical result.

Landmark Analysis and Immortal Time Bias

The principal attraction of landmark analysis is that it prevents future exposure classification from being used to assign earlier survival time.

Consider again a responder analysis. Without landmarking, a patient who responds at Week 12 might be classified as a responder from Day 0. With a Week-12 landmark, that patient becomes a responder only for the post-Week-12 analysis.

versus:

The latter avoids attributing pre-landmark survival to an exposure that had not yet been established.

Landmark Analysis Is Not the Same as a Time-Dependent Cox Model

Landmark analysis is one solution to a time-dependent exposure problem, but it is not the only one. A time-dependent Cox model can allow exposure status to change during follow-up.

Landmark Analysis in R

A basic landmark analysis can be implemented by first restricting the dataset to patients who reach the landmark, creating post-landmark survival time, and then fitting Kaplan–Meier and Cox models.

library(survival)

landmark_week <- 12

landmark_data <- subset(
  data,
  landmark_eligible == 1
)

landmark_data$time_from_landmark <-
  pmax(time_weeks - landmark_week, 0)

fit_km <- survfit(
  Surv(time_from_landmark, status) ~ disease_control,
  data = landmark_data
)

fit_cox <- coxph(
  Surv(time_from_landmark, status) ~
    disease_control +
    age +
    stage +
    performance_status,
  data = landmark_data
)

summary(fit_cox)

The exact implementation depends on how the source data define eligibility, events, censoring, and landmark exposure.

Landmark Analysis in SAS

The same conceptual workflow can be implemented in SAS.

data landmark;
    set analysis_data;

    if landmark_eligible = 1;

    time_from_landmark =
        max(time_weeks - 12, 0);
run;

proc lifetest data=landmark plots=survival;
    time time_from_landmark * status(0);
    strata disease_control;
run;

proc phreg data=landmark;
    class disease_control(ref='0')
          stage
          performance_status;

    model time_from_landmark * status(0) =
          disease_control
          age
          stage
          performance_status;

    hazardratio disease_control;
run;

The key programming requirement is not the software syntax. It is ensuring that the analysis dataset actually represents the landmark population and that survival time begins at the landmark.

Common Mistake: Keeping the Original Survival Time

A frequent programming error is to classify patients at the landmark but retain the original time from study entry. For example:

/* Incorrect conceptual approach */

proc phreg data=landmark;
    model original_time * status(0) =
          landmark_exposure;
run;

This mixes a post-landmark exposure definition with a baseline time origin.

For a standard landmark analysis, the time scale should be constructed so that the landmark becomes time zero for the analysis.

Common Mistake: Defining Exposure Using Future Information

Suppose the landmark is Week 12. It would be inappropriate to define "landmark responders" using a response that occurs at Week 20.

The exposure definition must respect the information available by the landmark.

Using future information defeats the purpose of the landmark design and can reintroduce serious bias.

Common Mistake: Choosing the Landmark After Looking at Survival

Investigators may be tempted to evaluate many possible landmarks and select the one that produces the strongest treatment effect.

For example, they might examine 4, 8, 12, 16, and 24 weeks and report only the landmark with the smallest p-value.

This creates an implicit multiplicity and selection problem.

Common Mistake: Ignoring Selection at the Landmark

Landmark analysis does not make selection bias disappear. Patients who die before the landmark are excluded. Consequently, the landmark population can differ systematically from the original study population.

The resulting hazard ratio should therefore not automatically be interpreted as the effect that would have been observed if every patient had survived to the landmark.

The analysis is conditional on reaching the landmark.

Advantages of Landmark Analysis

• It provides a straightforward framework for post-baseline exposures.
• It avoids attributing pre-landmark survival to an exposure determined later.
• It is relatively easy to explain clinically.
• Kaplan–Meier and Cox methods can be used with familiar software.
• It is useful when the scientific question concerns prognosis after a defined clinical milestone.
• The landmark population and exposure definition can be made explicit and reproducible.

Limitations of Landmark Analysis

• Patients who experience the event before the landmark are excluded.
• A late landmark can substantially reduce the sample size.
• The result applies conditionally to patients who reach the landmark.
• The choice of landmark can affect the estimated association.
• Exposure information may be simplified into a single landmark status.
• Changes occurring after the landmark may not be adequately represented.
• Residual confounding remains possible in observational studies.

Landmark Analysis vs. As-Treated Analysis

It is important not to confuse landmark analysis with simply dividing patients according to treatment received during follow-up.

An as-treated analysis can incorrectly classify exposure and may introduce immortal time bias if the timing of treatment is ignored.

Landmark analysis instead establishes a common analysis time origin and defines exposure using information available at that time.

Landmark Analysis vs. Conditional Survival

Landmark analysis is closely related to the concept of conditional survival. If \(S(t)\) is the probability of surviving beyond time \(t\), then the probability of surviving an additional \(u\) units of time given survival to \(L\) is:

Landmark analysis similarly conditions the analysis on patients who have reached a specified time.

The distinction is that landmark analysis often additionally compares groups defined by exposure or clinical status observed by the landmark.

Recommended Reporting Structure

A publication or clinical study report using landmark analysis should make the following details explicit.

Sensitivity Analyses

Sensitivity analyses are especially useful because landmark results can depend on the selected landmark. Potential analyses include:

• an alternative clinically reasonable landmark;
• different definitions of exposure at the landmark;
• adjustment for important prognostic factors;
• alternative handling of patients with missing landmark assessments;
• comparison with a time-dependent Cox analysis;
• analyses restricted to prespecified clinical subgroups.

Sensitivity analyses should be motivated scientifically rather than chosen solely because they produce a preferred result.

A Practical Landmark Analysis Workflow

Key Interpretation

Suppose a landmark Cox model produces: