Avoiding Immortal-Time Bias in Landmark Cohorts

Why the Time Origin Matters

A landmark analysis asks about future event risk among people who meet specified eligibility criteria at a chosen time. The landmark is the time origin for that prediction: information measured later cannot be used as though it had been available earlier, and survival before the landmark is not post-landmark follow-up. If those distinctions are ignored, a predictor or exposure can appear to have an event-free period that was guaranteed by how the analysis group was defined.

That guaranteed period is the source of immortal-time bias. For example, if people are classified as “treated” because they begin treatment at month 6, they must have remained alive and eligible until month 6 to enter that group. Counting months 0 to 6 as treated follow-up credits the treatment group with time during which its members could not yet have been treated. Landmarking avoids this particular error when the cohort, predictor availability, and follow-up start are all aligned at the landmark.

The target is conditional: for example, the probability of an event by \(L+h\) among people who are event-free and eligible at \(L\), given information available by \(L\). This differs from the risk from the original time origin in the full baseline cohort. Conditioning on landmark survival changes the population being described; as discussed in the earlier material on selection induced by landmark survival, that selection can affect who remains in the cohort. Immortal-time bias is a separate problem: it arises when pre-landmark time is incorrectly attributed to post-landmark predictors or follow-up.

Eligibility, Predictor Timing, and Follow-Up

A defensible construction starts by fixing the landmark and writing down the eligibility rule before examining post-landmark outcomes. For a single terminal event, a common rule is that the subject is known to be alive and event-free at \(L\). A subject with an event before \(L\) is not eligible for this conditional prediction. A subject censored before \(L\) is not known to have survived event-free to \(L\), so should not be treated as an eligible survivor. If an event occurs exactly at \(L\), decide in advance whether eligibility is assessed just before or just after that time and apply the rule consistently.

For an eligible subject, define the residual follow-up time \(Y_i=\min(T_i,C_i)-L\), and the post-landmark event indicator \(\delta_i=I(T_i\leq C_i)\). These quantities describe only follow-up after \(L\). If the event was observed before \(L\), the subject is excluded rather than assigned a negative or positive post-landmark time. If censoring occurred before \(L\), the subject’s status at \(L\) is unknown and cannot be assumed event-free.

A landmark predictor can be a value, change, slope, or other summary constructed from data available by \(L\), as covered in the earlier material on landmark predictors. Its measurement time must not extend beyond the landmark. A value recorded at a visit after \(L\), or treatment received after \(L\), is future information relative to the prediction time. Assigning people to groups using such information and then starting their follow-up at \(L\) or earlier creates a look-ahead problem; it does not create a valid landmark predictor.

Conditional Risk and the Time Reset

Let \(S(t)=P(T>t)\) denote event-free survival from the original origin. For a fixed landmark \(L\), the probability of remaining event-free for a further \(h\) time units, conditional on being event-free at \(L\), is

This follows directly from conditional probability. Since the event-free set \(\{T>L+h\}\) is contained in \(\{T>L\}\), for \(S(L)>0\),

Thus, conditional event risk over the next \(h\) time units is \(1-S(L+h)/S(L)\). In an analysis indexed by residual time \(u\), the landmark cohort’s survival curve starts at \(u=0\), which represents calendar time \(L\). It must not be interpreted as though those selected subjects had been at risk from the original origin with their landmark predictors already known.

What Backdating an Exposure Does

A common form of immortal-time bias occurs when exposure status is defined by a future event, such as starting treatment after enrollment, but is assigned retrospectively from baseline. The treated group then contains only people who survived and remained eligible long enough to start treatment. Their pre-treatment time is guaranteed to be event-free with respect to the event that would have prevented treatment initiation. Crediting this interval as treated follow-up can dilute the treated group’s event rate.

Estimating Conditional Risk From Survival

The survival ratio also shows why a baseline survival probability is not the same quantity as a landmark risk. If survival estimates are available from a cohort with consistent time origin and appropriate handling of censoring, their ratio gives the estimated conditional survival from \(L\) to \(L+h\). This calculation does not remove selection into the landmark cohort; it expresses the risk conditional on surviving to the landmark.

Diagnostics, Alternatives, and Limitations

Before interpreting a landmark analysis, check the data construction as well as the fitted model. The time origin and risk sets must agree: at residual time zero, all included subjects are eligible at \(L\), and all subsequent event times are measured from \(L\). Check that no post-landmark measurement defines a predictor, no pre-landmark event is coded as a post-landmark outcome, and no time before \(L\) appears in the post-landmark person-time. Verify that censoring before \(L\) has not been mistaken for documented survival to \(L\). These checks catch common implementation errors even when the analysis software runs without warnings.

A landmark model and a time-dependent Cox model address different constructions. A landmark model describes risk conditional on being eligible at a chosen time, using predictor information available then. A time-dependent Cox model can represent an exposure or covariate changing during follow-up, provided its timing is encoded correctly. A joint model uses longitudinal and event information through a different modeling framework. None of these approaches licenses assigning future information to earlier risk time: the chosen method must match the target question and preserve the correct temporal ordering.

Landmarking also has limits. Its conclusions apply to the population eligible at the chosen landmark, not automatically to the original cohort. A single landmark does not describe people who experience the event before it, and a different landmark can define a different eligible population and prediction target. If a predictor is unavailable for some eligible people, excluding them or handling missingness requires a clearly stated approach; silently selecting only those with later measurements can create additional selection and timing problems. These limitations are not repaired by resetting the clock.