Changing Target Populations Across Sequential Landmarks

Why the Target Population Can Change

A prediction made at a landmark applies to people who are eligible to be predicted at that time, not automatically to the people who formed the original study population. At a later landmark, some individuals may have entered an absorbing event state, died, been lost to observation, or failed another eligibility rule. Those who remain can differ systematically from those eligible earlier. Consequently, a difference between two average landmark predictions may reflect a changed population as well as changed predictions within comparable people.

This distinction matters whenever sequential predictions are reported side by side. A lower predicted risk at a later landmark does not, by itself, imply that the underlying process has become safer. The later cohort may simply contain a different mix of states, histories, or covariates. The first step in interpreting any such summary is to state precisely who is included and what probability is being averaged.

Define the Landmark Cohort and Its Prediction

Let \(t_l\) denote landmark \(l\), and let \(Y(t)\) be an individual's state at time \(t\). Define \(E_l=1\) when the individual meets the prespecified eligibility criteria at \(t_l\). For example, eligibility might require being alive, not yet in an absorbing event state, and under observation. The definition must be applied consistently at every landmark; otherwise, differences can arise merely because the rules changed.

The conditioning information \(H_l\) may include the current state, covariates, and relevant observed history. It must not include information from after \(t_l\). The eligible population and the prediction horizon also need to be explicit: a one-year prediction from \(t_l\) concerns a different calendar endpoint from a one-year prediction from \(t_{l+1}\).

Eligibility is often selective rather than random. For instance, people who remain free of an absorbing event until a later landmark have necessarily had a different event history from those who experienced that event earlier. Conditioning on later eligibility can therefore change the distribution of prognostic factors, even if those factors were balanced at baseline. This is a feature of the target population, not automatically a defect in the prediction calculation.

How Composition Changes an Average

Suppose \(X\) is a categorical characteristic recorded in comparable form at each landmark, and \(p_l(x)\) is the predicted probability of a specified future state for people in stratum \(X=x\) at landmark \(l\). The crude average for that landmark is

where \(F_l(x)=P(X=x\mid E_l=1)\) is the stratum proportion among those eligible at \(t_l\). Thus, \(\psi_l\) can change because the stratum-specific predictions \(p_l(x)\) change, because the eligible cohort's composition \(F_l(x)\) changes, or both.

One exact decomposition makes the two components visible. For two landmarks 1 and 2, add and subtract \(\sum_x p_1(x)F_2(x)\):

The first term is the change associated with applying the earlier stratum-specific predictions to the later cohort's composition. The second is the change in stratum-specific predictions, averaged over the later composition. This decomposition is an accounting identity, not a causal attribution. Its interpretation requires comparable strata and a common, clearly defined prediction target at both landmarks.

Standardizing to a Fixed Reference Distribution

A common way to compare conditional predictions across landmarks is to average them over the same reference distribution \(G(x)\), rather than over each landmark's own cohort distribution. The standardized summary is

where \(G(x)\) is fixed across the comparisons. This answers a different question from the crude cohort average: it compares the landmark-specific stratum predictions after holding the stratum mix constant. It does not turn a conditional prediction into an unconditional prediction for everyone in the original study population.

Standardization requires that the strata be defined comparably, that predictions be available for the reference strata at each landmark, and that the reference distribution be substantively appropriate. If a stratum has no eligible individuals at one landmark, its prediction there may not be estimable from the data without additional modeling assumptions. Extrapolating into such a stratum should be identified as extrapolation, not presented as a data-supported comparison. When current state or history matters, these features must also be handled in the definition of the reference distribution or in the conditioning set; standardizing only a baseline covariate may leave important composition differences unresolved.

Landmark-Conditional Predictions Versus a Fixed Population

A fixed baseline population is a different target. If the question concerns the state occupied at a calendar endpoint among everyone in the baseline population, the summary should retain everyone in its denominator and account for their state at that endpoint, including absorbing states. People who experienced an absorbing event before a later landmark are not simply omitted from this fixed-population question.

By contrast, a landmark-cohort prediction is conditional on eligibility at its landmark. It describes future outcomes among those who meet that landmark's eligibility rules. The two summaries can both be useful, but they answer different questions and their probabilities should not be compared as if they had the same denominator.

Assumptions, Diagnostics, and Limitations

Interpretation depends first on clear eligibility and outcome definitions. Eligibility should be set using information available at the landmark, and predictions should use no future information. The target state must be defined consistently, including how competing absorbing states are represented. For cohort-average predictions, the averaging distribution must be stated: the observed eligible cohort, a fixed reference distribution, or another explicit target.

When estimates rely on observed outcomes, censoring and loss to follow-up also matter. If individuals with incomplete follow-up differ in ways related to their future state, simply averaging observed outcomes may not estimate the intended cohort probability. The analysis must use an appropriate method for its observation process and state its assumptions. A model-based prediction does not remove the need to examine whether relevant predictors and histories are represented in the data.

Useful diagnostics include reporting the number eligible at each landmark, the distribution of current states and key covariates, and the number of observations contributing to each stratum-specific estimate. Compare these distributions before interpreting changes in crude averages. Look for strata present at one landmark but absent at another, sparse strata, or marked shifts in state and history composition. These are warnings that standardization may depend heavily on extrapolation or that a simple composition decomposition omits relevant differences.

The main alternatives answer distinct questions. A crude landmark average describes each landmark's own eligible cohort. A standardized average compares predictions using a common reference mix, subject to overlap and modeling support. A fixed-baseline state probability describes occupation in a defined original population at a calendar time and includes outcomes before later landmarks. Reporting one does not substitute for another when the scientific question concerns a different target.

A common mistake is to treat the average at each landmark as if it described the same people, then explain any difference as an improvement or deterioration in prognosis. That reasoning ignores selection into later eligibility. The correction is to name the target population for each average, inspect how its composition changed, and use standardization or a fixed-population summary only when it answers the intended question and its assumptions are defensible.