Landmarking When State Occupation Depends on Earlier History

Why the Current State May Not Be Enough

At a landmark time, people can occupy the same state yet have arrived there by different paths. Those paths may carry information about what happens next. For example, two people may both be in an active-disease state, while one is experiencing a first episode and the other has returned to that state after a prior remission. If their subsequent state-occupation probabilities differ, a prediction based only on the current state averages over a distinction that matters.

The goal here is to define and use predictions that condition on both the current state and relevant information from before the landmark. The information must be available at prediction time: a history summary can describe earlier transitions, but it cannot use future events. This is a conditional prediction, not a claim that the earlier history caused the difference in outcomes.

Definition: Let $S(t)$ denote the state occupied at time $t$, let $L$ be a landmark time, and let $u\geq 0$ be a prediction horizon after $L$. Let $G_L$ be a prespecified summary of event history observed by $L$. For a person eligible at $L$, the history-conditioned state-occupation probability for state $k$ is $$ \pi_k(u\mid r,g;L) = P\{S(L+u)=k\mid S(L)=r,\ G_L=g,\ \text{eligible at }L\}. $$ Here eligibility includes being under observation and meeting the study's landmark risk-set definition. The state $r$, history summary $g$, landmark $L$, and horizon $u$ are part of the prediction target.

The history summary might indicate whether a person has had a prior relapse, the number of previous episodes, or whether a particular transition occurred before reaching the current state. The choice should be clinically or scientifically motivated and defined using information recorded by the landmark. It is not necessary to condition on every detail of the path; the practical question is whether the chosen summary captures history relevant to the prediction.

What the State-Only Prediction Averages

A prediction conditional only on the current state combines people with different histories who occupy that state at $L$. If $G_L$ has possible values $g$, the law of total probability gives

P\{S(L+u)=k\mid S(L)=r,\ \text{eligible at }L\} = \sum_g \pi_k(u\mid r,g;L) P\{G_L=g\mid S(L)=r,\ \text{eligible at }L\}.

Thus a state-only prediction is a mixture of the history-specific predictions, with weights determined by the history distribution among eligible people currently in state $r$. It can describe that mixture population correctly while being poorly calibrated for a particular history group. The mixture weights can also change across landmark times or target populations, changing the state-only prediction even if the history-specific probabilities remain similar.

Worked Example: Two Histories, One Current State

Suppose a hypothetical cohort contains 40 people who are eligible at a landmark and occupy the active-disease state. Twenty have a first episode at the landmark ($G_L=\text{first}$); the other 20 have returned to the active state after an earlier remission ($G_L=\text{return}$). All 40 have complete follow-up to 12 months. Among the first-episode group, 12 are in remission, 6 remain active, and 2 have died. Among the return group, 5 are in remission, 9 remain active, and 6 have died.

The history-specific 12-month probability of remission for the first-episode group is $$ \hat{\pi}_{\text{remission}}(12\mid \text{active},\text{first};L) = \frac{12}{20}=0.600. $$ The corresponding probabilities of remaining active and of death are $6/20=0.300$ and $2/20=0.100$.

For the return group, the estimated remission probability is $$ \hat{\pi}_{\text{remission}}(12\mid \text{active},\text{return};L) = \frac{5}{20}=0.250. $$ The probabilities of remaining active and of death are $9/20=0.450$ and $6/20=0.300$. Each set of three probabilities sums to one.

The observed cohort-wide remission proportion is $17/40=0.425$. That is the equal-weight mixture $(0.600+0.250)/2=0.425$, not the prediction for either history group. The difference between 0.600 and 0.250 shows why conditioning on the current state alone may conceal useful predictive information. These are hypothetical empirical proportions; their precision and generalizability would need separate assessment.

Assumptions and Estimation

The estimand is a conditional probability, but estimating it requires care. First, the state and history summary must be defined consistently, and the summary must be determined by information available at $L$. Second, there must be adequate support: the data need people with the relevant combination of current state and history. A prediction for a history group absent from the landmark risk set is not identified by those data without additional modeling assumptions.

With complete follow-up to a fixed horizon, a simple estimator within a history group is the observed proportion in state $k$ at $L+u$. With censoring before that horizon, the observed proportion among those still followed is generally not enough: if follow-up depends on future state occupation, it can be biased. An estimator must account for censoring under an appropriate assumption, such as censoring independent of the future state process conditional on the modeled landmark information and any other variables required by the chosen method. If that assumption is implausible, additional measured predictors or sensitivity analyses may be needed.

A transition-based estimator, such as an Aalen-Johansen estimate applied within history groups, also requires the assumptions appropriate to that estimator. In particular, a state-only Markov transition model within a group is not automatically valid just because the group was defined by earlier history. If relevant distinctions remain within a group, its transition probabilities may still depend on more of the path. A direct estimator of state occupation avoids requiring a Markov transition structure, but still needs suitable handling of censoring and enough data at the horizon.

Define the target.
Specify the landmark, eligible risk set, current state, history summary, target state, and prediction horizon.

Check support.
Count eligible people and observed outcomes in each current-state and history combination. Flag sparse or empty groups before fitting a model.

Choose an estimator and its assumptions.
Use complete-horizon proportions when follow-up is complete; with censoring, use an approach justified for the censoring pattern and the desired conditional estimand.

Evaluate the conditional predictions.
Compare predicted and observed state occupation within history groups where follow-up data allow, and assess whether the history summary leaves important patterns unexplained.

A Short Derivation: Why a Mixture Can Mislead for a Subgroup

Suppose two history groups have different probabilities $p_1$ and $p_2$ of occupying a target state at the horizon, and the first group makes up proportion $w$ of people currently in state $r$. The state-only probability is $p=w p_1+(1-w)p_2$. Unless $p_1=p_2$, or one group has weight zero, the mixture generally differs from each group-specific probability:

p-p_1=(1-w)(p_2-p_1),\qquad p-p_2=w(p_1-p_2).

For $0<w<1$ and $p_1\ne p_2$, both differences are nonzero. The state-only prediction therefore cannot equal both history-specific predictions. This is not a defect in averaging: the mixture answers a different question, namely the probability for the combined current-state population.

Worked Example: A History-Specific Risk Over Two Intervals

Consider a hypothetical two-state process: people are either event-free or have experienced the event, which is absorbing. Among people in the event-free state at the landmark, suppose the conditional event probabilities in the first and second six-month intervals are 0.20 and 0.30 for history group A. Assume these interval probabilities apply to those still event-free at the start of each interval. The 12-month event probability for group A is $$ 1-(1-0.20)(1-0.30)=1-(0.80)(0.70)=0.440. $$

For history group B, suppose the corresponding interval probabilities are 0.50 and 0.40. Its 12-month event probability is $$ 1-(1-0.50)(1-0.40)=1-(0.50)(0.60)=0.700. $$

The calculation multiplies the conditional probabilities of remaining event-free across intervals, then subtracts from one. The interval risks are stipulated here as hypothetical conditional risks; in an analysis they would need to be estimated from eligible people at risk in each interval, with censoring handled appropriately. The result illustrates that a shared landmark state does not force shared future occupation probabilities when earlier history is predictive.

Diagnostics, Comparisons, and Limitations

A useful diagnostic is to compare observed and predicted state occupation within each history group, rather than only across all people in the current state. Large within-group discrepancies may indicate model misspecification, an inadequate history summary, or too few observations for stable estimates. Also inspect group sizes, event counts, follow-up, and censoring patterns. Extremely different predictions from very small groups may be driven by sampling variability rather than reproducible history effects.

The main comparison is with prediction conditional on the current state alone. A state-only Markov prediction is simpler and may be appropriate when the current state adequately summarizes the information needed for future transitions. History-conditioned prediction relaxes that restriction for the specified history distinctions, at the cost of more parameters or smaller groups. The approaches answer different questions: the state-only prediction targets the mixture of histories, while the conditional prediction targets a defined history subgroup.

History conditioning is not a guarantee that all relevant dependence has been captured. A crude summary can combine distinct paths that have different futures; an excessively detailed history can produce sparse data and unstable predictions. A history-specific association is predictive, not necessarily causal: differences may reflect measured or unmeasured characteristics associated with both prior events and future outcomes. The prediction also applies to the landmark-eligible population and horizon defined in the estimand, not automatically to people in other states, at other times, or outside the study population.

Common pitfall: “Everyone is in the same state at the landmark, so one state-only prediction must be right for every person in that state.” The correction is that the state-only probability is a weighted average across histories. It is appropriate for the combined group, but it need not describe any particular history subgroup well.

Check Your Understanding

Answer each question using the stated landmark population and prediction horizon. Brief solutions follow.

At a landmark, 30 eligible people occupy state $r$. Twelve have history group A and 18 have history group B. At the horizon, 9 people in A and 6 in B occupy state $k$. Calculate each history-specific proportion and the overall state-only proportion.
In your own words, what population does a state-only prediction conditional on $S(L)=r$ describe when histories differ?
Why can the proportion in state $k$ among people still observed at the horizon be biased when some participants are censored earlier?
Name two diagnostics that could reveal problems with history-conditioned prediction.

Solutions. (1) Group A: $9/12=0.750$; group B: $6/18=0.333$. Overall: $(9+6)/30=0.500$, equivalently $(12/30)(0.750)+(18/30)(0.333)=0.500$, up to rounding. (2) It describes the eligible population currently in state $r$, mixing its history groups in their observed proportions. (3) Those still observed may differ in future state occupation from those censored; the observed subset need not represent the full landmark group. (4) Examples include checking group sizes and event counts, comparing observed and predicted occupation within groups, and examining censoring patterns across groups.

Landmarking When State Occupation Depends on Earlier History

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