The Landmark State-Occupation Estimand

What Is the Landmark State-Occupation Estimand?

At a landmark time, a prediction question concerns a selected group: individuals who are under observation then, together with whatever information about them is specified at that time. The target is not simply “the probability of an event.” It is the probability of occupying each possible state at a particular future horizon, within that landmark group. Stating the target precisely makes clear who the prediction concerns, what counts as an outcome, and when that outcome is assessed.

For example, a clinical question might ask: among patients under observation at month 6 who have a specified profile at that time, what is the probability of being in each disease state 12 months later? The answer is a distribution across states at month 18. It is not automatically the probability of having experienced a particular event at any time between months 6 and 18.

Definition: Let $X(t)$ denote the state occupied at time $t$, with possible states in a specified state space $\mathcal S$. Let $s$ be the landmark time, $\tau>0$ the prediction horizon measured from that landmark, and $R_s=1$ indicate that an individual is under observation and eligible for the target population at time $s$. Let $W_s$ denote the information specified at the landmark. For a specified value or stratum $w$, the landmark state-occupation probability for state $k\in\mathcal S$ is $$ \theta_k(s,\tau;w) = P\{X(s+\tau)=k \mid R_s=1,\ W_s=w\}. $$ The collection $\{\theta_k(s,\tau;w):k\in\mathcal S\}$ is the landmark state-occupation distribution.

The definitions of $R_s$ and $W_s$ must match the question. Eligibility may mean being enrolled and still observed at $s$; the specified information may include the current state or other information recorded by then. The information is fixed at the landmark, not selected using what happens afterward. More generally, if the target conditions on a specified set of landmark information rather than a single value $w$, the estimand is the conditional probability given that set.

The horizon $\tau$ is elapsed time from the landmark. The target time is therefore $s+\tau$, on the study’s time scale. Naming both times avoids ambiguity, especially when people reach the same landmark at different calendar or follow-up times. The state space must also be explicit. If death is one of the states, for example, its probability is part of the distribution rather than an outcome that disappears from the denominator.

Properties of the State-Occupation Distribution

Suppose the states are mutually exclusive and collectively exhaustive at the horizon: every eligible individual occupies exactly one state in $\mathcal S$ at time $s+\tau$. Then the state probabilities are nonnegative and sum to one. This follows because the events $\{X(s+\tau)=k\}$, for $k\in\mathcal S$, partition the possible outcomes.

\theta_k(s,\tau;w)\geq 0 \quad\text{for every }k\in\mathcal S, \qquad \sum_{k\in\mathcal S}\theta_k(s,\tau;w)=1.

The sum-to-one property is a useful check on both the estimand and any reported prediction. It depends on defining a complete state space and assessing state at the same horizon for all members of the target group. If the recorded categories omit a possible outcome, such as death or an unknown state, the categories may not form a complete partition. A set of probabilities that sums to less than one may then reflect an omitted outcome or unresolved follow-up, not a valid complete state-occupation distribution.

The conditioning matters as much as the state definition. A probability among everyone enrolled at baseline generally differs from one among those still under observation at $s$. A probability for one landmark information profile may differ from a probability averaged across all profiles. These are different estimands, not competing estimates of one universally defined number.

Worked Example: A Three-State Distribution at a Fixed Horizon

Consider a hypothetical group of 48 individuals who are eligible and under observation at month 6 and meet the same specified landmark-information criterion. Their state at month 18 is completely ascertained. The possible states are stable disease, progressed disease, and death. At month 18, 18 individuals are stable, 11 have progressed disease, and 19 have died.

Here $s=6$ months and $\tau=12$ months, so the target time is $s+\tau=18$ months. The observed proportions, used here to describe this fully observed hypothetical group, are:

\widehat\theta_{\text{stable}}=\frac{18}{48}=0.375, \qquad \widehat\theta_{\text{progressed}}=\frac{11}{48}=0.2292, \qquad \widehat\theta_{\text{death}}=\frac{19}{48}=0.3958.

The values sum to $0.375+0.2292+0.3958=1.000$, up to rounding. In this group, 37.5% occupy the stable-disease state at month 18, 22.9% occupy the progressed-disease state, and 39.6% occupy the death state. These proportions describe state at month 18; they do not say when a transition occurred.

Conditioning, Averaging, and Interpretation

A landmark estimand may be conditional on a specific profile or may average over profiles. The distinction can be shown by partitioning the landmark-eligible population into mutually exclusive information strata $W_s=w_1,\ldots,w_m$. If each stratum has positive probability in that population, the overall state probability is a weighted average of its stratum-specific probabilities:

P\{X(s+\tau)=k\mid R_s=1\} = \sum_{j=1}^{m} P\{X(s+\tau)=k\mid R_s=1,W_s=w_j\} P\{W_s=w_j\mid R_s=1\}.

This identity is the law of total probability applied within the landmark-eligible group. Each stratum-specific probability is weighted by the proportion of eligible individuals in that stratum, not by an arbitrary or baseline-population weight. Consequently, a population’s overall state distribution can change when its mix of landmark profiles changes, even if the stratum-specific probabilities remain the same.

Worked Example: A Profile-Specific Probability

In a hypothetical landmark cohort, 24 individuals are under observation at month 3 and have a specified profile recorded at that time. At month 9, 15 are in state A, 5 are in state B, and 4 are dead. State at month 9 is known for all 24.

The landmark horizon is $\tau=6$ months. For this profile, the three state-occupation proportions are:

\widehat\theta_A=\frac{15}{24}=0.625, \qquad \widehat\theta_B=\frac{5}{24}=0.2083, \qquad \widehat\theta_{\text{death}}=\frac{4}{24}=0.1667.

Thus, the proportion occupying state B at month 9 is 20.8% among individuals who met the specified profile and were under observation at month 3. It is not necessarily 20.8% among everyone enrolled at baseline. The conditioning profile and landmark eligibility define the denominator and are part of the interpretation.

Occupation at a Horizon Is Not Event Incidence

A state-occupation probability asks which state an individual occupies at one specified time. An event-incidence quantity instead asks whether an event occurred by a specified time, whether or not the individual occupies the associated state at the endpoint. These coincide only under additional structural conditions, such as a transition into a state that is absorbing and remains occupied thereafter. In a multi-state process with recovery or further transitions, they can differ substantially.

Worked Example: Current Occupation Versus Ever Entering a State

Suppose, hypothetically, that 20 individuals are eligible at a landmark and their complete histories are known through the prediction horizon. At the horizon, 8 occupy state A, 6 occupy state B, and 6 occupy the death state. During the interval, 13 individuals entered state B at least once, including some who later returned to state A.

The probability represented by the observed state-occupation proportion for B is $6/20=0.300$, or 30.0%. The proportion who ever entered B is $13/20=0.650$, or 65.0%. The two quantities answer different questions. The first describes state at the horizon; the second describes whether a transition into B occurred at any time during the interval.

The endpoint categories are exhaustive because $8+6+6=20$. The “ever entered B” category is not an alternative endpoint state: it overlaps with other endpoint categories when individuals can leave B. Treating it as one of three mutually exclusive endpoint states would therefore be incorrect.

Justification and Assumptions for a Well-Defined Target

The estimand is a conditional probability, so defining it does not itself require a transition model or a causal assumption. It does require a clearly specified target. In particular, the state process must be well defined at $s+\tau$, the landmark eligibility rule must be explicit, and the information used for conditioning must be available at $s$. The state categories must also be mutually exclusive; for probabilities across the full state space to sum to one, they must be exhaustive at the horizon.

The distinction between defining and estimating the target is important. With complete outcome ascertainment for every member of a finite hypothetical group, the observed proportions directly describe that group. In a real study, some members may be censored or otherwise have unknown state at the horizon. The fraction whose state is observed is then not automatically an estimate of the target probability: it may be distorted if those with known outcomes differ systematically from those without them. Estimation from incomplete follow-up requires appropriate methods and assumptions; merely changing the denominator to those with observed outcomes does not establish validity.

A further condition applies when the target is stated for a particular value of $W_s$, especially a continuous profile: there must be a meaningful conditional distribution at that value in the population of interest. In data analysis, predictions at profiles unsupported by the observed landmark cohort require extrapolation. This is a limitation of the available information, not a change in the mathematical definition.

Justification: For each state $k$, define $I_k=1\{X(s+\tau)=k\}$. If exactly one state is occupied at the horizon, then $\sum_{k\in\mathcal S} I_k=1$. Taking conditional expectations given landmark eligibility and information yields $$ \sum_{k\in\mathcal S}P\{X(s+\tau)=k\mid R_s=1,W_s=w\} = E\left\{\sum_{k\in\mathcal S} I_k\mid R_s=1,W_s=w\right\} =1. $$ The same argument gives nonnegativity because each indicator is nonnegative. This verifies the defining distribution properties without specifying how the probabilities are modeled or estimated.

Common Pitfalls and Alternative Targets

Several related quantities are useful, but they should not be substituted for the landmark state-occupation estimand without changing the stated question.

Using the wrong denominator. The target is conditional on eligibility at $s$ and the stated landmark information. Using everyone enrolled at baseline instead answers a different, baseline-population question.
Conditioning on future survival or observation. The target does not, by definition, restrict the group to people who remain alive or observed through $s+\tau$. If death is a state, it belongs in the endpoint distribution. Restricting to people with an observed future state can introduce selection and changes the target population.
Using information from after the landmark. A later measurement cannot be part of $W_s$ for a prediction made at $s$. Conditioning on it changes the question and can use information unavailable at the time of prediction.
Calling occupation a cumulative incidence. Occupancy at one horizon is not generally the probability of ever entering a state during the interval. The two agree only in settings where the event and endpoint-state definitions make them equivalent.
Reporting a single event probability when the outcome is multi-state. A probability for one state alone may be meaningful, but the complete state-occupation distribution includes every possible state and should satisfy the sum-to-one check.

A marginal state probability averages over landmark profiles; a profile-specific probability conditions on them. A probability of ever experiencing an event summarizes history over an interval; a state-occupation probability describes status at its endpoint. A transition rate or hazard describes event occurrence over time rather than the endpoint state distribution. Each can be appropriate, but the target, conditioning set, and time reference must be named before results are compared.

The estimand also has limitations. It is tied to its landmark eligibility rule, information set, and prediction horizon, so it does not automatically transport to a different landmark cohort or time horizon. It is a probabilistic summary, not a guarantee for a particular individual and not, by itself, a causal effect. Its interpretation depends on consistent state definitions and accurate ascertainment at the horizon.

Check Your Understanding

Answer each question by naming the target and showing the relevant calculation where needed.

At a landmark, 30 eligible individuals have complete follow-up to the horizon: 12 occupy state A, 9 occupy state B, and 9 occupy death. What are the three state-occupation proportions?
In a group of 40 landmark-eligible individuals, 14 occupy state C at the horizon, while 22 entered state C at least once during follow-up. What are the two proportions, and why are they different targets?
Explain why a probability conditional on being under observation at the landmark need not equal the probability among everyone enrolled at baseline.
Suppose the reported probabilities for three states are 0.40, 0.35, and 0.10. Give one interpretation issue that should be checked before accepting these as a complete state-occupation distribution.

Solutions. (1) The proportions are $12/30=0.400$ for A, $9/30=0.300$ for B, and $9/30=0.300$ for death; they sum to 1. (2) The horizon-occupation proportion is $14/40=0.350$, whereas the ever-entered proportion is $22/40=0.550$. They differ because some individuals may enter C and later leave it. (3) Landmark eligibility selects people under observation at that time, while the baseline population also includes those who are no longer under observation. The groups and conditioning events differ. (4) The probabilities sum to $0.85$, not 1. Check whether a state is omitted, some endpoint states are unknown, or the quantities use inconsistent conditioning or denominators.