Conditioning Information at a Landmark

Why Conditioning Information Defines the Prediction

A dynamic prediction is not fully specified by naming an outcome and a future horizon. It also depends on who is being predicted for, and what is known about that person at the landmark. Two predictions of the same future state can differ because one is conditional only on a person's current state, another also uses current covariates, and a third uses the path of states and events observed so far.

The distinction matters for interpretation and for comparing predictions. A probability conditional only on current state generally averages over the covariate and history mix of the target population. A probability conditional on a particular covariate value or history describes a narrower group. These are different estimands, not competing ways of writing the same quantity.

The target population \(A_T=1\) must be specified before interpreting the probability. It could, for example, consist of individuals meeting a stated eligibility condition at \(T\). This tutorial concerns the information conditioned on within that population, not how to construct an analysis risk set. The state, covariates, and history in the conditioning statement refer to information available no later than \(T\). The future state \(Z(T+u)\) is the outcome.

A history \(H_T\) may include the current state, previous states, event times or types, and time-varying covariate measurements, depending on what was recorded and declared relevant. If it contains the full state path through \(T\), then it includes \(S_T\). It may also contain \(X_T\), but only if those covariates are part of the recorded history. By contrast, the conditioning set \((S_T,X_T)\) need not contain earlier state changes or event times.

Three Common Conditioning Targets

A state-conditioned target, \(P\{Z(T+u)=k\mid A_T=1,S_T=s\}\), describes the future state distribution among eligible individuals who occupy state \(s\) at the landmark. Unless the target population has further restrictions, it averages over their covariates and earlier histories. Thus, the target can depend on how those covariates and histories are distributed among people in state \(s\).

A state-and-covariate-conditioned target, \(P\{Z(T+u)=k\mid A_T=1,S_T=s,X_T=x\}\), additionally fixes the covariate value \(x\). It does not, by itself, fix all earlier history. If individuals with the same \(s\) and \(x\) have different relevant histories, this target still averages over those histories as represented in the eligible population.

A history-conditioned target, \(P\{Z(T+u)=k\mid A_T=1,H_T=h\}\), conditions on the specified recorded path through \(T\). It can distinguish people who occupy the same state now but arrived there by different routes. Its specificity comes with a practical cost: detailed histories may define small groups, and some histories may be absent from the data.

These targets can be nested, but only when their information sets really are nested. For example, if \(H_T\) includes both \(S_T\) and \(X_T\), conditioning on \(H_T\) is at least as specific as conditioning on \((S_T,X_T)\). A covariate-only target and a history-only target are not necessarily nested. Each may contain information absent from the other.

Marginalization and the Role of History

The averaging in the example is an instance of a general identity. If \(X_T\) is discrete, a target that does not condition on \(X_T\) can be obtained by averaging the covariate-specific target over the covariate distribution in the population that is already conditioned on. For continuous covariates, the sum is replaced by an integral. The weights must correspond to that target population and conditioning set; using a different population's weights defines a different marginal target.

The same logic applies to histories. A probability conditional on current state can average over multiple histories ending in that state. If history matters for future state occupation, the history-specific probabilities need not be equal. Consequently, state-only prediction is not automatically a prediction that applies uniformly to everyone currently in that state.

This identity is a decomposition, not a claim that the covariate or history causes the future outcome. It also does not prescribe how to estimate the conditional probabilities or the averaging weights. The target is defined first; estimation is a separate task.

What Counts as Information at the Landmark?

For a prediction intended to be made at \(T\), every conditioning variable must be known by \(T\). A measurement taken after \(T\), a later event, or the future outcome itself cannot be used as if it were landmark-time information. Conditioning on such information changes the question: it describes a subgroup defined partly by what happens after the prediction is meant to be made.

The definition of \(X_T\) should therefore specify both the covariate and its measurement time. A baseline measurement and a value updated at the landmark are not interchangeable labels. Similarly, a history-conditioned target must say which events and measurements are included in \(H_T\), and what observation window ends at the landmark. Ambiguous labels such as “adjusted for history” do not identify a unique target.

Before a target is used, check that its conditioning information is well-defined for the target population and that the relevant groups have support. If a requested covariate value or history does not occur among eligible individuals, its conditional probability is not identified from those data without additional assumptions. Sparse groups can also make estimates unstable even when the group is not empty.

Assumptions, Comparisons, and Failure Modes

Conditioning targets are mathematical quantities, but their use requires clear assumptions and definitions. The state system must assign a meaningful state at the horizon; the landmark and horizon must be the same for the people being summarized; and the target population and conditioning variables must be specified. To interpret an estimated conditional probability as a prediction for a group, that group must be represented in the data, and the estimation procedure must adequately capture its outcome distribution. These conditions do not make the prediction causal.

The main alternatives differ in the question they answer. Conditioning only on state gives a broader summary and averages over covariates and histories. Adding covariates gives a more tailored conditional target, while conditioning on recorded history can distinguish different routes to the same current state. A marginal target for a chosen population can also be obtained by averaging conditional predictions over a specified covariate or history distribution. Such averaging is useful for population summaries, but the chosen distribution must be reported because another choice can yield another answer.

A useful check is to write the conditioning statement in words and then compare it with the intended use. If the intended use is an individual prediction based on current state and measured covariates, a state-only target may be too broad. If the aim is a population-level state-occupation probability, a history-specific target may be unnecessarily narrow unless it will be averaged back over a stated history distribution.

Common warning signs include reporting a probability without naming the target population, treating a state-only probability as universal across covariate groups, or using a post-landmark measurement in a prediction said to be available at \(T\). Another failure mode is assuming that current state summarizes all relevant history. That may be a modeling assumption in a particular application, but it does not follow from the definition of conditioning on state. Diagnostics should therefore compare the declared target with the actual data collection times, subgroup support, and intended prediction use.