Using Landmark Covariates for State-Occupation Prediction

From Landmark Information to a Conditional Prediction

A state-occupation prediction answers a forward-looking question: given what is known about an individual at a landmark, what is the probability that the individual will occupy each possible state after a specified amount of time? Covariates measured at that landmark can distinguish predictions for people who are currently in the same state. They do so by entering the transition models, not by changing the meaning of the state-occupation estimand.

Let \(s\) be the landmark time and \(u>0\) the prediction horizon measured from that landmark. Write \(X(t)\) for the state occupied at time \(t\), and let \(Z(s)=z\) denote the vector of covariate values measured and available at the landmark. If the individual occupies state \(j\) at \(s\), the prediction for state \(k\) is conditional on that current state and the recorded covariates. Where the model requires additional history, such as elapsed time in the current state for a semi-Markov model, that information must also be included in the conditioning.

For a Markov model whose post-landmark transition rates depend on the landmark covariates, write \(q_{jk}^{(s)}(v;z)\) for the instantaneous transition rate from state \(j\) to state \(k\), at elapsed time \(v\) after the landmark. The diagonal entry is \(q_{jj}^{(s)}(v;z)=-\sum_{k\ne j}q_{jk}^{(s)}(v;z)\). These rates form a transition intensity matrix \(Q^{(s)}(v;z)\). For a finite-state Markov model, the conditional state-occupation vector is obtained by applying the transition matrix generated by these covariate-specific rates to the row vector concentrated in the landmark state:

The product integral is written schematically for continuously varying rates; in practice, its numerical calculation uses small time increments or an equivalent solution of the transition-probability equations. The key operation for this tutorial is to evaluate the model at \(z\), then calculate the transition matrix and state probabilities. The prediction is conditional on the specified covariates; it is not a causal effect of changing them.

How Covariates Enter the Calculation

Transition-specific models describe how covariates relate to particular transitions. For example, a proportional intensity model may specify

Here \(q_{jk,0}^{(s)}(v)\) is the baseline transition rate and \(\boldsymbol{\beta}_{jk}\) is the coefficient vector for transition \(j\to k\). The coefficients can differ across transitions: a covariate associated with a higher rate of entering an illness state need not have the same association with a rate of death. For a categorical covariate, the value \(z\) is represented using the model's specified indicator coding; for a continuous covariate, the recorded value is inserted using the fitted scale and functional form.

The model must match the prediction question. Its assumptions include a clearly defined landmark risk set, valid measurement of the covariates by the landmark, adequate representation of the relevant history, and correctly specified transition models. Estimation also requires sufficient data to support the transitions and covariate values being used. If censoring prevents observing later states, the estimation procedure must handle censoring appropriately, under its stated assumptions. These conditions are not verified merely by inserting values into a fitted model.

The landmark covariates are held at their recorded values for the prediction calculation unless the model explicitly specifies a different post-landmark covariate path. Thus a model using \(Z(s)\) makes a prediction conditional on landmark values; it does not automatically account for later changes in a covariate. When dependence on earlier history is important, conditioning only on the current state and \(Z(s)\) may be inadequate. For semi-Markov predictions, for instance, the elapsed time in the current state may also be needed to determine the transition distribution.

Why the Transition Structure Matters

For multiple transient and absorbing states, the probability of occupying a state is not generally obtained by treating each transition separately and adding its rate. An individual may pass through an intermediate state before the horizon, and the chance of doing so depends on the time spent in preceding states. The transition matrix calculation accounts for these pathways. The following derivation shows this for an illness-death model with constant rates.

Worked Predictions Across Discrete Intervals

If transition probabilities are modeled over discrete intervals, the same conditional principle applies. For each interval, evaluate its transition matrix at the landmark covariate values. Multiply the interval-specific matrices in chronological order, then select the row corresponding to the state occupied at the landmark. The resulting row gives the conditional probabilities of occupying each state at the end of the intervals.

Diagnostics, Alternatives, and Limitations

A covariate-specific prediction can be numerically valid but unreliable. Before interpreting it, check that the covariate values are within the range represented in the landmark data and that the relevant transitions have adequate support for people with similar values. Examine whether the fitted covariate form is plausible, whether transition rates or probabilities behave sensibly across covariate values, and whether the model represents the history needed for the process. Large predictions based on sparse transition data or extrapolation beyond observed covariate values deserve particular caution.

Also check the calculation itself. Transition rates must be nonnegative off the diagonal, and transition probabilities must be between zero and one. For a model in which the listed states exhaust all possibilities, the predicted state probabilities should sum to one, apart from rounding. A failure of these checks may indicate an incorrect matrix order, a coding error, or an inconsistent model. Good arithmetic does not establish that the model assumptions hold.

A marginal Aalen-Johansen estimate can summarize state occupation for a landmark risk set without distinguishing predictions by an individual's covariates. It is useful for a population-level summary, but it does not by itself provide different conditional predictions for different covariate values. Transition-specific models provide a route to such conditional predictions, at the cost of modeling assumptions about each transition. Their predictions are associational and conditional: they do not imply that intervening to change a covariate would cause the predicted probabilities to change accordingly. Unmeasured prognostic factors, model misspecification, and limited support can all affect the prediction.

• Mistake: Insert a landmark covariate into one transition model, then interpret the resulting probability as the chance of ever entering that state. Correction: A state-occupation probability concerns the state at the specified horizon; transition pathways and later exits must be included.
• Mistake: Treat the landmark covariate value as if it described the individual's future covariate trajectory. Correction: The prediction conditions on the recorded value at \(s\). Future covariate changes are represented only if the model explicitly accounts for them.
• Mistake: Call an individualized conditional prediction a causal effect. Correction: It describes a model-based probability conditional on observed information, not the result of an intervention.