Why Predictions Change at a New Landmark
A prediction made today is conditional on what is known today. At a later landmark, the person may be in a different state, and new covariate or history information may have been recorded. The prediction for a future time should then be recalculated using that new information. It is not generally the same as the earlier forecast, because the earlier forecast averaged over possible paths to the later landmark, while the updated prediction conditions on the path information that actually occurred.
For example, a prediction made while a person is in a disease-free state must account for the possibility of disease onset before the prediction horizon. If the person is subsequently observed to have entered a disease state, the relevant question changes: what are their future state probabilities, given that they are now in that state? The answer is a new conditional prediction, not the old forecast with its probabilities carried forward unchanged.
The information \(\mathcal{H}_s\) can include the current state, baseline and updated covariates, the landmark time, and any relevant observed history. In a Markov model, the current state and specified covariates are sufficient for predicting the future under the model. If prediction depends on earlier history beyond that information, the conditioning set or model must retain the relevant history. A prediction should never condition on information that was not yet known at its landmark.
Recalculating With Transition Probabilities
For a finite-state Markov model, write \(P(s,t)\) for the transition probability matrix from time \(s\) to time \(t\). Its \((i,j)\) entry is \(P(X(t)=j\mid X(s)=i)\). If the state at landmark \(s\) is observed to be \(i\), the state-occupation prediction at \(t\) is row \(i\) of \(P(s,t)\). Equivalently, it is \(e_iP(s,t)\), where \(e_i\) is a row vector with a one in position \(i\) and zeros elsewhere.
When an earlier prediction spans two intervals, it averages over the possible states at the intermediate time. For a time-homogeneous Markov model, or more generally when the transition matrices correctly represent the two intervals, the Chapman–Kolmogorov relation gives \(P(s_0,t)=P(s_0,s_1)P(s_1,t)\). If covariates or other relevant information change, the second-interval transition law must reflect the information assumed for that forecast. At \(s_1\), once the current state and updated information are observed, the new prediction uses those observations directly.
Identify the observed state, updated covariates, landmark time, and any history needed by the prediction model.
Use the transition law from the new landmark to the requested future time, not a row of the old prediction vector as though it were the new starting state.
For a Markov model, select the transition-matrix row for the observed state. Apply the appropriate updated covariates or history if they affect the transition law.
Confirm that the probabilities are between zero and one, refer to the intended future horizon, and include all possible states under the model.
Worked Example: Recalculating After Observing the Intermediate State
Hypothetical data. Consider three states: 0 (no disease), 1 (disease), and D (death, absorbing). At landmark \(s_0\), a person is observed in state 0. Suppose the estimated transition matrices for the next two intervals are
The model assumes the stated transition probabilities apply to these intervals, and the individual’s state is observed without error at \(s_1\). At \(s_0\), the forecast for time \(t\) starts in state 0 and averages over the possible states at \(s_1\). The probability of state 0 at \(t\) is
The corresponding probabilities of state 1 and death are
Thus the original forecast at \(t\) is \((0.430,0.305,0.265)\). Now suppose the person is actually observed in state 1 at \(s_1\). The updated prediction is row 1 of \(P(s_1,t)\): \((0.050,0.650,0.300)\). In context, conditional on being in state 1 at \(s_1\), the model predicts a 65.0% probability of still occupying state 1 at \(t\), and a 30.0% probability of having reached death by \(t\). These are conditional predictions for the person now observed in state 1; they are not the original probabilities for the group forecast at \(s_0\).
Why the Updated Prediction Is a New Conditional Quantity
The calculation above follows directly from conditioning on the intermediate state. Before \(s_1\), the state is not yet known, so the forecast combines the possible states at \(s_1\), weighted by their probabilities. After observing state \(k\) at \(s_1\), only that observed state is the starting point for the individual’s updated prediction.
For each future state \(j\), add the probability of reaching each intermediate state \(k\) and then moving from \(k\) to \(j\): \(P(X(t)=j\mid\mathcal{H}_{s_0})=\sum_k P(X(s_1)=k\mid\mathcal{H}_{s_0})P(X(t)=j\mid X(s_1)=k,\mathcal{H}_{s_0})\), when the stated conditional transition law applies.
If \(X(s_1)=k\) is observed, the updated quantity is \(P(X(t)=j\mid\mathcal{H}_{s_1})\). Under a Markov model with the required covariates specified, this is the \((k,j)\) entry of the transition matrix from \(s_1\) to \(t\).
This also explains an important qualification. If covariates observed at \(s_1\) were unknown at \(s_0\), the earlier prediction cannot simply insert their eventual values. An earlier prediction must use a prespecified assumption about future covariates or average over their possible values using a suitable model. Once the covariates are observed, the later prediction can condition on them. The two predictions answer different questions because their conditioning information differs.
Worked Example: Incorporating a Newly Observed Covariate
Hypothetical data. Consider states 0 (no event), 1 (event), and D (death). At a new landmark, a person is observed in state 0. A fitted model gives the following probabilities of occupying the three states over the next year, for two values of a newly recorded binary covariate \(z\):
Suppose the earlier prediction used \(z=0\), because that was the value available then. Its state-0 row predicted probabilities \((0.80,0.15,0.05)\). At the new landmark, the person is still in state 0 but is observed to have \(z=1\). The updated prediction uses the \(z=1\) row: \((0.55,0.30,0.15)\). The arithmetic is direct: for example, the predicted one-year probability of state 1 is \(0.30\), or 30.0%, conditional on the new landmark information. The probabilities remain nonnegative and sum to \(0.55+0.30+0.15=1\).
This update is justified only if the model’s covariate-specific transition probabilities are appropriate for the new landmark and the covariate is measured before the future prediction interval. If \(z\) were measured after the outcome process of interest, using it this way could introduce future information and would not represent a valid prediction at the landmark.
Relevant History and the Prediction Model
A later landmark may reveal more than a current state and a baseline covariate. For instance, the amount of time already spent in the current state may predict subsequent transitions. In that setting, the prediction must use a model that conditions on the relevant sojourn time or history; simply selecting a current-state row from a model that omits this information may be inadequate. The time origin and prediction horizon must also be clear: a prediction “for the next year from \(s_1\)” differs from a prediction “at calendar time \(t\)” if \(t\) is not one year after \(s_1\).
Worked Example: Updating a Sojourn-Time-Conditioned Prediction
Hypothetical data. Suppose a person occupies transient state 1 at a landmark, and death D is the only other possible state. A semi-Markov prediction model uses the time already spent in state 1. For a specified future horizon, assume its estimated state probabilities are \((0,0.60,0.40)\) when the sojourn time is 2 months and \((0,0.45,0.55)\) when it is 4 months, with entries ordered as states 0, 1, and D. These probabilities are assumed to apply to the same prediction horizon and to be appropriate for the person’s observed history.
An earlier prediction made when the known sojourn time was 2 months assigned a 40.0% probability to death over that horizon. At the next landmark, suppose the person is still in state 1 and the recorded sojourn time is now 4 months. The updated prediction uses the 4-month condition: the death probability is 0.55, or 55.0%, and the probability of remaining in state 1 is 0.45, or 45.0%. The increase is 15 percentage points. It is a change in a model-based conditional prediction, not evidence that the person’s sojourn time caused an increase in risk.
Assumptions, Diagnostics, and Alternatives
A successive-landmark prediction is only as sound as its conditioning information and transition model. In practice, check that the person’s state and covariates are available by the landmark; that the prediction horizon is measured from the intended origin; and that the model is suitable for the state and history being conditioned on. A Markov transition model requires that the specified state and covariates capture the history relevant to future transitions. If that assumption is not plausible, a model that includes additional history, such as sojourn time, may be needed. Estimated probabilities also rely on adequate data to support the relevant states, covariate patterns, and time interval.
Useful warning signs include a prediction that starts from a state the person does not occupy at the new landmark, a covariate value recorded after that landmark, a mismatch between the model’s time scale and the stated horizon, or probabilities outside \([0,1]\). A sum different from one beyond rounding can indicate that states have been omitted, the wrong matrix row was used, or calculations were combined inconsistently. A large change from the earlier forecast is not by itself an error: the new state or covariate may be informative. But an implausible change warrants checking the input information, model specification, and numerical calculation.
The main alternative is to carry forward the earlier prediction unchanged. That answers what was predicted using the earlier information; it does not answer the updated conditional question. Transition-based landmark recalculation, as shown here, uses a model for transitions between states. A direct model could instead predict state occupation at a chosen horizon from landmark information without constructing all transition probabilities. These approaches target related quantities but make different modeling choices. Regardless of approach, the prediction at each landmark must use only information available then and must state the population and horizon to which it applies.
Check Your Understanding
Use the information in each question to identify the correct conditional prediction or calculation.
- At an early landmark, a model predicts probabilities for three states at a later time. At that later landmark, the person is observed in state 2. What information should determine the new prediction?
- For a two-interval forecast, the probability of an intermediate state is 0.40. Conditional on that state, the probability of a future event is 0.25. The other intermediate state has probability 0.60 and a conditional event probability of 0.10. Calculate the earlier forecast’s event probability.
- An updated three-state prediction is \((0.62,0.28,0.10)\). State one numerical check and explain what the prediction means if the entries are ordered as no event, event, death.
- A covariate is measured after the new landmark but is inserted into a prediction described as being made at that landmark. What is the problem?
Solutions. (1) Use the observed state at the new landmark, together with covariates and any relevant history available by that time, in the transition or direct prediction model. Do not treat the old forecast’s distribution over possible states as the person’s observed state. (2) Apply the law of total probability: \(0.40(0.25)+0.60(0.10)=0.10+0.06=0.16\). The earlier forecast assigns a 16.0% event probability. (3) All entries must be between zero and one and their sum is \(0.62+0.28+0.10=1.00\). Conditional on the information at that landmark, the predicted probabilities are 62.0% for no event, 28.0% for event, and 10.0% for death at the stated horizon. (4) The prediction uses future information unavailable at its stated landmark, so it is not a valid prediction conditional on information available then and may be biased by information leakage.