Two Routes to a Landmark Prediction
At a landmark time \(s\), a prediction may ask which state a person will occupy after a further \(u\) time units, given information available at \(s\). One approach models that future state directly as a categorical outcome. Another models transitions between states and combines those transition probabilities to obtain the future state distribution. Both can produce probabilities that sum to one, but they do not make the same modelling commitments.
The distinction matters when a prediction is used for decisions. A direct model can target the desired horizon without describing every route to the outcome. A transition-based model describes those routes, and can therefore generate predictions at multiple horizons or for alternative starting states. Its predictions depend, however, on whether the transition model captures the relevant history and time structure.
For a fixed landmark profile, a direct model treats \(X(s+u)\) as a \(K\)-category response. For example, a multinomial regression can model each category probability as a function of the variables in \(\mathcal{G}_s\). A transition-based model instead estimates how the process moves among states over time, then combines the estimated transitions to obtain the distribution at \(s+u\).
What Each Approach Assumes
A direct landmark model requires the conditional distribution of the state at the chosen horizon to be adequately modelled for the target population. Its prediction is specific to the horizon, the landmark information used as predictors, and the population represented by the training data. The model need not explain which intermediate transitions produced the future state. If the state at the horizon is unavailable for some people because of censoring, the analysis also needs an appropriate method and assumptions for that missing outcome; simply treating those people as if their state were observed is not justified.
A transition-based prediction requires a model for the transitions used in the calculation. In a time-inhomogeneous Markov model, the transition distribution over an interval depends on the state at the start of that interval and the modelled covariates, but not on omitted earlier history once those variables are conditioned on. If transition risks also depend on time already spent in the current state, a state-only Markov model may be inadequate; duration or other relevant history must be represented, as in an appropriate semi-Markov specification. Transition estimation also requires suitable assumptions about censoring and the observation process.
These assumptions are not interchangeable. A direct model can be well specified for one horizon even when a simple Markov transition model is not. Conversely, a well-specified transition model can provide coherent predictions at multiple horizons, even when a separate direct model would be unstable because few observations are available at one particular horizon.
How Transition Predictions Are Assembled
Suppose transitions are modelled across two successive intervals, from \(s\) to \(s+u_1\), then from \(s+u_1\) to \(s+u_1+u_2\). Let \(P_1\) and \(P_2\) be the corresponding transition-probability matrices. Each entry \((P_m)_{ij}\) is the probability of occupying state \(j\) at the end of interval \(m\), conditional on starting that interval in state \(i\) and the specified covariate information. Each row sums to one. If the Markov conditioning assumptions hold, the final distribution is obtained by matrix multiplication.
The entry in row \(i\), column \(j\) of \(P_1P_2\) sums the probabilities of all possible intermediate states. More generally, the same calculation follows from the law of total probability:
Each term corresponds to a possible state \(k\) at the intermediate time. The multiplication adds the probability of reaching \(k\) in the first interval and then reaching \(j\) in the second. The sum accounts for every possible intermediate state. This factorization is valid under the stated conditional transition model; if earlier history still predicts the next transition but is omitted, the assembled prediction can be wrong.
Worked Examples
Worked Example: Comparing Predictions for One Horizon
Consider three states: A (a non-absorbing state), B (a second non-absorbing state), and D (an absorbing state). For one hypothetical landmark profile and a horizon of one year, a direct multinomial model predicts probabilities \(0.62\), \(0.28\), and \(0.10\), respectively. A separately fitted transition model gives the following one-year transition row for someone in A at the landmark: \(0.70\), \(0.20\), and \(0.10\).
The direct prediction is \(62\%\) for state A, \(28\%\) for state B, and \(10\%\) for state D. Its entries sum to \(0.62+0.28+0.10=1.00\). The transition-based prediction from A is \(70\%\), \(20\%\), and \(10\%\), also summing to \(1.00\).
The two predictions differ by \(0.08\) for A, \(-0.08\) for B, and \(0\) for D. Both are distributions over the same three states at one year, but that alone does not establish that they estimate the same conditional quantity. Before treating the difference as evidence of a modelling problem, check that both use the same landmark population, covariates and history, horizon, and censoring strategy. If they do, disagreement can reflect different fitted models or finite-sample variation; it does not, by itself, show which prediction is more accurate.
Worked Example: Multiplying Transition Matrices
Suppose the hypothetical states are A, B, and absorbing state D. Over the first interval, the transition matrix is
Over the second interval, suppose the matrix is
A hypothetical person starts in A, so the initial distribution is \(\boldsymbol{\pi}(s)=(1,0,0)\). After the first interval, \(\boldsymbol{\pi}(s)P_1=(0.80,0.15,0.05)\). Multiplying by \(P_2\), the probability of A at the end of the second interval is
The probability of B is
and the probability of D is
Thus the final state-occupation distribution is \((0.5615,0.2635,0.1750)\). It sums to \(0.5615+0.2635+0.1750=1.0000\). In this calculation, the probability of D includes people already in D after the first interval as well as people entering D from A or B in the second interval.
Worked Example: When Omitted History Changes the Target
Suppose a hypothetical direct landmark analysis is restricted to people currently in B who have been in B for less than one month. At a six-month horizon, observed outcomes for 100 people in this target group are: 32 in A, 48 in B, and 20 in D. A direct estimate of the state-occupation probabilities is therefore \(32/100=0.32\) for A, \(48/100=0.48\) for B, and \(20/100=0.20\) for D.
Now suppose a transition model conditions on current state B but omits time already spent in B. For this same profile, it predicts \(0.50\) for A, \(0.35\) for B, and \(0.15\) for D. These values sum to \(1.00\), but they differ from the direct estimates by \(+0.18\), \(-0.13\), and \(-0.05\), respectively.
A possible explanation is that time in B predicts subsequent transitions: for example, people newly entering B may have different transition risks from people who have remained there longer. If so, conditioning on current state alone does not adequately represent the history of this target group. This example does not prove that duration caused the discrepancy; it shows why the conditioning sets must be checked before comparing outputs. A transition model that includes relevant duration or history could address this concern, while the direct estimate is explicitly for the selected group.
Practical Comparison and Failure Modes
| Feature | Direct landmark model | Transition-based prediction |
|---|---|---|
| Modelled quantity | State at a specified future horizon | Transitions over one or more intervals, combined to give future state occupation |
| Intermediate paths | Not modelled explicitly | Represented through the sequence of transition probabilities |
| Useful when | A particular horizon and target group are the main focus | Several horizons, starting states, or transition pathways are of interest |
| Key vulnerability | Limited data at the selected horizon or a misspecified horizon-specific relationship | Misspecified transition structure or omitted history needed to predict transitions |
| Interpretation | Conditional probability of occupying a state at the selected horizon | Conditional probability derived from the fitted transition model and its assumptions |
A useful comparison begins by aligning the estimands: the same landmark time, prediction horizon, target population, conditioning information, state definitions, and handling of censoring. If any of these differ, the predictions need not agree even if both models are correctly fitted for their own targets. For instance, a direct model restricted to people with a particular recent history cannot be fairly compared with a transition prediction averaged over a broader set of people in the same current state.
After alignment, inspect whether each prediction is supported by the data. A direct model may be unstable at a horizon with few observed outcomes, or may extrapolate for covariate profiles poorly represented in the training sample. A transition model can accumulate error across intervals: even modest errors in estimated transitions can alter the final distribution when multiplied repeatedly. Check that every transition row is a valid probability distribution and that the resulting state-occupation probabilities are nonnegative and sum to one. Those checks detect arithmetic or implementation errors, but do not establish that the model is well specified.
A transition model that depends only on the current state can fail when transition probabilities depend on sojourn time or other earlier history. A direct model can avoid that particular factorization, but it still needs the relevant history represented if predictions vary with that history. Neither approach removes the need to define the target clearly or justify how incomplete follow-up is handled. For inference, uncertainty in fitted coefficients or transition probabilities should also be carried through to the final predictions; a single probability vector does not convey its own precision.
The choice is therefore not simply between a “more flexible” and a “more realistic” method. Direct models are often attractive for a prespecified horizon and target population; transition models are attractive when transitions themselves or predictions at multiple times are needed and their assumptions are defensible. A substantive transition interpretation should not be inferred merely because a model was built from transition probabilities.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one sentence, distinguish the quantity directly estimated by a horizon-specific multinomial model from the quantities estimated by a transition-based approach.
- A transition matrix has rows that sum to one. Does this guarantee that the final state-occupation prediction is correct? Explain.
- Starting in A, a person has probability \(0.6\) of being in A and \(0.4\) in B at an intermediate time. In the next interval, the probabilities of reaching D are \(0.1\) from A and \(0.3\) from B. What is the probability of being in D at the end of that interval, assuming the transition model is appropriate?
- Give two features that must match before directly comparing a direct landmark prediction with a transition-based prediction.
Solutions. (1) The direct model estimates the conditional state distribution at a specified horizon; the transition approach estimates interval-specific transition distributions and combines them to derive that state distribution. (2) No. Row sums check that each row is a probability distribution, but do not show that the estimated transitions, conditioning history, or censoring assumptions are appropriate. (3) By total probability, \(0.6(0.1)+0.4(0.3)=0.06+0.12=0.18\), so the predicted probability is \(0.18\). (4) Examples include the target population, landmark time, horizon, conditioning covariates and history, state definitions, and censoring strategy.