From Landmark Predictions to Trajectories
A landmark prediction is often reported at one selected future horizon: for example, the predicted probability of being in a particular state six months after a clinical assessment. A single number answers that specific question, but it does not show how the prediction develops at shorter or longer horizons. A state-probability trajectory displays the predicted probability for each state across a range of future horizons, making the timing and direction of changes visible.
The plot describes probabilities for a specified individual, conditional on information available at a specified landmark. It is not a forecast of the individual's realized path. A person occupies one state at a given time; the curves describe model-based probabilities over possible future paths for people with the same conditioning information.
The prediction target must be stated precisely: specify the landmark, the information conditioned on, the state definitions, and the time scale. For an individual-level display, \(Z_L\) may include the individual's covariates and current state; in history-dependent or semi-Markov settings it may also include relevant earlier history or sojourn time. The model and its prediction procedure must use that information consistently.
How to Read the Curves
At each horizon, a curve gives the predicted probability of occupying its named state at that horizon. It does not give the probability of ever having visited that state, nor the probability of following a particular sequence of transitions. The latter questions involve different quantities. State-occupation probabilities are point-in-time predictions.
The shape depends on the state structure. For a transient state, its probability can rise and later fall: individuals may enter it and then leave for another state. For an absorbing state, under a model in which entry is permanent, its occupation probability cannot decrease as the horizon increases. A curve that changes direction is not automatically an error; it may be exactly what the allowed transitions imply. Conversely, a model that disallows exit from a state should not produce a declining probability of occupying that state.
A sound display uses a common, clearly labelled horizontal scale and vertical probability scale for all states being compared. Mark the landmark or horizon zero, label the unit of time, identify each curve, and state whether the horizontal axis is elapsed time since the landmark or absolute calendar/age time. A small table of selected horizons can accompany a plot when exact values matter, but a table alone can make the overall timing harder to see.
Worked Example: Interpreting Three State Trajectories
Suppose a hypothetical model predicts the states stable, complication, and dead for one person assessed at month 0. The model's point predictions at selected horizons are:
| Months after landmark | Stable | Complication | Dead |
|---|---|---|---|
| 0 | 1.000 | 0.000 | 0.000 |
| 3 | 0.720 | 0.240 | 0.040 |
| 6 | 0.560 | 0.330 | 0.110 |
| 12 | 0.380 | 0.370 | 0.250 |
To interpret the display, follow each column over time. The predicted probability of being stable declines from 1.000 at the landmark to 0.380 at month 12. The probability of occupying the complication state first increases, from 0.000 to 0.370. It need not represent a permanent outcome: people can enter and leave this transient state under the model. The probability of being dead increases from 0.000 to 0.250; this direction is consistent with death being absorbing.
At month 6, for example, the relevant reading is 0.330 for complication: the model predicts a 33.0% probability of occupying that state exactly six months after the landmark, conditional on the stated landmark information. It is not a 33.0% probability of having experienced a complication at any time by month 6.
These values are explicitly hypothetical model outputs, not observations from a real study. The table shows selected points; a trajectory plot would connect or otherwise represent predictions over the full horizon range supported by the model. It would not justify treating the person's future state as known.
Calculating Trajectory Points From a Transition Matrix
For a time-homogeneous Markov model with a fixed state space, the state-probability vector at elapsed horizon \(u\) can be obtained from the transition matrix. This provides a direct way to calculate points for a trajectory. The assumptions matter: transitions depend on the current state rather than unrepresented earlier history, and the transition probabilities do not change with calendar time over the interval considered. If these conditions are inappropriate, a different prediction model is needed.
This result follows by conditioning on the state at the preceding unit. For any destination state \(k\), the probability of occupying \(k\) after one more unit is the sum, over all possible preceding states \(j\), of the probability of being in \(j\) times the probability of transition from \(j\) to \(k\). In vector notation, this is right multiplication by \(P\). Repeating the argument \(m\) times gives \(P^m\). Thus plotting state probabilities amounts to evaluating the prediction at a sequence of horizons and displaying the resulting state-specific components.
Worked Example: Computing Points for an Illness-Death Trajectory
Consider a hypothetical time-homogeneous Markov model with healthy, ill, and dead states. The one-year transition matrix is
Rows give the current state and columns the state one year later. The model permits healthy-to-ill and healthy-to-dead transitions, ill-to-dead transitions, and remaining in the same state; death is absorbing. Assume the individual is healthy at the landmark, so \(\boldsymbol{p}(0)=(1,0,0)\). The assumptions for using powers of this matrix are that the transition structure remains constant across years and the Markov property is suitable for the prediction.
After one year, the prediction is the first row of \(P\):
For two years, multiply the one-year vector by \(P\). For example, the two-year probability of being ill is the sum of the routes through healthy and ill at year one:
The other two components are
Thus \(\boldsymbol{p}(2)=(0.490,0.375,0.135)\). Repeating the multiplication gives \(\boldsymbol{p}(3)=(0.343,0.4225,0.2345)\). Plotting these vectors at years 0, 1, 2, and 3 gives discrete trajectory points for each state. The ill-state probability rises from 0.250 at year 1 to 0.4225 at year 3, while the dead-state probability rises from 0.050 to 0.2345. The calculations are conditional predictions under the stated matrix model, not a claim that a particular individual will follow those transitions.
Keeping the Landmark and Time Axis Clear
Elapsed horizon and absolute time are related but not interchangeable. If the landmark is at calendar month 6, then elapsed horizon \(u=3\) months refers to absolute calendar month \(L+u=9\). A plot against elapsed horizon starts at zero for each landmark; a plot against absolute time places points on the calendar-time scale. Labeling the axis prevents a reader from mistaking one for the other.
A trajectory is also tied to its conditioning information. If predictions are recalculated at a later landmark, the later prediction conditions on information available then, such as a newly observed state or updated covariates. It is a new conditional prediction, not simply the continuation of the original curve. Comparing the two can be useful, but the display should mark their distinct landmarks and avoid implying that they share the same conditioning set.
Worked Example: Showing a Prediction Updated at a Later Landmark
Suppose a hypothetical person is assessed at month 0 while healthy. A fixed-landmark model predicts a 0.600 probability of occupying the healthy state at month 6 and a 0.300 probability of occupying the ill state then. At month 6, the person is observed to be ill. A new prediction is made from that later landmark, conditional on being ill at month 6 and on the covariates then available. It predicts a 0.500 probability of being ill and a 0.500 probability of being dead at month 12.
A useful display can place the original predictions at absolute months 6 and 12 alongside the updated predictions at absolute month 12. For the updated prediction, month 12 is six months after its own landmark: \(12-6=6\). The original month-12 prediction concerns the state at month 12 conditional on month-0 information; the updated month-12 prediction conditions on the month-6 information. They answer different conditional questions and need not agree.
For example, if the original model predicted a 0.200 probability of death at month 12, that value should not be replaced by the updated model's 0.500. The latter is conditional on the person being observed ill at month 6; the former averages over possible states at month 6 under the original landmark prediction. A plot should distinguish the two origins, such as by separate line styles and explicit landmark labels.
Diagnostics, Alternatives, and Limitations
Before interpreting a trajectory, check that the state definitions and prediction target remain the same across horizons; that the horizontal axis and time units are explicit; and that the plotted values are model predictions at the stated landmark. Check whether the model allows the transitions suggested by the curve shapes. For example, a declining curve for a state declared absorbing signals a coding, plotting, or model-specification problem. A sharp kink may reflect a genuine time-varying transition pattern, but it can also result from sparse prediction support, coarse time intervals, or an implementation error and merits checking.
Do not draw conclusions from line smoothness alone. Connecting predictions calculated at a few discrete horizons can suggest intermediate values that were not evaluated. Use a step display if the prediction method is inherently discrete, or show evaluated points if interpolation is not justified. Restrict the plotted range to horizons for which the model and data provide meaningful support; extending a line far beyond that range can give a false impression of precision.
| Display or summary | What it emphasizes | Main caution |
|---|---|---|
| State-probability trajectories | How point-in-time occupation predictions change across horizons | Curves are conditional model predictions, not realized paths |
| Selected-horizon table | Exact numerical predictions at named times | Can hide the timing and shape between reported horizons |
| Competing-risk cumulative incidence curves | Probability of experiencing a specified event by a given time | Not the same quantity as occupation probability for a transient state |
A trajectory is descriptive of the fitted prediction function. It does not, by itself, demonstrate predictive accuracy, causal effects, or certainty about an individual's future. Point predictions also do not show parameter uncertainty. The number of plotted decimals should not imply more precision than the model and data warrant; reporting three or four significant figures is generally adequate for numerical examples.
A common mistake is to read each line as a record of the individual's eventual course: “the complication curve rises, so this person will become ill and then recover.” That conclusion is not supported. The curve reports the probability of occupying the complication state at each horizon, conditional on the landmark information. It neither specifies the individual's realized state nor, by itself, identifies which sequence of transitions would produce that occupation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A transient state's predicted occupation probability is 0.20 at month 2, 0.38 at month 6, and 0.25 at month 12. Does the decline after month 6 necessarily indicate an error? Explain.
- A landmark is at calendar month 8. What absolute time corresponds to an elapsed horizon of 5 months?
- In a Markov model with one-unit transition matrix \(P\) and landmark state vector \(\boldsymbol{p}(0)\), write the state-probability vector at horizon 3.
- Why should an updated prediction from a later landmark not automatically be drawn as a continuation of the original prediction?
Solutions. (1) No. If the state is transient, probability can rise as people enter and fall as they leave; the model's permitted transitions should be checked. (2) Month \(8+5=13\). (3) \(\boldsymbol{p}(3)=\boldsymbol{p}(0)P^3\), under the stated time-homogeneous Markov model. (4) The later prediction conditions on new information, so it has a different landmark and conditioning set and answers a different conditional question.