State-Occupation Probabilities At A Fixed Horizon

What Is the Probability of Occupying a State at a Chosen Time?

In a multi-state model, a subject occupies exactly one state at a given time. A fixed-horizon question asks for the probabilities of being in each possible state at a specified time \(t\), given the subject’s state at an earlier time \(s\). For example, after starting in a disease-free state, what are the probabilities of still being disease-free, being ill, or having died by week 12? These probabilities form a distribution across states at that horizon.

The answer is not generally the probability of one particular transition. A subject could occupy a destination state at time \(t\) after following several different paths, and the subject could also have left that state again before \(t\). The fixed-horizon probability concerns the state occupied at \(t\), not the full history of transitions.

The term “state-occupation probability” emphasizes the state occupied at the chosen time. It does not mean the probability of ever entering that state, the probability of making a direct transition to it, or the proportion of time spent in it. Those are different quantities. Throughout, probabilities are conditional on the stated initial state unless an initial distribution is explicitly given.

From an Estimated Matrix to Occupation Probabilities

Suppose an estimator, such as the Aalen-Johansen estimator, provides an estimated transition-probability matrix \(\widehat{P}(s,t)\). For a known initial state \(i\), the estimated fixed-horizon occupation probabilities are obtained by selecting row \(i\):

This row-extraction rule is simple, but the interpretation depends on keeping the origin state and horizon clear. The entry \(\widehat{P}_{ij}(s,t)\) estimates the probability of occupying state \(j\) at time \(t\), conditional on occupying state \(i\) at time \(s\). It is not necessarily the probability of a direct \(i\)-to-\(j\) transition during the interval.

How Intermediate States Contribute

When transition probabilities are constructed across successive time intervals, matrix multiplication combines all possible intermediate states. Suppose \(s<u<t\). The general law-of-total-probability decomposition, conditional on both the intermediate state and the initial state, is

This expression is valid without assuming that the future depends only on the state at time \(u\): it retains \(X(s)=i\) in the first conditional probability. Under a Markov model, that first factor reduces to \(P(X(t)=j\mid X(u)=k)\). In that setting, the decomposition becomes the Chapman-Kolmogorov multiplication rule. The qualification matters: dropping the initial-state condition in the first factor is not justified by the law of total probability alone.

For estimated transition matrices under a model where this multiplication rule applies, the calculation is \(\widehat{P}(s,t)=\widehat{P}(s,u)\widehat{P}(u,t)\). The order matters: rows index origin states and columns index destination states, so the first interval’s matrix is on the left. The entry in row \(i\), column \(j\) adds the contributions from every possible state \(k\) at the intermediate time.

Calculating Occupancy from Event-Time Updates

In discrete event-time computation, the estimated transition matrix to a horizon is obtained by multiplying the successive estimated update matrices in time order. Each update describes the transition probabilities at one event time, with rows summing to one. If the initial state is fixed at \(i\), the \(i\)th row of the resulting product gives the horizon-specific occupation probabilities. This calculation assumes the states form a complete set for the analysis and that the estimated updates are appropriate for the transition model and data.

Initial Distributions, Checks, and Interpretation

If subjects do not all start in the same state, let \(\boldsymbol{\pi}\) be their row vector of initial-state probabilities, where \(\pi_i=P(X(s)=i)\). The corresponding state distribution at time \(t\) is \(\boldsymbol{\pi}P(s,t)\), and an estimated distribution is \(\boldsymbol{\pi}\widehat{P}(s,t)\) when \(\boldsymbol{\pi}\) is specified. This is a weighted average of the origin-state rows, not a new transition probability conditional on one particular origin.

Several checks help identify errors. Each estimated matrix row should contain probabilities between 0 and 1 and sum to 1, apart from negligible numerical rounding. A product of row-stochastic transition matrices should also be row-stochastic. At a fixed horizon, the selected row must correspond to the stated initial state. A failure of these checks can indicate reversed multiplication order, an incorrect row, omitted destinations, or inconsistent update matrices. These checks verify arithmetic and probability accounting; they do not establish that the model or estimator is appropriate for the data.

The usual interpretation also depends on the model and estimation conditions supporting the transition-probability estimates. The states must be defined consistently, and the event-time updates and risk sets must represent the process being studied. If the assumptions needed for the chosen estimator fail, a row can still sum to one while its entries do not estimate the intended probabilities. Uncertainty in the estimated entries is a separate issue from calculating and interpreting the point estimates.

Common Pitfall: Occupancy Is Not a Direct Transition

A common mistake is to read \(\widehat{P}_{ij}(s,t)\) as the estimated probability of a direct \(i\)-to-\(j\) transition. The correct interpretation is the probability of being in state \(j\) at \(t\), conditional on being in state \(i\) at \(s\). As the second example shows, this can include paths through intermediate states. Another distinct quantity is the probability of ever visiting state \(j\) by time \(t\); a subject who visits \(j\) and leaves before \(t\) may count toward that probability but not toward occupancy of \(j\) at \(t\). When the question concerns a particular path, a direct transition, or ever entering a state, the corresponding probability must be defined and estimated for that event rather than inferred from a single fixed-horizon matrix entry.

A fixed-horizon summary is useful because it puts all possible states on a common probability scale at a clinically meaningful time. It can show how the distribution of patients across states changes over time and can support clear comparisons between starting states or specified initial populations. Its limitation is equally important: a single horizon does not describe when transitions occurred or how many transitions a subject experienced along the way.