Conditional Transition Probabilities From Each Origin State

Why the Origin State Matters

A transition-probability matrix contains probabilities for moving between every pair of states over a stated time interval. Its rows are not interchangeable: each row answers a question conditional on a different origin state. For example, the probability of being in state 2 at month 12 may be quite different for someone who began in state 0 than for someone who began in state 1.

The key to interpreting any entry is to read its row first. The row identifies the state at the start of the interval; the column identifies the state at the end. A probability without its origin state and time interval is incomplete.

This conditional probability describes a group defined by its state at time \(s\), not a probability conditional on having followed one particular path before \(s\). If the process is Markov, its future transition distribution, given its present state and time, does not additionally depend on its earlier path. That condition supports interpreting the matrix as a transition law that depends on the current state and time. Without it, \(P\{X(t)=j\mid X(s)=i\}\) remains a well-defined conditional probability, but it can average over different earlier histories among individuals in state \(i\).

For an estimated transition matrix to have the intended state-probability interpretation, the state definitions and time points must be clear, and the underlying estimation assumptions must be appropriate. In particular, a Markov interpretation requires the Markov condition just described. Earlier material on the Aalen–Johansen estimator gives the estimation framework; here the emphasis is on reading the resulting entries correctly.

Reading a Row and Selecting an Entry

Consider a hypothetical illness-death model with states 0 (alive without the illness), 1 (alive with the illness), and 2 (dead). Suppose the estimated transition-probability matrix from baseline to 12 months is

The first row applies to people in state 0 at baseline. For example, the entry in row 0 and column 2 is 0.08, so the estimated probability of being dead at 12 months, conditional on being alive without the illness at baseline, is 0.08. The second row applies instead to people already in state 1 at baseline: its row-1, column-2 entry is 0.35.

The diagonal entries are probabilities of occupying the same state at the end of the interval as at its start. Thus, 0.72 is the estimated probability of being in state 0 at 12 months given state 0 at baseline. It does not say that 72% of everyone in the study is in state 0 at 12 months; it concerns only the specified origin group.

Each row is a probability distribution over the possible destination states. If the state set is \(\{0,1,\ldots,K\}\), then for every origin \(i\),

The row total is one because, conditional on being in state \(i\) at time \(s\), the process must occupy exactly one of the defined states at time \(t\). This property is also a useful check on an estimated matrix. Small departures from one may result from rounding the displayed entries; large departures suggest a calculation, transcription, or model-output problem.

Calculating Probabilities Across Successive Intervals

A requested interval may span multiple periods for which separate transition matrices are available. When the Markov condition holds, the probability of moving from \(i\) to \(j\) over the full interval can be calculated by considering each possible state \(k\) at the intermediate time. The process must be in some such state at that time, so the origin-to-destination probability is the sum of the probabilities for the possible intermediate routes.

In matrix form, this is \(P(s,u)=P(s,t)P(t,u)\). The first matrix acts over the earlier interval and the second over the later interval. Keeping the order and the time labels straight is essential: an entry in the product still has a specific origin row and destination column.

Checks, Alternatives, and Common Pitfalls

Before reporting a selected transition probability, verify the origin state, destination state, and interval. Then check that the selected entry is nonnegative and that the row is a probability distribution. For a multi-interval calculation, confirm that adjacent matrices refer to consecutive intervals and that their multiplication order matches the time order. If states are ordered differently in two matrices, a numerically valid product can still answer the wrong question.

The main alternative to reading a transition-probability matrix is to work directly with transition intensities or event-time updates. Intensities describe instantaneous transition rates; a transition matrix describes probabilities over an interval. The matrix is often easier for answering questions such as “what is the probability of state \(j\) at month 12, given state \(i\) at baseline?” Intensity-based calculations can be useful when the underlying rates are the quantities of interest or when constructing the transition matrix. They are not interchangeable: a rate is not itself an interval probability.

A further distinction is between a conditional row and an overall probability that combines people from different initial states. A row answers a question for one specified origin. If an overall probability for a population with several origin states is required, the origin groups must be combined using their relevant proportions. That population-level quantity is not any single row entry.

• Wrong row: selecting a destination column correctly but using the wrong origin row changes the conditioning group. State the origin in words before looking up the entry.
• Ignoring the interval: a probability from baseline to 12 months is not automatically the probability for months 6 to 18. Report both endpoints.
• Treating an entry as an unconditional proportion: a row entry is conditional on its origin state. Do not describe it as a percentage of everyone in the study.
• Adding probabilities from different routes without accounting for the intermediate state: each route must include both the probability of reaching its intermediate state and the probability of the subsequent transition.
• Overinterpreting an estimate: a transition probability is not a causal effect, and a point estimate alone does not describe its uncertainty.

These interpretations rely on the state system being sufficiently well defined for the question, on using the correct time-specific matrix, and—when interpreting the matrix as a history-independent transition law—on the Markov condition. Sparse data in an origin state, model misspecification, or poorly defined states can make estimates unstable or their interpretation less useful. Rounding can also cause displayed row totals to differ slightly from one, so calculations should use unrounded estimates when available.