Transition Probabilities In Models With Absorbing States

Why Absorbing States Need Special Attention

Some states represent outcomes from which a person cannot return during the modeled follow-up. Death is a familiar example; in a competing-risks model, each event type may be represented by its own absorbing state. Once a subject enters such a state, the subject remains there. This simple structural feature has a useful consequence: for a fixed starting state, the probability of occupying an absorbing state can only stay the same or increase as the horizon moves forward.

That property is more than a visual check. It helps interpret the columns of a transition-probability matrix, distinguish a cause-specific cumulative probability from an event-free probability, and detect calculation errors. The discussion uses the row convention: rows index the starting state, columns index the state occupied at the later time.

Here \(q_{ij}(u)\) denotes the transition intensity from state \(i\) to state \(j\), for \(i\ne j\). The diagonal intensity is \(q_{ii}(u)=-\sum_{j\ne i}q_{ij}(u)\), so the row of the intensity matrix for an absorbing state is entirely zero. A transition-probability matrix has nonnegative entries and each row sums to one. Its diagonal entries are probabilities of still occupying the same state, not intensities.

The interpretation of \(P_{ia}(s,t)\) is conditional on the state at the starting time. It is not automatically the probability for a population with multiple starting states. If the population has initial-state probabilities \(\pi_i\), its overall probability of occupying \(a\) at \(t\) is \(\sum_i \pi_i P_{ia}(s,t)\).

Why the Absorbing-State Probability Accumulates

Assume a finite-state, time-inhomogeneous Markov model with transition intensities that are well-defined and integrable over the follow-up interval. The Markov assumption means that, conditional on the current state, the future transition behavior does not additionally depend on the earlier path. Under these conditions, the transition probabilities evolve according to the forward equation \( \frac{\partial}{\partial t}P(s,t)=P(s,t)Q(t)\), at times where the derivative exists. This is the continuous-time counterpart to updating a transition matrix as follow-up advances.

For a fixed origin state \(i\) and absorbing destination \(a\), the derivative of the relevant matrix entry is the \(i,a\) entry of \(P(s,t)Q(t)\). Because the absorbing-state row of \(Q(t)\) is zero, the diagonal term \(q_{aa}(t)\) contributes nothing. Thus, for almost every \(t\),

Every term in the sum is nonnegative: \(P_{ih}(s,t)\) is a probability, and \(q_{ha}(t)\) is a nonnegative transition intensity. The derivative is therefore nonnegative. Integrating over any interval from \(t_1\) to \(t_2\), where \(s\le t_1\le t_2\), gives the more explicit result

The increase comes from probability flowing into \(a\) from states outside \(a\). Probability already in \(a\) remains there; it does not flow back out. The same reasoning applies when there are several absorbing states, separately to each absorbing-state column.

In an event-time product-integral calculation, the corresponding argument is discrete. At an event time \(u\), let \(D(u)\) be the row-stochastic update matrix and write \(P(s,u)=P(s,u-)D(u)\). If \(a\) is absorbing, its row in \(D(u)\) has a 1 in column \(a\) and zeros elsewhere. For any origin \(i\), the change in its probability of occupying \(a\) is

The update therefore preserves accumulation at each event time as well as over continuous intervals. This calculation assumes the transition structure truly treats \(a\) as absorbing and that the update matrix is a valid probability update.

Worked Calculations

Reading and Checking the Matrix

For a fixed starting state, follow a row of \(P(s,t)\) as \(t\) changes. The entry in an absorbing-state column should be nondecreasing, while the row as a whole continues to sum to one. Probability assigned to that absorbing state is no longer available to any other state at later times. If there are multiple absorbing causes, probability can accumulate in each cause column; the sum across those columns is not generally one because some probability may remain in transient states.

In a competing-risks setting, the probability of occupying a cause-specific absorbing state by time \(t\), starting event-free, is the cumulative incidence for that cause under the model. It is not generally obtained by taking one minus a Kaplan–Meier estimate that treats other causes as censoring. That complement can allocate probability differently because it does not retain the other causes as absorbing outcomes in the state system.

Useful checks for a computed matrix include:

• Each entry lies between zero and one, and each row sums to one, allowing for rounding.
• For a state declared absorbing, its row has probability one of remaining in that state at every later horizon.
• For each origin state, the probability in an absorbing-state column does not decrease across later horizons.
• The probability increase at an event update agrees with probability entering that state from the other states.

A decrease beyond rounding tolerance is a warning to check the multiplication order, state labels, update matrices, and whether the state is in fact absorbing. A decrease can be legitimate only if the modeled state is not absorbing—for example, if recovery or re-entry is possible. Monotonicity should be checked for the same origin state and time scale; comparing different starting states does not create such a requirement.

Scope and Limitations

The accumulation result follows from the definition of an absorbing state; it does not require constant intensities. The constant-intensity formulas in the examples do require constant transition intensities and the stated transition structure. Event-time computations require valid probability-update matrices. Estimating such matrices from observed data additionally depends on the assumptions needed for the estimation procedure, including appropriate handling of censoring; an absorbing-state label alone does not guarantee that an estimate is unbiased.

The Markov formulation also matters: the current state must be adequate for the transition model being used. If transition behavior depends on unrepresented history, a model using only the recorded state may not describe the desired transition probabilities well. Finally, the probabilities describe modeled state occupancy for a defined origin state and horizon; by themselves they do not establish causal effects or explain why transitions occur.