Duration-Dependent Transition Intensities

Why Duration Can Matter

Two people can occupy the same state at the same calendar time but have different transition risks because they have spent different amounts of time there. A recently diagnosed patient and a patient who has remained in a disease state for years, for example, need not have the same instantaneous risk of leaving that state. A model that assigns one fixed transition rate to everyone in the state cannot represent this difference.

A duration-dependent transition intensity makes the risk of moving from state \(i\) to state \(j\) depend on the elapsed time since entry into \(i\). This is a natural extension of the Markov model, in which transition intensities depend on the current state but not on how long it has been occupied. The earlier material on the Markov property and sojourn time establishes why elapsed time can be part of the information needed to describe future transitions.

The definition is conditional on still being in state \(i\) at duration \(u\). For small \(h\), the transition probability is approximately \(\alpha_{ij}(u)h\); it is not the probability of eventually entering \(j\), nor a probability that remains fixed over a long interval. If state \(i\) has several possible destinations, each destination has its own intensity. Under the usual assumption that simultaneous transitions have probability of smaller order than \(h\), the probability of any exit in \((t,t+h]\) is approximately the sum of the destination-specific intensities multiplied by \(h\).

From Intensities to Exit Risk

Suppose the possible exits from state \(i\) are to states in a set \(J_i\). The total exit intensity at duration \(u\) is $$ \lambda_i(u)=\sum_{j\in J_i}\alpha_{ij}(u). $$ It describes the instantaneous rate of leaving \(i\), conditional on still occupying \(i\) at duration \(u\). The destination-specific intensities describe how that instantaneous exit rate is allocated among destinations.

A useful consequence is the relationship between the total intensity and the probability of remaining in state \(i\). Assume the intensities are nonnegative and locally integrable, and that the model's transition mechanism is specified by these intensities conditional on the current state and duration. Write \(S_i(u)\) for the probability of no exit from \(i\) during the first \(u\) units after entry, for an individual entering at duration zero under the model. Over a short additional interval \(h\), conditional on remaining in \(i\) until duration \(u\), the chance of exit is \(\lambda_i(u)h+o(h)\). Thus $$ S_i(u+h)=S_i(u)\{1-\lambda_i(u)h+o(h)\}. $$ Subtract \(S_i(u)\), divide by \(h\), and let \(h\) decrease to zero. At durations where the derivative exists, this gives \(S_i'(u)=-\lambda_i(u)S_i(u)\). Since \(S_i(0)=1\), integrating the differential equation yields $$ S_i(u)=\exp\left\{-\int_0^u\lambda_i(v)\,dv\right\}. $$

This derivation also shows why an intensity and a fixed-interval probability should not be confused. The probability of at least one exit by duration \(u\) is \(1-S_i(u)\), which depends on the accumulated total intensity, not just its value at the endpoint. If there are multiple possible destinations, the same total exit probability applies, but the destination of an exit depends on the separate intensities.

What Different Duration Patterns Mean

The shape of \(\alpha_{ij}(u)\) gives a conditional, instantaneous description of how the transition risk changes as state duration increases. A rising curve indicates that, among individuals still in state \(i\) at each duration, the instantaneous rate of transition to \(j\) is higher at greater durations. A falling curve indicates a lower instantaneous rate at greater durations. Neither statement says that every individual follows the same path or that an individual’s risk necessarily changes causally because time has passed.

A duration effect can be monotone or nonmonotone. Risk may rise soon after entry, peak, and then decline; it may fall initially and later rise. Such patterns can arise when the mechanisms associated with leaving a state change over the course of the sojourn. A flexible model should allow the data and scientific context to support the shape rather than impose a trend by default.

Assumptions, Model Choices, and Diagnostics

A duration-only model makes a substantive sufficiency assumption: once the current state and the elapsed time in it are known, the transition intensities do not additionally depend on calendar time or on the earlier route through the state system. It also assumes that entry into a state resets the duration clock for that state. The clock-reset and clock-forward conventions distinguish which time scale is used; this tutorial's duration-dependent formulation uses time since the relevant entry. If prior history or calendar time matters beyond duration, a duration-only specification may be inadequate.

A constant intensity is the special case \(\alpha_{ij}(u)=c_{ij}\) for each transition. It is simpler and corresponds to the familiar Markov specification when rates depend only on the current state. A duration-dependent model is more flexible, but flexibility has a cost: it needs enough observed transitions across the duration range to estimate the shape. A piecewise-constant intensity can be easier to communicate and estimate, while a smooth parametric curve can be compact but risks imposing a wrong shape. Flexible smoothers can represent nonmonotone patterns but may be unstable where data are sparse.

Useful checks include plotting estimated transition rates or smoothed transition counts against duration, and examining how many individuals remain at risk at long durations. Compare observed transitions with model-based expected transitions across duration bands. Large discrepancies, systematic residual patterns, or conclusions that change sharply when the duration grouping or smoothing choices change are warnings of poor fit. Pay particular attention to the longest durations: few individuals may remain there, so an apparently dramatic late increase or decrease can be driven by very little information.

A further limitation is that duration patterns in observed data need not identify a changing individual-level mechanism. Individuals who remain in a state for a long time may differ from those who leave earlier; unmeasured heterogeneity can therefore contribute to an observed duration pattern. The intensity is a conditional model quantity for those still in the state, not by itself a causal effect of waiting longer. Scientific interpretation should account for relevant covariates and study design.

Common Pitfalls

• Calling the intensity a probability. An intensity has units of inverse time and can exceed 1. Multiply it by a sufficiently short interval to obtain a first-order probability approximation.
• Interpreting a rising intensity as a rising cumulative probability. Cumulative exit probability rises over time whenever exit is possible; the duration effect describes whether the instantaneous rate rises or falls.
• Using one intensity for all destinations without checking the question. The sum describes any exit. Separate transition intensities are required to describe destination-specific risk.
• Ignoring duration reset. If duration is time since entry, it restarts when the process enters the state again. Using time since the study began instead describes a different time scale.
• Trusting a tail pattern with little support. Few individuals at risk or few observed transitions at long durations make estimates there uncertain and potentially unstable.