Phase-Type Expansions For Non-Markov Sojourns

Why Represent a Sojourn With Hidden Phases?

A continuous-time Markov model has exponential holding times when the state is fully specified. If a clinically meaningful state instead has a non-exponential sojourn distribution, its duration contains information about how long the individual has already remained there. One way to retain a Markov description is to distinguish several unobserved transient phases within that state. Conditional on the current phase, the next transition has an exponential waiting time; the observed sojourn ends when the process leaves the collection of phases.

This construction is called a phase-type representation. It is useful when the target model needs a finite-state Markov formulation but the scientifically relevant state duration is not exponential. The phases are modeling states, not necessarily clinical events that are directly observed.

The initial distribution \(\boldsymbol{\alpha}\) specifies the phase at entry to the represented state. While the process is in phase \(r\), it may move to another transient phase according to the off-diagonal rates of \(S\), or be absorbed at rate \(s_r\). The diagonal entry \(S_{rr}\) is the negative total rate of leaving phase \(r\), including both transitions to other phases and absorption.

Sojourn-Time Distribution and Its Derivation

Let \(\mathbf{p}(t)\) be the row vector whose \(r\)th element is the probability that the process is in transient phase \(r\) at time \(t\), having started with distribution \(\boldsymbol{\alpha}\). The continuous-time Markov forward equation is \(\mathbf{p}'(t)=\mathbf{p}(t)S\), with \(\mathbf{p}(0)=\boldsymbol{\alpha}\). Its solution is \(\mathbf{p}(t)=\boldsymbol{\alpha}e^{St}\). The event \(T_{\mathrm{abs}}>t\) is exactly the event that the process is still in some transient phase, so summing the phase probabilities gives the survival function. Absorption at time \(t\) occurs at the total rate represented by \(\mathbf{s}\), giving the density.

Here \(h(t)\) is the instantaneous absorption hazard among processes that have not yet been absorbed. Unless the phase structure reduces to a single exponential holding time, this hazard will generally vary with elapsed time. The variation arises because the mix of phases among individuals still in the state can change over time.

If all eigenvalues of \(S\) have negative real parts, then \((-S)^{-1}\) exists and the absorption-time moments are \(\mathrm{E}(T_{\mathrm{abs}}^k)=k!\boldsymbol{\alpha}(-S)^{-k}\mathbf{1}\) for positive integers \(k\). In particular, the mean is \(\boldsymbol{\alpha}(-S)^{-1}\mathbf{1}\). This condition also ensures that the transient probability tends to zero, as required for eventual absorption.

From Absorption to a Multi-State Model

To embed a phase-type sojourn in a multi-state process, replace a clinical state \(i\) by its transient phases \((i,1),\ldots,(i,m_i)\). Transitions among these phases follow the transient subgenerator for state \(i\). An exit from a phase represents a transition out of clinical state \(i\). If the destination is \(j\), the phase-specific exit rates must specify both how quickly the exit occurs and where the process goes.

For a state with possible destinations \(j=1,\ldots,J\), let \(\mathbf{t}_{ij}\) be the column vector of rates from each phase of \(i\) to destination \(j\). The within-state phase subgenerator \(S_i\) must satisfy \(\sum_j\mathbf{t}_{ij}=-S_i\mathbf{1}\). On entry to \(i\), assign a phase according to \(\boldsymbol{\alpha}_i\). If the destination state \(j\) itself has phases, an exit to \(j\) enters phase \(s\) with probability \(\alpha_{j,s}\). Thus the expanded-chain rate from \((i,r)\) to \((j,s)\) is \(t_{ij,r}\alpha_{j,s}\). If \(j\) is absorbing, the exit goes directly to that absorbing state.

This rate construction makes the expanded process Markov: its next-transition rates depend on the current phase, not on the time already spent in that phase or the preceding path. When phases are not observed, the clinical-state process generally has non-exponential holding times and is not itself Markov.

Checking the Representation and Choosing an Approach

A phase-type construction imposes a specific structure, so its assumptions and fit should be checked rather than inferred from the fact that its transition rates are exponential. At minimum, verify that the initial probabilities are nonnegative and sum to 1, that off-diagonal entries of each transient subgenerator are nonnegative, and that the destination exit-rate vectors sum to the total rate of leaving the transient phases. Check also that absorption is certain. If the fitted model gives a nonzero probability of remaining in the phases forever, it does not define a proper finite sojourn distribution.

In practice, compare the implied survival curve and hazard with the intended or observed sojourn behavior, paying attention to early and late durations as well as the mean. A sharp mismatch can indicate too rigid a phase structure, an unsuitable initial phase distribution, or an omitted dependence on history or covariates. Different phase structures can also produce the same or very similar observable sojourn distributions, so the hidden phases may not be identifiable as unique scientific entities.

A direct semi-Markov specification describes the holding-time and destination distributions without requiring exponential phases. It can express the target duration law directly, but does not in general give a finite-state continuous-time Markov representation using only the clinical states. A phase-type representation instead supplies such a finite Markov construction, at the cost of choosing a phase structure and estimating its rates. Markovization by state augmentation is a broader modeling strategy: phase-type expansion is one specific way to use augmented states to represent a sojourn distribution.

Limitations and Practical Interpretation

The expanded chain is Markov only when the phase is included in the current state and the specified rates are adequate. If entry-phase probabilities or phase transitions depend on prior history, that dependence must be represented in the model rather than silently assumed away. Likewise, if an individual re-enters a clinical state, the model must specify the phase assignment on re-entry; using the same entry distribution each time is an assumption, not a consequence of the phase-type construction.

The hidden phases can be useful computational devices, but they should not automatically be interpreted as distinct biological stages. A good representation can approximate a desired sojourn law while remaining finite and Markov, yet it can require several phases and its parameters may be difficult to estimate from limited observations. The relevant check is whether the implied duration and destination behavior is suitable for the scientific question, not whether every phase has an independent clinical label.