The Clock-Reset Mistake In Semi-Markov Models

Why the Clock Origin Matters

A transition hazard is always defined relative to a time scale. In a semi-Markov model, the relevant clock for a transition out of a state is often time spent in that state. The clock begins when the person enters the current state, not automatically when they enter the study. Confusing these origins can assign a person the wrong transition hazard and therefore produce incorrect predicted risks.

Consider an illness-death process with states 0 (illness-free), 1 (ill), and 2 (dead), where 2 is absorbing. A person who enters state 1 four years after study entry has spent zero years in state 1 at that moment. At a later study time of six years, that person has spent two years in state 1. If death risk depends on duration of illness, the relevant hazard at that point is evaluated at duration two, not at study time six.

In this tutorial, study time \(t\) runs from study entry, while duration \(u\) resets to zero at entry into the current state. For an illness-death path that begins in state 0 at study entry and enters state 1 at time \(s\), the duration in state 1 at time \(t\geq s\) is \(u=t-s\). A hazard written as a function of \(t\) instead of \(u\) is not automatically wrong: it represents a different assumption, namely dependence on calendar or study time rather than duration in the current state.

Hazards and Risks on the Reset Clock

The transition-specific hazard \(\lambda_{ij}(u)\) is the instantaneous rate of transition from state \(i\) to state \(j\) at duration \(u\), conditional on having remained in state \(i\) up to that duration and on the stated history and covariates. Its units are inverse time. If the process can leave state \(i\) for several destinations, the total exit hazard is the sum of the destination-specific hazards. The probability of still occupying state \(i\) through duration \(u\), conditional on entry into that state, follows from that total hazard.

The risk integral accounts for the fact that a person can make the transition to \(j\) at duration \(v\) only if they have remained in state \(i\) until \(v\). For a single possible exit, \(S_i(u)\) reduces to \(\exp\{-\int_0^u\lambda_{ij}(v)\,dv\}\), and the probability of that exit by duration \(u\) is \(1-S_i(u)\).

For an illness-death path with illness at study time \(s\) and death at time \(t\), the relevant state-1 duration at death is \(t-s\). Under a duration-based model, the path's state-1 contribution to the transition likelihood has the form \(\lambda_{12}(t-s)\exp\{-\int_0^{t-s}\lambda_{12}(v)\,dv\}\). If the person is censored while still ill at duration \(c\), that transition contributes survival through duration \(c\), rather than a death hazard at the censoring time.

Deriving the Risk Calculation

The competing-exit formula follows by considering a short duration interval of width \(du\). Given survival in state \(i\) to duration \(u\), the probability of an exit to \(j\) in that interval is approximately \(\lambda_{ij}(u)\,du\). The probability of reaching \(u\) without leaving state \(i\) is \(S_i(u)\), so the unconditional probability of an exit to \(j\) in that interval is approximately \(S_i(u)\lambda_{ij}(u)\,du\). Summing over small intervals and taking the limit gives the integral above.

For piecewise-constant hazards, the calculation can be done interval by interval. If the hazards for exits \(j\) and \(k\) are constant within an interval of length \(d\), with total hazard \(q=\lambda_{ij}+\lambda_{ik}\), then survival through the interval is \(e^{-qd}\). The probability of exit to \(j\) during it, conditional on being at risk at its start, is \((\lambda_{ij}/q)(1-e^{-qd})\). This follows by integrating \(e^{-qv}\lambda_{ij}\) from \(v=0\) to \(d\).

Competing Exits Must Share the Same Duration Scale

If multiple transitions can leave state \(i\), each cause-specific hazard must be evaluated at the same duration \(u\) for the same person. The total hazard determines the chance of remaining in state \(i\); the cause-specific hazard determines which exit occurs. The following hypothetical calculation includes remission as an additional exit from illness to illustrate the competing-exit arithmetic. It is an extension of the basic illness-death structure, not a required feature of that structure.

Recognizing and Correcting the Mistake in Data

A practical check is to compute both times explicitly for every transition out of a state. For a person entering illness at study time \(s\), a death at \(t\) has state-1 duration \(t-s\), and censoring at \(c\) has state-1 follow-up duration \(c-s\). A model described as duration-dependent should use those durations to define its hazard and the period at risk.

Assumptions, Diagnostics, and Alternatives

A duration-based calculation is appropriate when the modeled hazard is defined as a function of time since entry into the current state, possibly conditional on specified history or covariates. It also requires accurate transition-entry times and correct identification of the current state. For interpreting a fitted hazard as a conditional transition rate, the risk set must consist of individuals still in that state and under observation at the relevant duration. Censoring assumptions must be appropriate for the estimation method; for example, independent censoring conditional on modeled information is commonly required for standard nonparametric or likelihood-based risk estimation.

Diagnostics begin with the data representation: verify that each state-entry time is recorded and that each subsequent transition or censoring time is converted to a nonnegative duration. Plotting or tabulating events by both study time and state duration can reveal whether a purported duration effect was accidentally indexed by study time. Compare records with different entry times: if a duration-specific hazard is intended, individuals at the same duration should be assessed on that duration scale, while any calendar-time effect should be represented separately rather than silently substituted for duration.

A clock reset is not universally correct. A calendar-time model uses \(\lambda_{ij}(t)\) when risk depends on the time origin of the study, calendar date, or another absolute-time scale. An age-time model might instead use attained age. A duration model uses \(\lambda_{ij}(u)\). These answer different questions; the choice is scientific and must be explicit. A model may also include both duration and calendar-time information, but that requires a specification capable of distinguishing their contributions.

Common warning signs include durations that continue increasing across a state transition, a change point applied at the same study time to people who entered the state at different times despite a stated duration effect, and risk sets grouped by study time when the target hazard is duration-specific. A numerical disagreement between duration and calendar-time calculations is not itself proof that one is wrong: it shows that they encode different assumptions. The error is using one clock while interpreting results as if the other had been used.

The duration clock also has limitations. Duration-only hazards may be inadequate if the same time in state has different risks for people with distinct prior histories, or if calendar time affects risk. Conversely, adding history or calendar-time structure without sufficient data can make estimates unstable. The conditional history and time scale therefore need to be stated alongside any transition-risk result.