Two Clocks, Two Different Questions
A transition risk can change because a person has remained in a state for a long time, or because the transition process changes over the calendar. These explanations are not interchangeable. For example, the chance of leaving a hospital state might rise with days since admission, while a new treatment protocol introduced in a particular month might change transition rates for everyone currently in that state. A model that uses only one clock can attribute one pattern to the other.
The distinction concerns the time scale indexing a transition intensity. It is not a distinction between two kinds of outcomes. For a person currently in state \(h\), the transition intensity to state \(j\) is an instantaneous rate conditional on being in \(h\) immediately before time \(t\) and on the relevant observed history. Assume a continuous-time process with no simultaneous transitions. Write \(S_h(t)\) for the time of the most recent entry into the current visit to state \(h\), and define the duration in that visit as \(u=t-S_h(t)\). Calendar time \(t\) is measured from a specified common origin, such as the start of follow-up.
In a duration-only specification, the intensity for a transition is a function of elapsed time in the current state, such as \(\lambda_{hj}(u)\). People at the same duration in the state have the same modeled transition intensity, subject to any other included covariates, even if they are observed on different calendar dates. In a calendar-time-only specification, it is a function of \(t\), such as \(\alpha_{hj}(t)\). At the same calendar time, people currently in state \(h\) have the same modeled intensity even if they entered that state at different times. These statements concern the model's conditional intensity, not necessarily the observed crude transition frequency.
A combined specification can allow both clocks to matter: \(\lambda_{hj}(u,t)\). For clarity, examples below suppress covariates and assume that, conditional on the stated intensity, no additional history affects the transition rate. If covariates are present, comparisons must hold their values fixed or account for them explicitly. A duration-only model is semi-Markov in the usual sense; a calendar-time-only model can be a time-inhomogeneous Markov model because its future transition rates depend on the current state and known calendar time, not on the time spent there.
From an Intensity to a Conditional Risk
An intensity is a local rate, not the probability of a transition over a whole interval. To translate one into a risk, specify how the rate behaves during the interval. Consider first a state with just one possible exit and a deterministic intensity \(\lambda(u,t)\). Conditional on entering the interval in the state, the probability of remaining there through its end is the exponential of minus the integrated intensity along the interval. The probability of exiting is one minus that survival probability.
The duration argument advances as calendar time advances: if the current visit began at \(S_h(t)\), then at a later time \(v\) the duration is \(v-S_h(t)\). With several possible destinations, let \(\lambda_{h+}(v)\) be the sum of the intensities over all exits from \(h\). The probability of no exit through \(t+\Delta\) is \( \exp\{-\int_t^{t+\Delta}\lambda_{h+}(v)\,dv\}\). The probability of exiting specifically to \(j\) is instead the integral of the probability of still being in \(h\) just before \(v\), multiplied by \(\lambda_{hj}(v)\). Thus one cause-specific intensity alone does not generally determine the probability of that destination when competing exits are possible.
Worked Example: Same Duration, Different Calendar-Time Risk
Suppose a hypothetical study follows people in state \(h\), with one possible exit. The calendar-time-only intensity is \(0.020\) per week before week 10 and \(0.035\) per week from week 10 onward. Assume it is constant within each of those periods, and assume no covariates or other history alter the rate. Compare two people who have each spent 2 weeks in \(h\): one is observed at calendar week 3, the other at week 11.
The first person is in the period with intensity \(0.020\) per week. The second is in the period with intensity \(0.035\) per week. Their durations match, but their modeled rates differ because their calendar times differ. For the next week, each person remains within the same calendar period, so the one-week risks are:
Under these assumptions, the one-week exit risks are about 1.98% at week 3 and 3.44% at week 11. These are conditional risks for people in \(h\) at the specified times, not probabilities that apply to everyone in the study. The example illustrates calendar-time dependence: equal duration does not imply equal intensity.
Worked Example: Same Calendar Time, Different Durations
Now suppose, hypothetically, that the only exit intensity is duration-dependent: it is \(0.010\) per week before 4 weeks in state \(h\) and \(0.030\) per week from 4 weeks onward. There are no covariates or other history effects. At calendar week 10, person A entered \(h\) at week 2 and has duration 8 weeks; person B entered at week 8 and has duration 2 weeks.
At week 10, A's intensity is \(0.030\) per week and B's is \(0.010\) per week. Over the next week, A remains above the 4-week duration threshold. B reaches duration 3 weeks and remains below it. Therefore the one-week conditional risks are:
The corresponding risks are about 2.96% for A and 0.995% for B, despite both being observed at the same calendar time. The difference follows from their durations in the state, not from the calendar week.
When Both Clocks Matter—and When They Cannot Be Separated
A combined model can express separate duration and calendar effects. For example, the intensity might increase at a duration threshold and also increase after a policy change. The following additive model is deliberately simple; it represents a nonnegative, piecewise-constant rate and is not a claim that effects must be additive in general.
Worked Example: A Joint Duration and Calendar Effect
Suppose a hypothetical transition from \(h\) has intensity \(\lambda(u,t)=0.010+0.020I(u\geq 4)+0.015I(t\geq 10)\) per week, where \(I(\cdot)\) equals 1 when its condition holds and 0 otherwise. Assume the rate is constant over each one-week interval considered, there is one possible exit, and no covariates or unmodeled history affect the rate.
At calendar week 11, compare a person with duration 2 weeks and a person with duration 6 weeks. Both have the calendar-time indicator equal to 1. The first person's rate is \(0.010+0+0.015=0.025\) per week; the second person's rate is \(0.010+0.020+0.015=0.045\) per week. Their one-week risks are:
At duration 2 weeks, compare calendar week 8 with calendar week 11. The duration indicator is 0 in both cases, while the calendar indicator changes from 0 to 1. The rates are \(0.010\) and \(0.025\) per week, giving one-week risks \(1-e^{-0.010}=0.009950\) and \(1-e^{-0.025}=0.02469\). In this specified model, comparisons at fixed calendar time reveal the duration term, while comparisons at fixed duration reveal the calendar term. These are model-based contrasts; the data must contain people supporting the comparisons for the effects to be estimated reliably.
There is a basic identification problem when entry times do not vary. If everyone enters \(h\) at the same calendar time \(s_0\), then throughout a visit \(u=t-s_0\). A duration function \(g(u)\) can be rewritten as a calendar-time function \(a(t)=g(t-s_0)\). The observed intensity pattern alone cannot determine which clock generated it.
A Short Identification Argument
This equivalence follows directly from substituting the common entry time. Suppose the observed intensity among people in \(h\) is represented by a duration-only function \(g(u)\). If every visit begins at \(s_0\), then at each observed time \(t\), duration is \(t-s_0\), so the same intensity can be written as a calendar-time function:
Conversely, any calendar-time function \(a(t)\) on this observation pattern can be expressed as a duration function \(g(u)=a(u+s_0)\). The two descriptions make identical predictions for these data. Variation in state-entry times can break this exact equivalence by creating observations with the same duration at different calendar times and the same calendar time at different durations. But variation alone does not guarantee a useful comparison: there must be adequate overlap, accurate entry times, and no uncontrolled cohort differences that are mistaken for a time-scale effect.
The calendar origin must also be meaningful and defined consistently. Shifting the origin changes the numerical values of \(t\), though it need not change a model that is transformed accordingly. Calendar time may represent study time, attained age, or time since a policy began; these are different scientific questions. Calling a variable “calendar time” without stating its origin can obscure the interpretation.
Model Choice, Checks, and Limits
| Specification | Intensity depends on | Interpretation | Main caution |
|---|---|---|---|
| Duration-only | Time since the latest entry into the current state | Risk changes as the state visit continues | May miss calendar events or cohort changes |
| Calendar-time-only | Absolute time on a stated scale | Risk changes with the common calendar | May miss duration patterns within state visits |
| Combined | Both duration and calendar time | Allows each clock to contribute, conditional on the other | Needs data with adequate overlap to distinguish the effects |
A practical check is to ask what comparisons the observed data actually support. Plotting or tabulating transitions by both duration and calendar period can reveal empty or sparse combinations, abrupt changes, or apparent patterns that occur only in one entry cohort. Such summaries are descriptive: differences in entry cohort, covariates, censoring, or observation practice may explain them. Formal model diagnostics and tests require additional choices and are distinct from defining the time scale.
The main alternatives answer different questions. A duration-only model is suitable when the transition process is tied to the length of the current state visit and calendar conditions do not add relevant information. A calendar-time-only model suits a process whose rates change over the calendar while being conditionally independent of state duration. A combined model is more flexible, but it can be difficult to estimate when duration and calendar time are strongly aligned. If recent transitions or other history matter beyond these clocks, neither a duration-only nor a two-clock specification necessarily captures that dependence.
These models describe conditional transition rates; they do not establish that a calendar event caused a rate change. A policy change may coincide with changes in patient mix, care, or measurement. Likewise, a duration association does not show that remaining in the state itself causes risk to rise. Interpretation depends on the design, measured covariates, observation process, and assumptions about unmeasured differences.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two people have each spent 5 weeks in state \(h\), but one is observed before and one after a calendar-time rate change. Which time scale can explain different intensities if the model has no covariates?
- A duration-only intensity is \(0.040\) per week and remains constant for the next 2 weeks. What is the conditional probability of an exit during those 2 weeks?
- Everyone in a data set enters a state at calendar week 6. Explain why duration and calendar-time effects cannot be separated from the observed intensity pattern alone.
- At the same calendar time, people with longer durations show higher estimated transition rates. Name one reason this pattern may fail to establish a duration effect.
Solutions. (1) A calendar-time effect can explain different intensities despite equal durations; the model's rate changes across the calendar threshold. (2) Under the stated constant-intensity, single-exit assumption, the risk is \(1-e^{-0.040(2)}=1-e^{-0.080}=0.07688\), or about 7.69% over 2 weeks. (3) For everyone, duration is \(u=t-6\), so every duration pattern can be rewritten as a calendar-time pattern, and vice versa; the observed data do not distinguish the explanations. (4) Longer-duration people may differ in cohort, covariates, observation, or selection into remaining in the state. The comparison is descriptive unless these issues and the model assumptions are addressed.