Event-Order Dependence In Multi-State Paths

Why Event Order Can Matter

A person can arrive in the same state after different sequences of events. Those sequences may carry different information about what happens next, even when the person has spent the same amount of time in the current state. A model based only on current state cannot distinguish the paths; a semi-Markov model based on current state and sojourn duration still cannot distinguish them if the durations match. Event-order dependence gives the transition risk an additional dependence on the sequence of earlier events.

The purpose here is to make that dependence precise and to show how to specify it. The focus is not on accumulating time or exposure across states: it is on whether the order of events in a path predicts the next transition.

Definition: Let \(X(t)\) denote the state occupied at time \(t\), and let \(H_{t-}\) denote the observed event history immediately before \(t\): the states visited and the times of transitions up to, but not including, \(t\). If the process is in state \(j\) at \(t\), its sojourn duration is \(s=t-T\), where \(T\) is the time it most recently entered \(j\). An event-order effect is dependence of a next-transition distribution on the order of events in \(H_{t-}\), beyond any dependence on the current state and sojourn duration.

For a transition from state \(j\) to state \(k\), the history-dependent transition intensity is

\lambda_{jk}(t\mid H_{t-}) = \lim_{\Delta t\downarrow 0} \frac{P\{X(t+\Delta t)=k\mid X(t)=j,H_{t-}\}}{\Delta t}.

The intensity is a conditional instantaneous rate, typically expressed per unit time. It is not itself a probability. For a sufficiently short interval \(\Delta t\), the conditional probability of a \(j\to k\) transition is approximately \(\lambda_{jk}(t\mid H_{t-})\Delta t\), provided the intensity remains finite and the probability of multiple jumps in that interval is negligible.

Constructing Matched Histories

Consider a process with states 0, 1, 2, and 3. Suppose the allowed transitions include \(0\to1\), \(0\to2\), \(1\to2\), \(2\to1\), \(1\to3\), and \(2\to3\). After entry to state 3, suppose transitions \(3\to4\) and \(3\to5\) are possible. The paths

0\to1\to2\to3 \qquad\text{and}\qquad 0\to2\to1\to3

end in the same state, but the order of states 1 and 2 is reversed. If both paths enter state 3 at time 8 weeks and are observed at time 10 weeks, each has current state 3 and sojourn duration 2 weeks. They differ in event order, not current state or time spent there.

To model this contrast, define \(O(H_{t-})=1\) when the path entered state 3 by \(0\to1\to2\to3\), and \(O(H_{t-})=0\) when it entered by \(0\to2\to1\to3\). This is a deliberately specific history summary. In an application, its definition must be determined before fitting the model and must be computable from the recorded path.

Worked Example: Matching State and Duration While Reversing Order

Two hypothetical individuals follow the allowed transitions above. Individual A visits state 1 at week 2, state 2 at week 5, and state 3 at week 8. Individual B visits state 2 at week 2, state 1 at week 5, and state 3 at week 8. Both are observed at week 10, with no further transition by that time.

For A, the history ending in state 3 is \(0\to1\to2\to3\); for B, it is \(0\to2\to1\to3\). Both entered state 3 at week 8 and are currently in state 3 at week 10. Their durations are therefore \(10-8=2\) weeks. Thus \(O=1\) for A and \(O=0\) for B, while current state and duration are identical.

This construction isolates the intended contrast: a model whose predictors are only current state and duration assigns these two histories the same transition risk, whereas a model that includes \(O\) can assign different risks. The example does not establish that order actually predicts risk; that is a modeling question for data from the application.

Specifying an Event-Order Effect

A useful starting point is a duration-dependent intensity model with a multiplicative event-order effect. For the two exits from state 3, one possible specification is

\lambda_{34}(s\mid H)=\lambda_{34,0}(s)\exp\{\beta_{34}O(H)\}, \qquad \lambda_{35}(s\mid H)=\lambda_{35,0}(s).

Here \(\lambda_{34,0}(s)\) and \(\lambda_{35,0}(s)\) are baseline intensities as functions of duration \(s\) in state 3. The coefficient \(\beta_{34}\) describes the event-order contrast for the \(3\to4\) transition. This particular specification allows order to change the intensity of the \(3\to4\) transition and assumes no order effect on \(3\to5\). A different scientific question could motivate an order term for that transition too.

Conditional on the same duration, the \(3\to4\) intensity ratio for the two specified orders is

\frac{\lambda_{34}(s\mid O=1)}{\lambda_{34}(s\mid O=0)} = \exp(\beta_{34}).

This is a conditional rate ratio, not a ratio of probabilities of ever making the transition. The probability of a particular exit also depends on the other possible exits and on how intensities vary over time.

Worked Example: Interpreting an Order Coefficient

Suppose, hypothetically, that the \(3\to4\) baseline intensity is \(0.05\) per week at a given duration and the fitted intensity for histories with \(O=1\) is \(0.20\) per week. Under the multiplicative model, the estimated coefficient is

\hat{\beta}_{34} = \log\left(\frac{0.20}{0.05}\right) = \log(4) \approx 1.386.

The corresponding conditional intensity ratio is \(\exp(1.386)\approx4.00\). At that duration, the instantaneous rate of transition \(3\to4\) for the \(0\to1\to2\to3\) history is four times the rate for the \(0\to2\to1\to3\) history, under this model. The rates are in transitions per week; the ratio has no units.

This is a conditional association under the specified model. It does not show that changing event order would cause an individual’s transition rate to change. Other features of the histories or individuals could account for an observed difference.

From Intensities to Next-Transition Probabilities

To see how a rate contrast affects next-transition risks, suppose that after entry to state 3 the two cause-specific intensities are constant over the next \(\tau\) weeks for a fixed history group: \(\lambda_{34}=a\) and \(\lambda_{35}=b\). Assume these are the only possible exits during this interval. Conditional on the history at entry, the probability of remaining in state 3 through duration \(u\) is \(S(u)=\exp\{-(a+b)u\}\): over a short interval the total exit rate is \(a+b\), so survival solves \(S'(u)=-(a+b)S(u)\), with \(S(0)=1\). A transition to state 4 at duration \(u\) has density \(S(u)a\). Integrating that density gives

P(\text{exit to 4 by }\tau\mid H) = \int_0^\tau a\exp\{-(a+b)u\}\,du = \frac{a}{a+b}\left(1-\exp\{-(a+b)\tau\}\right).

The competing exit matters: an individual who first transitions to state 5 cannot subsequently make a first exit from state 3 to state 4. The next example calculates this probability for the two event orders.

Worked Example: Comparing Two-Week Risks With Competing Exits

Suppose, hypothetically, that for \(O=1\), the constant intensities are \(a=0.20\) per week for \(3\to4\) and \(b=0.10\) per week for \(3\to5\). For \(O=0\), let them be \(a=0.05\) and \(b=0.10\) per week. Calculate the probability of a \(3\to4\) transition within 2 weeks for each history.

For \(O=1\), the total exit rate is \(0.20+0.10=0.30\) per week. Therefore,

P(3\to4\text{ by 2}\mid O=1) = \frac{0.20}{0.30}\left(1-e^{-0.30(2)}\right) = 0.6667(1-e^{-0.60}) \approx 0.3008.

For \(O=0\), the total exit rate is \(0.05+0.10=0.15\) per week, so

P(3\to4\text{ by 2}\mid O=0) = \frac{0.05}{0.15}\left(1-e^{-0.15(2)}\right) = 0.3333(1-e^{-0.30}) \approx 0.0864.

Under these hypothetical constant-intensity specifications, the two-week probability of a \(3\to4\) transition is about 0.301 for the first order and 0.0864 for the second. The corresponding cause-specific intensities differ fourfold, but the two-week probabilities do not: each probability also incorporates the competing \(3\to5\) transition and the time available to exit. The calculation assumes constant intensities over the two weeks and no other exits.

Assumptions, Diagnostics, and Alternatives

An event-order model is only interpretable if its histories and risk sets are defined consistently. In particular, the analyst needs sufficiently accurate transition times and a rule for identifying the relevant sequence. The intensity model must be appropriate for the intended comparison: the example above assumes a multiplicative order effect that does not vary with duration, and the probability calculation additionally assumes constant cause-specific intensities over its stated interval.

For an estimated order contrast to be supported by the data, both order groups need observations at comparable durations in state 3, with enough transitions and follow-up to estimate the relevant intensities. Sparse histories can produce unstable estimates; if one order is nearly absent at some durations, the comparison there relies heavily on model extrapolation. Check that paths are coded in chronological order, that entry to the current state is identified correctly, and that the order indicator is not accidentally defined using future events. Examine whether the assumed duration pattern and order effect are plausible across the observed follow-up. These checks do not prove the model correct, but they can reveal coding errors, poor overlap, and obvious lack of fit.

The main alternatives make different information available to the model:

Specification	Information used for the next-transition risk	Can distinguish matched histories above?
Markov	Current state	No
Semi-Markov	Current state and time since entry to it	No, if state and duration match
Recent-transition model	Current state and selected recent transitions	Only if its selected history terms distinguish the orders
Event-order model	Current state, duration, and a specified order summary	Yes, when the summary separates them

The recent-transition approach is one particular way to represent history. An event-order model makes the order contrast explicit; it may use just one indicator rather than the entire path. Neither approach automatically captures every relevant feature of a history. A fully history-conditioned intensity is conceptually broad, but often difficult to estimate because many distinct histories may have little or no data. A parsimonious summary such as \(O\) is easier to estimate, at the cost of treating paths with the same summary as equivalent for the modeled transition.

Common pitfall: “Both individuals are currently in state 3 and have been there for two weeks, so their next-transition risks must be equal.” That conclusion follows only if the model makes current state and duration sufficient for the risk. If the intensity depends on event order, equal state and duration do not imply equal risks. Conversely, observing different risks across orders does not establish a causal effect of rearranging events; it is a conditional predictive contrast unless stronger causal assumptions and a suitable design justify more.

The practical importance of this distinction is that a model can appear to describe sojourn duration correctly yet still miss systematic differences between paths. An order term can make those differences explicit and testable, but it does not remove the need to justify which histories are compared, how the order summary is chosen, and whether the observed data support the comparison.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two paths both enter state \(j\) at week 6 and are observed in \(j\) at week 9. One arrived by \(0\to1\to j\), the other by \(0\to2\to j\). What do they share, and what differs?
In a model \(\lambda_{jk}(s\mid H)=\lambda_{jk,0}(s)\exp\{\beta O(H)\}\), what is the conditional intensity ratio for \(O=1\) versus \(O=0\)?
With constant exits \(a=0.10\) and \(b=0.10\) per week, calculate the probability of exit by cause \(a\) within 1 week.
Why can a semi-Markov model still assign equal risks to two paths whose event order differs?

Solutions. (1) Both have current state \(j\) and duration \(9-6=3\) weeks; their preceding event orders differ. (2) The ratio is \(\exp(\beta)\), holding duration and other modeled conditions fixed. (3) Using \(a/(a+b)\{1-\exp[-(a+b)\tau]\}\), the probability is \(0.10/0.20\{1-e^{-0.20(1)}\}=0.5(1-0.8187)\approx0.0906\), or 9.06% within 1 week. (4) A semi-Markov specification conditions on current state and sojourn duration, not necessarily on the earlier order of events; if those two inputs match, it gives the same risk unless additional history information is included.