Direct and Indirect Routes to an Endpoint

Why distinguish routes to an endpoint?

Knowing that a person reached an endpoint does not necessarily tell us how they got there. In an illness-death model, for example, a person may die while still in the initial state or may first become ill and die later. Both histories end in death, but they represent different observed sequences of transitions. Distinguishing those routes can describe clinical progression, identify where events occur along a pathway, and clarify what an endpoint-only summary leaves out.

The route is a feature of the person’s observed state history, not simply another name for the endpoint. A route-specific probability therefore answers a different question from the overall probability of reaching the endpoint. It also does not, by itself, explain why one route occurred or establish a causal mechanism.

Definitions and the route decomposition

Consider the three-state illness-death structure: state 0 is alive and not ill, state 1 is alive and ill, and state 2 is dead. The allowed transitions are \(0\to1\), \(0\to2\), and \(1\to2\). State 2 is the endpoint. For a person starting in state 0, the direct route is \(0\to2\); the indirect route is \(0\to1\to2\).

Let \(D(t)\) denote the event that a person follows the direct route and reaches state 2 by time \(t\), and let \(I(t)\) denote the event that the person follows the indirect route and reaches state 2 by time \(t\). If everyone starts in state 0, transitions are limited to those listed above, and state 2 is absorbing, then these events are disjoint. They also exhaust the ways of reaching state 2 by \(t\). Thus:

This is a partition of the endpoint event by observed route. It does not require the two route probabilities to be equal, nor does it imply that one route causes the other. The equality depends on the route definitions and allowed transitions: if a model permits other ways to reach state 2, they must be included as additional route events.

Why route probabilities depend on timing

The indirect route requires two transitions to occur in the right order before the follow-up deadline: first \(0\to1\), then \(1\to2\). A person who becomes ill close to \(t\) has less time to reach death by \(t\) than a person who becomes ill early. Consequently, multiplying an overall probability of illness by an overall probability of death does not generally give the indirect-route probability.

A precise expression can be written using the time and type of the first transition out of state 0. Let \(S\) be the time of that first transition, and let \(J\) identify its destination. For the indirect route, the first transition must be to state 1. By conditioning on when that transition occurs, the probability of the indirect route by \(t\) is:

This is the law of total probability: sum the probability of entering state 1 at each possible time \(s\), weighted by the conditional probability of reaching state 2 by \(t\) after that entry. The conditional probability retains the relevant history; no assumption that it is constant across entry times is required. In continuous time, the integral notation represents accumulation across the possible entry times.

The direct-route probability, in contrast, is the probability that the first transition out of state 0 is to state 2 by \(t\). The two route calculations therefore refer to different transition histories. Route-specific probabilities can be estimated or described without asserting that intermediate-state entry is random or independent of later transitions.

Interpreting routes in treatment settings

A treatment setting can use the same route distinction when an endpoint may occur either directly or after an intermediate clinical state. For example, state 0 might mean alive without a documented response, state 1 alive after response, and state 2 death. Death in state 0 is then direct; death after response is indirect. The scientific meaning depends on how states and transitions were defined: a response state should be based on a clear assessment rule, and follow-up times should make the sequence interpretable.

Choosing between route-based and endpoint-only summaries

An endpoint-only summary, such as the probability of death by a specified time, is compact and may be the primary clinical outcome. A route-based summary adds detail by separating that probability into mutually exclusive histories. Neither is universally preferable: the choice should match the scientific question, the state definitions, and the information available in follow-up.

The route categories must be specified before interpreting the numbers. If people can visit the intermediate state more than once, recover, or reach the endpoint by additional sequences, one must decide whether routes are defined by the first visit, the complete history, or some other rule. In such settings, the simple two-term decomposition may not be exhaustive, and histories may not fall into just one category unless the rules make them mutually exclusive.

Diagnostics, assumptions, and common pitfalls

Before calculating route probabilities, check that the starting state, endpoint, intermediate state, and follow-up horizon are explicit. Verify that the listed routes are allowed by the state structure and that they cover all endpoint histories under consideration. For a simple proportion calculation, also check that each person is counted once, follow-up is complete to the chosen time, and the route counts sum to the endpoint count. If follow-up is censored, the observed fraction of the original cohort with a recorded event is not generally the endpoint probability; appropriate methods for censored multi-state data are needed.

A route analysis is also vulnerable to state misclassification and timing errors. For example, a response recorded after an endpoint cannot count as an intermediate state on the path to that endpoint. Investigators should check that event dates preserve the claimed order and that ambiguous or simultaneous transitions have a prespecified handling rule. If the observed route counts fail to add to the total endpoint count, investigate missing histories, unrepresented transitions, and inconsistent definitions before interpreting the discrepancy.

• Common mistake: multiplying marginal probabilities. “Twenty percent become ill and 25% die, so 5% follow the indirect route” is not justified unless the second percentage is the appropriate conditional probability among those who entered illness, over the remaining follow-up. Use the conditional group and time horizon that correspond to the path.
• Common mistake: treating route differences as mechanisms. A smaller indirect-route proportion does not prove that a treatment prevented death by preventing response or illness. Route categories describe observed histories, and post-treatment state membership can select different groups.
• Common mistake: assuming two routes always cover the endpoint. If additional transitions are allowed, the two-route sum can omit endpoint histories. Enumerate the allowed paths and include every relevant route.
• Common mistake: calling the route probability an endpoint transition probability. The probability of a whole path from initial state to endpoint is not the same quantity as the probability of one particular transition out of a state.

The main limitation is that route-specific summaries inherit the quality and granularity of the state definitions. Combining clinically distinct situations into one state can hide meaningful differences; splitting states too finely can leave sparse route counts. Route probabilities also depend on the time horizon: a person entering the intermediate state late may not have enough observed time to reach the endpoint by the cutoff. Report the horizon and state definitions alongside the probabilities.