Initial States, Entry Rules, and Eligibility

Why the Starting Point Needs a Rule

A multi-state analysis follows a process through states, but the scientific time origin and the start of observation need not coincide. A participant may enter observation after the time origin, and different participants may be in different states when they enter. If those details are left implicit, the state diagram can suggest that everyone began in one state when the data do not support that claim.

The entry rule also defines the population being described. For example, a study of people eligible at age 40 who are observed only when they attend a clinic after age 50 is not automatically a study of all eligible people from age 40 onward. It directly observes a selected group from their later entry times. The scientific question, eligibility criteria, and observation schedule therefore need to be stated together.

Eligibility criteria are rules for membership in the analysis population: for instance, meeting an age range, having a specified diagnosis at screening, or having a recorded state at enrolment. The initial state must be defined relative to a time point. It can mean the state at the scientific origin, or, in an analysis explicitly defined from entry, the state at each person’s entry time. Those two choices describe different starting distributions and may answer different questions.

Eligibility, Entry, and the Observable Process

A useful analysis specification names (1) the eligible population, (2) the time origin, (3) the rule that determines \(L_i\), (4) which state is recorded at entry, and (5) when observation ends. Write \(C_i\) for the last time at which follow-up can provide an observed state. In a simple setup, a person is observable at \(t\) only if they have entered and follow-up has not ended; more generally, \(O_i(t)\) records whether the state is in fact observed, including any gaps in ascertainment.

The risk set for a transition out of state \(j\) at time \(t\) includes eligible people whose observation covers \(t\) and whose state immediately before the transition is \(j\). With \(X_i(t-)\) denoting the state just before \(t\), a corresponding set is:

The convention at event and censoring times must be stated when exact ties matter. The expression above focuses on observation immediately before the event time; it does not imply that someone whose follow-up ended earlier remains at risk. In particular, entry before \(t\) alone is not enough for inclusion.

If a state is observed for everyone currently under observation at \(t\), the empirical state proportion among those observed is:

This is a descriptive proportion among the eligible people whose states are observed at \(t\). It is not automatically an estimate of the state proportion in the full target population: entry and follow-up may select who is observed. Methods that generalize beyond the observed group need assumptions about that selection, as well as appropriate handling of incomplete follow-up.

Delayed Entry and Multiple Entry States

Delayed entry occurs when observation begins after the chosen scientific time origin, so \(L_i>0\). For example, if time is measured from diagnosis but a person is recruited six months later, that person has delayed entry on the diagnosis-time scale. Their known state at recruitment is \(X_i(L_i)\); their earlier state history is not observed merely because the analysis uses diagnosis as time zero.

Delayed entry changes who can contribute information at each time. Someone entering at time 6 cannot be included in a risk set at time 4. Someone whose observation ended at time 4 cannot be included at time 6. When estimating quantities for a population broader than those observed after entry, the analyst must consider the entry and follow-up mechanisms. A standard condition for generalizing transition experience after entry is that, conditional on relevant measured variables and the entry state, entry is not informative about the subsequent process being estimated, with adequate representation across the covariate and entry-time ranges. This condition does not recover unobserved pre-entry histories by itself. If entry depends on the process in ways not accounted for, analyses among entrants may not represent the target population.

A multi-state cohort may also have several possible entry states. This is not a problem if the states and allowed transitions are defined clearly. Participants can begin follow-up in different nodes of one state structure; the initial or entry-state distribution then records how many enter in each state. A participant observed entering in state \(B\) should not be assigned a prior transition from state \(A\) unless that transition is known from data or required by the scientific definition. A transition that occurred before observation is not made observable by drawing it in the diagram.

Different starting states can also imply different transition opportunities. At a given time, only participants observed in a state contribute to the risk set for transitions out of that state. Combining all entry states into one group may be acceptable for a clearly defined overall summary, but it does not make them one state or make their outgoing transitions interchangeable.

Justifying the Entry-Time Risk Set

The risk-set rule follows directly from the question “who was under observation and in state \(j\) just before \(t\)?” Eligibility must hold, entry must have occurred, observation must cover the time, and the current pre-transition state must be \(j\). Multiplying the corresponding indicators makes the inclusion rule explicit:

Summing this indicator over participants counts exactly the members of \(\mathcal{R}_j(t)\). Dropping \(O_i(t-)\) would count people who entered but were no longer observed; dropping the state indicator would combine people who have different transition opportunities. This logic also explains why entry-state proportions and later state proportions have different denominators: entry proportions concern the cohort at entry, while a later observed proportion concerns people with an observed state at that later time.

Choosing an Entry Convention and Checking Its Limits

Several conventions can be appropriate, but they answer different questions. Retaining a common scientific time origin and accounting for delayed entry preserves the original time scale. Resetting each participant’s clock to zero at entry instead describes time since entry; it does not automatically answer a question about age or time since diagnosis. Restricting eligibility to one entry state simplifies the starting distribution but narrows the population. Retaining multiple entry states preserves a broader cohort but requires state-specific risk sets and transparent reporting of the entry-state distribution.

Before analysis, check that entry times and entry states are recorded consistently, that the stated eligibility rules match the data, and that there are observed participants in the states and time ranges needed for the intended comparisons. A state proportion at \(t\) requires observed states at \(t\); an entry before \(t\) is not sufficient. If some entry states have no follow-up or some time ranges contain no eligible observed participants, the corresponding state-specific quantities may not be estimable from those data. Sparse entry states can also make summaries unstable.

A particularly consequential limitation is that the state history before \(L_i\) is generally unavailable under delayed entry. The analysis can describe the known entry state and subsequent observed process, but it cannot determine the unobserved route into that state without additional information or assumptions. Further, even complete follow-up after entry does not by itself justify conclusions about people who were eligible but never entered. The intended target population and the assumptions supporting any generalization should therefore be reported explicitly.

Common Mistakes

• Treating entry state as the state at the scientific origin. A recorded state at recruitment does not reveal the state at an earlier time unless earlier information establishes it.
• Counting everyone who entered in a later observed-state denominator. The correction is to include only people whose state is actually observed at the time in question, using an observation indicator such as \(O_i(t)\).
• Assuming delayed entry is harmless because entry time is known. Knowing \(L_i\) identifies who could be observed when; it does not establish that entrants represent everyone in the target population.
• Combining multiple entry states as if they were identical. Report their distribution and use the state-specific risk set for each transition.
• Changing the clock without changing the interpretation. Time since entry and time since the scientific origin are different scales and can define different estimands.