Delayed Entry in Event-History Likelihoods

Why Entry Time Changes the Likelihood

In event-history data, observation may begin only after a subject has already experienced some event-free or multi-state history. A subject who enters a study at time \(L\) is not observed from the origin of the time scale. Treating that subject as at risk from time zero adds exposure that was not observed and, more importantly, ignores the fact that inclusion may require the subject to have reached the entry time in a specified state.

The appropriate likelihood describes the observed path conditional on the entry information. It accumulates exposure only after entry, and it includes transition factors only for transitions observed after entry. This is the event-history likelihood version of delayed entry, also called left truncation. The earlier material on event-history likelihoods supplies the basic intensity-based path likelihood; here the essential change is the conditioning and the lower limit of observation.

Assume a continuous-time multi-state process with transition intensity \(\alpha_{rs}(t)\) from state \(r\) to state \(s\), conditional on the history used by the model. For clarity, the examples below use constant or state-based intensities and no time-varying covariates. Let \(N_{irs}(t)\) count observed \(r\)-to-\(s\) transitions by subject \(i\). The at-risk indicator for transitions out of state \(r\) is

The strict inequality at the lower limit excludes an event at entry from the ordinary post-entry path contribution. The state is evaluated just before \(t\), so a transition at \(t\) is attributed to the state from which it occurs. The upper limit includes an observed transition at the end of follow-up; if follow-up ends by censoring, there is no transition factor at that censoring time. In continuous-time likelihood integrals, changing an endpoint alone does not change the integrated exposure, but the convention matters for transition factors and for a transition coincident with entry.

The Conditional Path Likelihood

Suppose the observed path begins in state \(x_i\) at entry, follows a known sequence of states, and is observed until \(C_i\). Conditional on the entry information, the path likelihood has a factor for each observed transition and an exponential factor for remaining in the current state between transitions. In counting-process notation it is

The product ranges only over observed transitions strictly after entry. The sum in the integral is the total instantaneous exit intensity from the state occupied just before time \(t\). Thus, if a subject occupies state \(r\) over an interval, all possible exits from \(r\) contribute to the probability of having no transition during that interval. An absorbing state has no outgoing transition intensities and contributes no exit exposure.

This expression is a conditional likelihood for the observed segment, not a claim that the unobserved pre-entry path had likelihood one. It conditions on the starting information and does not use that earlier path. If the model’s intensity after entry depends on unobserved pre-entry history beyond the state at entry, the state alone is not sufficient conditioning information: the likelihood must also account for the relevant history, or integrate over its conditional distribution.

Why the Lower Limit Is Entry: A Short Derivation

Consider a short interval of length \(dt\) after entry, while the process is in state \(r\). Conditional on the model history, the probability of no transition in that interval is \(1-\sum_{s\ne r}\alpha_{rs}(t)dt+o(dt)\), while the density contribution for a transition to \(s\) at time \(t\) is \(\alpha_{rs}(t)\). Multiplying the no-transition contributions across the observed intervals and the transition-density contributions at the observed jumps gives the path likelihood.

For a single interval \((a,b]\) spent in state \(r\), the no-transition probability is the limit of products of terms \(1-\alpha_r(t_j)\Delta t_j+o(\Delta t_j)\), where \(\alpha_r(t)=\sum_{s\ne r}\alpha_{rs}(t)\). Taking logarithms, using \(\log(1-u)=-u+o(u)\), and letting the partition become fine gives

For a delayed-entry path, the first observed interval begins at \(a=L_i\), not at the time origin. This yields the post-entry exposure integral. A transition factor is then multiplied in at each observed jump time. This argument assumes the stated intensities describe the process conditional on the information at risk and that the observation and censoring scheme permits this conditional path likelihood.

Multi-State Risk Limits in Practice

For multi-state data, delayed entry changes the risk process separately for every possible origin state. At any post-entry time, a subject contributes to the risk set for exits from the state occupied immediately before that time. A subject who enters in state \(A\) contributes to the \(A\)-state risk set after entry; after an observed transition to \(B\), the subject contributes to the \(B\)-state risk set instead. No exposure is assigned to states or times before entry.

Conditioning on Selection and the Entry Design

A useful general principle is that likelihoods for selected subjects must match the sampling mechanism. If inclusion requires an event-free subject to survive until a fixed entry time \(L\), the post-entry event-time density is \(f(t)/P(T>L)\) for \(t>L\). In a competing-risks setting, with cause-specific intensities \(\lambda_k(t)\), the conditional density for an observed cause-\(k\) event at \(t>L\) is

The denominator is the probability of the entry condition, and cancellation leaves only post-entry exposure. For a general multi-state selection rule, the corresponding conditioning event might be \(X(L)=r\), rather than simply “no event by \(L\).” If the state at entry is conditioned upon and the post-entry model is valid conditional on that state and observed history, the conditional path likelihood starts from that state and uses only post-entry transitions and exposure. If instead the analysis uses a full-from-origin model for selected subjects, it must normalize by the probability of the selection event under that model.

Assumptions, Diagnostics, and Alternatives

A conditional delayed-entry likelihood is appropriate only when its conditioning and observation assumptions match the study design. In particular, the entry time and inclusion rule must be handled correctly. Conditioning on entry is often justified when entry is fixed or when, given modeled information, the entry process is independent of the event process apart from the stated eligibility condition. If entry depends on unobserved prognosis or on the event process in a way not captured by the conditioning information, the observed subjects may be selected in a parameter-dependent way. A post-entry likelihood alone may then be inadequate; a model for the entry or selection process, or an appropriate joint model, may be needed.

Censoring must also be non-informative for the likelihood being used, conditional on the modeled history and entry information. If dropout depends on unobserved future event risk, treating the censoring time as a simple endpoint can distort inference. The process must have the stated transition intensities on the intervals where the subject is at risk, and the observed states and transition times must be recorded consistently.

Several checks help detect an incorrectly constructed likelihood:

• Check the lower limit. No risk exposure or transition factor should be added before \(L_i\) for a likelihood of the observed segment.
• Check the state at each jump. Use \(X_i(t-)\) to identify the transition’s origin state and the appropriate risk indicator.
• Check the entry condition. State whether the subject had to be event-free, in a particular state, or otherwise eligible at entry. Include the selection normalization if using a full-from-origin likelihood for selected observations.
• Check coincident events. If a transition is recorded at entry, the ordinary \((L_i,C_i]\) path convention does not automatically describe it. Specify how the entry design treats that event.
• Check follow-up termination. A censoring endpoint contributes exposure up to censoring, not an event factor. An observed transition endpoint contributes its transition factor.

An event-time distribution without atoms gives probability zero to an event at a fixed time. Thus, if entry is fixed or suitably independent of the event time, and the event-time distribution has no atoms, an event exactly at entry has probability zero. Atomlessness alone does not justify that conclusion when entry depends on the event time: for example, if the design sets \(L=T\), then \(P(T=L)=1\). If the design permits or induces coincidence, its treatment must be specified rather than silently dropping or including the transition.

Ignoring delayed entry and starting every risk process at time zero is generally not a valid alternative: it assigns unobserved pre-entry exposure and fails to condition on inclusion. A conditional post-entry likelihood is the natural alternative when the target is the transition process after entry, conditional on entry information. A full likelihood that models the pre-entry trajectory and divides by the probability of selection is useful when the selection mechanism itself is part of the inferential target or cannot be ignored. Neither approach corrects informative entry automatically; the required model depends on how entry was generated.

Common Pitfall