Delayed Entry in the Joint-Model Likelihood

Why Delayed Entry Changes the Likelihood

In a time-to-event study, a subject may become observable only after the time origin. For example, a participant might enter a registry several years after diagnosis, provided they are still alive at that point. Their observed follow-up begins at entry, but the event-time scale still begins at diagnosis. The subject was not at risk in the observed sample before entry, and survival to entry is part of the sampling condition.

The event-process likelihood must therefore describe what is observed conditional on having survived event-free to entry. Treating entry as if it were the time origin, or using an ordinary contribution from time zero without conditioning, generally gives the wrong likelihood. This tutorial focuses on that adjustment; the earlier material on the subject-specific hazard and survival function supplies the hazard-based ingredients.

In a joint model, the event hazard may depend on subject-specific random effects \(b_i\), including through a latent longitudinal trajectory. A common form is

where \(h_0(t)\) is the baseline hazard, \(w_i\) denotes event-process covariates, and \(m_i(t;b_i)\) is the latent marker value. Define the conditional cumulative hazard and survival function by

All times in these definitions use the same origin. In particular, \(L_i\) is not silently redefined as zero.

The Conditional Event-Process Contribution

Given \(b_i\), the event-time density at \(Y_i\) is \(h_i(Y_i\mid b_i)S_i(Y_i\mid b_i)\). For a right-censored subject, the event-process contribution is the probability of surviving through \(Y_i\), namely \(S_i(Y_i\mid b_i)\). Both expressions must be conditioned on the subject having survived to \(L_i\).

The exponent contains only cumulative hazard after entry. For an observed event, the contribution is a density and has units of inverse time; for censoring, it is a survival probability and is dimensionless. The formula presumes \(Y_i\ge L_i\). If an observed record has \(Y_i<L_i\), it is inconsistent with the stated delayed-entry setup and should be investigated rather than accommodated by this expression.

Why the Survival Ratio Appears

The adjustment follows directly from conditional probability. For an observed event at \(Y_i\), conditional on \(b_i\), the density of the event time given survival to entry is the unconditional event density divided by the probability of surviving to entry. For censoring at \(Y_i\), the conditional probability of surviving to that time is likewise a ratio of survival probabilities.

Combining the event-density expression when \(\Delta_i=1\) with the survival expression when \(\Delta_i=0\) gives the result above. Equivalently, using \(S_i(t\mid b_i)=\exp\{-H_i(t\mid b_i)\}\), the ratio of survival probabilities is \(\exp\{-[H_i(Y_i\mid b_i)-H_i(L_i\mid b_i)]\}\). This derivation also shows why using \(S_i(Y_i\mid b_i)\) alone counts survival before entry even though the likelihood is conditional on having survived that period.

Integrating Over Random Effects

The contribution just derived is conditional on \(b_i\). In a joint model, the random effects are latent, so a marginal event-process likelihood must account for their distribution. Delayed entry changes the distribution of random effects among entrants: subjects with random effects associated with higher early hazards are less likely to survive to entry.

Suppose \(f(b)\) is the random-effects density in the source population and entry time \(L\) is fixed or conditioned upon. For an event-process-only calculation, the random-effects density among entrants is proportional to \(f(b)S(L\mid b)\). Normalizing it gives \(f(b\mid T>L)=f(b)S(L\mid b)/\int f(u)S(L\mid u)\,du\). Averaging the conditional delayed-entry contribution over this entrant distribution yields

The denominator is the source-population probability of surviving to entry, averaged over random effects. This marginal expression makes the selection effect explicit. In a full joint likelihood, longitudinal measurements and any modeled observation process must also be handled consistently with selection into observation; the event-process calculation alone does not specify those other factors.

Assumptions, Checks, and Alternatives

The delayed-entry contribution is appropriate when event times and entry times share a defined time scale, entry requires survival to \(L_i\), and the subject's event status is observed only from that entry point. The hazard model must describe the event process on this full time scale, including before entry, because the probability of surviving to entry determines who can be observed. For the event-process likelihood, censoring must also be conditionally independent under the model, so that a censoring time does not provide unmodeled information about the event time after conditioning on modeled quantities.

The adjustment further relies on an appropriate account of the entry mechanism. The expressions above condition on the stated survival-based eligibility and treat entry as fixed or conditioned upon. If entry time or selection depends on unmodeled event risk, covariates, or random effects beyond survival itself, this conditioning may not remove selection bias; the entry process may need explicit modeling. In particular, one should not assume that random effects among entrants retain their source-population distribution.

Useful checks include confirming \(Y_i\ge L_i\), verifying that all times use the intended origin, and checking that the cumulative hazard difference is nonnegative. A negative value of \(H_i(Y_i\mid b_i)-H_i(L_i\mid b_i)\) for \(Y_i\ge L_i\) signals a coding or model error for a nonnegative hazard. Compare the number and covariate profile of observed entrants with expectations under the assumed selection mechanism. Sensitivity to the entry assumptions matters when the probability of surviving to entry varies greatly across subjects.

Delayed entry is not left censoring. Under delayed entry, the subject is known to be event-free up to entry and then becomes at risk in the observed sample; under left censoring, an event has occurred by an observation time but its exact time is unknown. Nor is delayed entry generally handled by resetting every subject's clock to zero at entry: that changes the hazard time scale and can change the meaning of time-dependent covariates and baseline hazard. A landmark analysis can instead define a new target population and time origin at a chosen landmark, but that is a different analysis and estimand.