Delayed Entry in Competing-Risks Joint Models

Why Delayed Entry Changes the Likelihood

In many studies, follow-up does not begin at the time origin relevant to the event process. A participant may become eligible at time zero but enter the study later, after surviving without any event. Such a participant is included only because they remained event-free until their entry time. In a competing-risks joint model, that condition concerns every event cause, not just the cause eventually observed.

The time origin must remain the scientific origin of the modeled event process. If someone enters at time \(L_i\) and experiences cause \(k\) at time \(t_i>L_i\), the likelihood must reflect survival free of all causes from time zero to entry, conditional on inclusion. Simply treating \(L_i\) as a new time zero generally changes the model and discards information about the original time scale.

Cause-Specific Event Contributions

Suppose there are \(K\) mutually exclusive event causes. Let \(b_i\) denote subject \(i\)'s shared random effects, and let \(\lambda_{ik}(t\mid b_i)\) be the cause-\(k\) hazard at time \(t\), conditional on those random effects and the modeled covariates. The total hazard of any event is the sum of the cause-specific hazards:

Here \(S_i(t\mid b_i)\) is the probability of remaining free of all event causes through time \(t\), conditional on \(b_i\). For an exact event of cause \(k\) at time \(t_i\), the event-time density from the origin is \(S_i(t_i\mid b_i)\lambda_{ik}(t_i\mid b_i)\). For a right-censored observation at time \(C_i\), the event-process contribution is \(S_i(C_i\mid b_i)\).

Conditional on surviving to entry, the event contribution for an exact cause-\(k\) event is therefore \(\{S_i(t_i\mid b_i)/S_i(L_i\mid b_i)\}\lambda_{ik}(t_i\mid b_i)\). For censoring at \(C_i\), it is \(S_i(C_i\mid b_i)/S_i(L_i\mid b_i)\). These are conditional contributions after entry; they do not say that the subject’s event-time clock began at \(L_i\).

The survival in this expression is over the interval from entry to the observed time, and includes hazards for all causes. For a censored subject, replace \(t_i\) by \(C_i\) and omit the cause-specific hazard factor.

Conditioning When Random Effects Are Integrated Out

The conditional expression above is for fixed \(b_i\). In a joint model, random effects are usually unobserved and integrated over their population distribution \(p(b_i)\). A crucial point is that the random-effects distribution among entrants is generally not the original population distribution. Values of \(b_i\) associated with higher event risk are less likely to survive until \(L_i\).

Let \(Y_i\) denote the subject’s longitudinal measurements, and let \(f(Y_i\mid b_i)\) be their longitudinal likelihood conditional on the random effects. Assume the longitudinal and event processes are conditionally independent given \(b_i\) and covariates, and that the measurement process is appropriately handled. Let \(\tilde T_i\) be the observed event or censoring time, with \(\delta_i=k\) for an event of cause \(k\) and \(\delta_i=0\) for right censoring. The conditional likelihood contribution given entry at \(L_i\) is

For a censoring observation, all hazard exponents are zero, leaving the survival term. For an event of cause \(k\), exactly one hazard factor appears. The denominator is the marginal probability of surviving event-free to entry. The longitudinal likelihood belongs in the numerator because the observed measurements are part of the entrant’s observed data.

Equivalently, define the random-effects distribution among entrants by \(p(b_i\mid T_i>L_i)=p(b_i)S_i(L_i\mid b_i)/\int p(b_i)S_i(L_i\mid b_i)\,db_i\). Then the event contribution after entry is integrated against this selected distribution; if longitudinal data are also used, they update it further. This identity explains why it is generally wrong to integrate a fixed-\(b_i\) conditional event contribution against the unselected population distribution \(p(b_i)\).

Adding Longitudinal Measurements

Longitudinal measurements can provide information about \(b_i\), so a joint likelihood uses both those measurements and the entry-conditioned event data. One convenient factorization first weights \(b_i\) by the chance of reaching entry, then by the longitudinal likelihood. In particular, after observing \(Y_i\), the random-effects weights are proportional to \(p(b_i)S_i(L_i\mid b_i)f(Y_i\mid b_i)\). This is not the same as conditioning on entry alone, nor is it generally the same as using \(p(b_i)f(Y_i\mid b_i)\) without the survival-to-entry factor.

Assumptions, Diagnostics, and Alternatives

The likelihood requires a clear event-time origin and correctly specified cause-specific hazards over the relevant time scale. It assumes that inclusion follows the stated survival rule: subjects enter observation only after remaining free of every event cause through \(L_i\). Conditional on modeled covariates and random effects, the entry mechanism must not carry additional information about future event outcomes that the likelihood ignores. If entry is related to unmodeled risk, or if the process determining entry is itself informative, it may need to be modeled or addressed with a justified sampling design.

The event/censoring contribution also relies on appropriate right-censoring assumptions: conditional on the model variables, censoring should not convey unmodeled information about the event time or cause. The joint likelihood assumes the specified dependence between the longitudinal and event processes is adequate; conditional independence given random effects is a modeling assumption, not an automatic fact. Measurement times and missingness must likewise be handled consistently with the assumed longitudinal model.

Useful checks include verifying that every event or censoring time exceeds its entry time, confirming that the total hazard sums across all modeled causes, and checking that the survival denominator uses the subject’s actual entry time. Inspect entry times and risk-set construction for accidental time resets. Compare estimates under plausible alternative random-effects or hazard specifications, and assess whether fitted survival probabilities at entry are credible. Very late entry, sparse causes, or little longitudinal information can make the selected random-effects distribution weakly identified or numerically difficult to integrate.

A standard delayed-entry survival analysis conditions each subject’s risk contribution on being event-free at entry. The joint model adds longitudinal information and integrates over shared random effects; it does not remove the need for that conditioning. An analysis that treats entry as time zero may sometimes be appropriate if the scientific time scale truly restarts at entry, but that is a different model. Merely omitting the survival-to-entry conditioning is not an equivalent alternative: it treats the observed entrants as though they were a random sample from the original population.

• Common mistake: “The participant entered at year 2, so I can use \(S_i(t_i\mid b_i)\lambda_{ik}(t_i\mid b_i)\) and ignore earlier survival.” This is the event density from the origin, not the event contribution conditional on selection into observation at year 2.
• Correction: For fixed random effects use \(S_i(t_i\mid b_i)/S_i(L_i\mid b_i)\) times the cause-specific hazard. After integrating over random effects, account for the survival-selected distribution among entrants, as in the numerator-over-denominator likelihood.
• Another pitfall: Using only the hazard for the observed cause in the survival exponent. The exponent must use the total hazard across every competing cause, because any cause ends event-free survival.