Why the Observation Process Matters
An event-time record does not always reveal when the event occurred. A participant may remain event-free until follow-up ends, may already have had the event before first assessment, or may be known to have experienced it only between two assessments. In each case, the observed record is information about the event time, not necessarily an exact event time. A likelihood must represent that information as a probability or density for the observation actually made.
A different issue arises when participants enter observation only after surviving event-free to an entry time. This is left truncation, also called delayed entry. Such participants are included because they satisfied an entry condition. Their event-time likelihood must therefore be conditional on having survived to entry. Censoring describes incomplete knowledge of an event time; truncation describes selection into the observed sample.
For clarity, the formulas below condition on covariates as well as \(b\), even when the covariates are not written. Assume event times are continuous, so the probability of an event at exactly one specified instant is zero and an exact observation contributes a density. Assume the event-time model is correctly specified and that censoring or inspection information is noninformative conditional on the modeled covariates and individual effects: the observation mechanism supplies no additional information about \(T\) after conditioning on them. If censoring is informative, its process must be modeled rather than discarded from the likelihood.
Likelihood Contributions for Censoring
With an exact event at time \(t\), the conditional likelihood contribution is \(f(t\mid b)\). If observation ends at \(c\) while the participant is still event-free, the record says \(T>c\), so its contribution is \(S(c\mid b)\), not \(f(c\mid b)\). A left-censored record known to have occurred by \(l\) says \(T\leq l\) and contributes \(F(l\mid b)\). For an event known to occur after \(l\) and by \(u\), the contribution is the probability of that interval.
The interval formula follows because the event must have occurred by \(u\), but not by \(l\): \(P(l<T\leq u\mid b)=P(T\leq u\mid b)-P(T\leq l\mid b)\). The equivalent survival difference is often convenient because many event-time models specify \(S\) directly. The lower endpoint is excluded and the upper endpoint included; for a continuous event-time distribution, changing endpoint inclusion does not change the probability.
Worked Example: Three Censoring Records Under an Exponential Model
Suppose hypothetical event times follow an exponential model with constant hazard \(\lambda=0.20\) per day. Then \(S(t)=\exp(-0.20t)\), \(F(t)=1-\exp(-0.20t)\), and \(f(t)=0.20\exp(-0.20t)\). Assume the censoring or assessment schedule is noninformative.
Right-censored at day 4. The participant was event-free at day 4, so the contribution is \(S(4)=\exp(-0.20\times4)=\exp(-0.8)=0.4493\). This is the probability of remaining event-free beyond day 4, not the density of an event on day 4.
Left-censored by day 2. The event is known to have occurred by day 2, so the contribution is \(F(2)=1-\exp(-0.20\times2)=1-\exp(-0.4)=1-0.6703=0.3297\).
Interval-censored in \((2,5]\) days. The contribution is \(F(5)-F(2)=S(2)-S(5)=\exp(-0.4)-\exp(-1)=0.6703-0.3679=0.3024\). Each calculation uses the probability of the observed event-time information, rather than assigning an exact event time that was not recorded.
Delayed Entry and Left Truncation
Suppose a participant enters observation at time \(a\) only if the event has not yet occurred. The observed record is therefore conditional on \(T>a\). If the individual effect \(b\) is treated as known or conditioned upon, an exact event at \(t>a\) contributes \(f(t\mid b)/S(a\mid b)\); right censoring at \(c>a\) contributes \(S(c\mid b)/S(a\mid b)\). For an event known to occur in \((l,u]\), with \(a\leq l<u\), the contribution is \([F(u\mid b)-F(l\mid b)]/S(a\mid b)\). The denominator is the probability of satisfying the entry condition.
In a mixed-effects model, \(b\) is usually unobserved and has source-population density \(g(b)\). Left truncation changes which individual effects are represented among entrants: effects associated with longer survival are more likely to be included. Consequently, when the likelihood is conditional on inclusion, the appropriate contribution is a ratio of integrated probabilities, not generally an average of the fixed-\(b\) conditional likelihoods under the unselected density \(g\).
Here \(L_{\mathrm{obs}}(b)\) is the event-time density or probability for the observed record, including the fact that its event-time information is after entry. For example, it is \(f(t\mid b)\) for an exact event at \(t>a\), \(S(c\mid b)\) for right censoring at \(c>a\), and \(F(u\mid b)-F(l\mid b)\) for an interval \((l,u]\) with \(a\leq l\). The denominator is the source-population probability of being eligible to enter at \(a\).
This ratio also follows by first conditioning on the individual effect. Among entrants, the density of \(b\) is proportional to \(S(a\mid b)g(b)\). Averaging a conditional-on-entry contribution \(L_{\mathrm{obs}}(b)/S(a\mid b)\) over that selected density gives \(\int L_{\mathrm{obs}}(b)g(b)\,db/\int S(a\mid b)g(b)\,db\). Thus selection of individual effects is accounted for by the denominator after integration.
Worked Example: Exact Event After Delayed Entry in a Two-Point Mixed Model
Consider a hypothetical population with two equally common individual-effect groups. Their exponential hazards are \(\lambda_1=0.10\) per day and \(\lambda_2=0.40\) per day, with source probabilities \(0.5\) each. Participants enter only if event-free at day \(a=2\); one entrant has an exact event at day \(t=3\). Assume the entry rule depends only on survival under this model.
For group 1, \(S(2\mid b_1)=\exp(-0.20)=0.8187\); for group 2, \(S(2\mid b_2)=\exp(-0.80)=0.4493\). The source-population probability of being eligible is therefore \(\int S(2\mid b)g(b)\,db=0.5(0.8187)+0.5(0.4493)=0.6340\).
The exact-event density at day 3 is \(f(3\mid b)=\lambda_b\exp(-3\lambda_b)\). The integrated numerator is \(0.5[0.10\exp(-0.30)]+0.5[0.40\exp(-1.20)]\) \(=0.5(0.07408)+0.5(0.12048)=0.09728\) per day.
Thus the event-time contribution conditional on selection into the sample is \(0.09728/0.6340=0.1534\) per day. Its units are inverse days because the numerator is an event-time density and the denominator is a probability. Averaging \(f(3\mid b)/S(2\mid b)\) using the original 50:50 group weights would ignore that the high-hazard group is less likely to enter; it would not give the correct selected-sample likelihood.
Justification, Assumptions, and Practical Checks
For any event-time information \(A\) recorded after eligibility, the likelihood conditional on entry is \(P(A,T>a)/P(T>a)\). For a right-censored record at \(c>a\), \(A=\{T>c\}\) is already a subset of \(\{T>a\}\), so the conditional probability is \(S(c)/S(a)\). For an interval \((l,u]\) with \(l\geq a\), the event set is also a subset of survival to entry, giving \([F(u)-F(l)]/S(a)\). An exact event time uses densities: the conditional density at \(t>a\) is \(f(t)/S(a)\). Integrating the joint numerator and entry probability over the source distribution of \(b\) yields the mixed-effects ratio above.
Decide whether the information is an exact event, right censoring, left censoring or an event interval. Use the corresponding density or probability, not a substitute event time.
For a left-truncated record, verify that the participant was event-free at entry and that subsequent event or censoring information is compatible with that condition.
Use survival-to-entry conditioning. If individual effects are integrated out, include selection in the mixed-effects likelihood rather than treating the entrant effects as an unselected sample.
Useful checks include confirming that every interval has \(l<u\), every event or censoring time is after entry, and all contributions lie in their appropriate range: probabilities between zero and one, and densities nonnegative with units of inverse time. An interval contribution \(S(l)-S(u)\) must be nonnegative because survival cannot increase with time. A zero or extremely small entry-survival denominator can signal an incompatible model, implausible extrapolation, or a data-coding problem.
The formulas rely on the event-time model and on a noninformative observation or censoring process conditional on modeled information. They also assume the entry rule is correctly represented. If entry depends on an unmodeled factor related to event time, the simple truncation adjustment may not suffice. For repeated or clustered subjects, the relevant joint dependence structure must also be respected; multiplying individual contributions as if independent is justified only under the model's independence assumptions.
A common alternative is to analyze only exact events, discarding censored or interval-censored records. This loses information and can select a nonrepresentative subset. Treating censoring times as event times is worse: it asserts events occurred at times known only to mark the end of observation. The likelihood contributions above preserve the available information. Their limitation is that they do not repair a misspecified event-time distribution or informative censoring; those require additional modeling assumptions and data.
Common Mistakes
- Calling a right-censoring time an event time. The record says \(T>c\), so its contribution is \(S(c)\), not \(f(c)\).
- Confusing left censoring with left truncation. “Event occurred by \(l\)” contributes \(F(l)\); “entered only after surviving to \(a\)” requires conditioning on \(T>a\).
- Using a fixed-effect truncation correction after mixing. In a mixed-effects model, selection changes the effect distribution among entrants. The conditional likelihood is generally a ratio of integrals.
- Ignoring the observation mechanism without justification. The stated censoring formulas assume censoring is noninformative conditional on modeled information. Informative censoring requires modeling its dependence on the event process.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- Under a continuous event-time model, what likelihood contribution corresponds to a participant known to be event-free at time \(c\)? Why is it not an event density?
- With \(S(t)=\exp(-0.1t)\), calculate the contribution for an event known to occur in \((2,6]\) days.
- Explain in one sentence how left censoring differs from left truncation.
- For a mixed-effects model with entry at \(a\), why is the selected-sample likelihood generally a ratio of integrals rather than the integral of individual conditional-on-entry contributions under the original effect distribution?
Solutions. (1) It is \(S(c)\), because the observed information is \(T>c\), not an event at \(c\). (2) \(S(2)-S(6)=\exp(-0.2)-\exp(-0.6)=0.8187-0.5488=0.2699\). (3) Left censoring says the event occurred before a limit, whereas left truncation includes participants only after they have survived event-free to an entry time. (4) Entry selects individual effects according to their survival probabilities, so the correct conditional likelihood divides the integrated observed-record probability by the integrated probability of surviving to entry.