Conditioning the Likelihood on Delayed Entry

Why Delayed Entry Changes the Likelihood

In a joint model, a subject may enter observation after the time origin used for the event process. The subject is then observed only because the event has not occurred by the entry time. This selection matters: the event-time contribution must describe follow-up conditional on surviving to entry, rather than treating the subject as if observation began at time zero.

Shared random effects make the conditioning especially important. Subjects with random effects associated with higher event hazards are less likely to survive long enough to enter. Consequently, the distribution of random effects among entrants generally differs from their distribution in the source population. A correct likelihood accounts for both the conditional event process after entry and this selection of entrants.

Definition: Let \(T_i\) be subject \(i\)'s event time measured from a common time origin, \(L_i\) the delayed-entry time, \(\tilde T_i=\min(T_i,C_i)\) the observed event or censoring time, and \(\delta_i=I(T_i\le C_i)\) the event indicator. A subject is observed only if \(T_i>L_i\), so the observed follow-up must be modeled conditional on survival to \(L_i\). The subject's random effects are denoted \(b_i\), with source-population density \(p(b_i)\).

Write \(S_i(t\mid b_i)=P(T_i>t\mid b_i)\) for the event-free survival function and \(\lambda_i(t\mid b_i)\) for the event hazard, conditional on the random effects and any covariates. The usual event-or-censoring kernel at \(\tilde T_i\) is \(\{\lambda_i(\tilde T_i\mid b_i)\}^{\delta_i}S_i(\tilde T_i\mid b_i)\). With delayed entry, its conditional version is divided by \(S_i(L_i\mid b_i)\). For an event, this is the conditional event density after entry; for censoring, it is the conditional probability of remaining event-free until the censoring time.

Entrant Distribution and Subject Contribution

The random-effects distribution among subjects who survive to entry follows directly from Bayes’ rule. Conditional on a fixed entry time \(L_i\), survival to entry has probability \(S_i(L_i\mid b_i)\) for a subject with random effects \(b_i\). Thus the density of \(b_i\) among entrants is

p(b_i\mid T_i>L_i) = \frac{S_i(L_i\mid b_i)p(b_i)} {\int S_i(L_i\mid u)p(u)\,du}.

Let \(Y_i\) denote the observed longitudinal measurements, with conditional density \(f_Y(Y_i\mid b_i)\). Assume that, conditional on \(b_i\) and the modeled covariates, the longitudinal and event processes are independent, and that the longitudinal observation schedule is handled as noninformative. The entrant-conditional joint contribution is then

L_i^{\mathrm{entry}} = \int f_Y(Y_i\mid b_i) \frac{\{\lambda_i(\tilde T_i\mid b_i)\}^{\delta_i}S_i(\tilde T_i\mid b_i)} {S_i(L_i\mid b_i)} p(b_i\mid T_i>L_i)\,db_i.

Substituting the Bayes formula for the entrant distribution cancels the conditional survival denominator inside the integral. Equivalently, the contribution can be written as a ratio of integrals under the source-population random-effects distribution:

L_i^{\mathrm{entry}} = \frac{ \int f_Y(Y_i\mid b_i) \{\lambda_i(\tilde T_i\mid b_i)\}^{\delta_i} S_i(\tilde T_i\mid b_i)p(b_i)\,db_i }{ \int S_i(L_i\mid b_i)p(b_i)\,db_i }.

The numerator integrates the observed longitudinal and event-or-censoring contribution over source-population random effects. The denominator is the source-population probability of surviving to entry. These two forms are equivalent, but the first form must average over the entrant distribution, not over \(p(b_i)\). Averaging post-entry conditional event contributions over the source distribution would ignore the fact that survival to entry changes the composition of the observed sample.

Assumptions: This likelihood conditions on a known entry time and on survival to that time. The entry mechanism must not select subjects on additional unmodeled information about \(b_i\) or the event process. Event and censoring times must be conditionally independent given modeled covariates and random effects, or the censoring mechanism must otherwise be modeled. The longitudinal density and measurement schedule must be appropriate conditional on the random effects and the sampling scheme. If these conditions fail, the displayed contribution alone does not account for the selection or dependence.

Worked Example: Conditional Follow-Up for a Fixed Random Effect

Suppose a subject has a fixed conditional hazard of \(0.12\) events per year, so \(S(t\mid b)=\exp(-0.12t)\). The subject enters at year \(2\) and is observed event-free until censoring at year \(5\). Find the event-time contribution conditional on entry.

Here \(\delta=0\), so the hazard factor is raised to the zero power and equals 1. The conditional censoring contribution is

\frac{S(5\mid b)}{S(2\mid b)} = \frac{\exp(-0.12\times5)}{\exp(-0.12\times2)} = \exp[-0.12(5-2)] = \exp(-0.36) \approx 0.6977.

Thus, conditional on having survived to year 2, the probability of remaining event-free through year 5 is about 0.6977. If instead an event occurred at year 5, the conditional event density would be

0.12\frac{S(5\mid b)}{S(2\mid b)} = 0.12\exp(-0.36) \approx 0.08372\ \text{per year}.

The entry time does not reset the time origin of the hazard: the hazard is evaluated at the absolute event time, while the survival ratio conditions follow-up on survival from year 2.

Deriving the Conditioning Adjustment

The conditional event-or-censoring contribution for a fixed \(b_i\) follows from the definition of a conditional probability or density. For censoring at \(t>L_i\), the probability of surviving to \(t\), conditional on survival to \(L_i\), is

P(T_i>t\mid T_i>L_i,b_i) = \frac{P(T_i>t\mid b_i)}{P(T_i>L_i\mid b_i)} = \frac{S_i(t\mid b_i)}{S_i(L_i\mid b_i)}.

For an event at \(t>L_i\), the conditional density is the event density at \(t\), divided by the probability of surviving to \(L_i\):

f_{T_i}(t\mid T_i>L_i,b_i) = \frac{\lambda_i(t\mid b_i)S_i(t\mid b_i)} {S_i(L_i\mid b_i)}.

Combining these cases gives \(\{\lambda_i(t\mid b_i)\}^{\delta_i}S_i(t\mid b_i)/S_i(L_i\mid b_i)\). This derivation explains why merely using a survival contribution from time zero is not appropriate for a sample selected to have survived to entry. It also shows that the conditional event density is a density in time, whereas the censoring contribution is a probability.

Worked Example: Two Random-Effect Types Among Entrants

Consider a hypothetical source population with two equally common random-effect types. Their constant event hazards are \(0.05\) and \(0.20\) per year. Subjects enter at year \(5\), and a subject has an event at year \(8\). There are no longitudinal measurements in this example. Under an exponential event model, calculate the event density conditional on entry.

The source-population survival probability to entry is

\int S(5\mid b)p(b)\,db = 0.5e^{-0.05(5)}+0.5e^{-0.20(5)} = 0.5e^{-0.25}+0.5e^{-1} \approx 0.5733401.

By Bayes’ rule, the entrant weights of the low- and high-hazard types are

w_{\mathrm{low}} = \frac{0.5e^{-0.25}}{0.5733401} \approx 0.6792, \qquad w_{\mathrm{high}} = \frac{0.5e^{-1}}{0.5733401} \approx 0.3208.

The event occurs three years after entry. For the low-hazard type, its conditional event density is \(0.05e^{-0.05(3)}\); for the high-hazard type it is \(0.20e^{-0.20(3)}\). Averaging these densities over the entrant weights gives

0.6792(0.05e^{-0.15})+0.3208(0.20e^{-0.60}) \approx 0.06444\ \text{per year}.

The equivalent ratio-of-integrals calculation uses the absolute event time, year 8, in the numerator:

\frac{0.5(0.05e^{-0.05(8)})+0.5(0.20e^{-0.20(8)})} {0.5e^{-0.25}+0.5e^{-1}} = \frac{0.0369477}{0.5733401} \approx 0.06444\ \text{per year}.

A tempting but incorrect calculation averages the conditional event densities over the original, equal source-population weights: \(0.5(0.05e^{-0.15})+0.5(0.20e^{-0.60})\approx0.07640\) per year. This overweights the high-hazard type among entrants. The correct entrant weights are approximately \(0.6792\) and \(0.3208\), because fewer high-hazard subjects survive to year 5.

Including Longitudinal Measurements and Censoring

When longitudinal data are available, their conditional density stays inside the random-effects integral. The shared random effects connect the longitudinal and event processes: measurements can provide information about \(b_i\), while delayed entry also changes the distribution of \(b_i\) among observed subjects. The following calculation illustrates both components for a censored subject.

Worked Example: A Longitudinal Measurement With Delayed Entry and Censoring

Suppose a hypothetical population has two equally common random-effect types. Their constant event hazards are \(0.10\) and \(0.30\) per year. A subject enters at year \(2\), has one longitudinal measurement \(Y=0.2\), and is censored at year \(4\). Conditional on type, let \(Y\sim N(\mu,1)\), with means \(0\) and \(1\), respectively. Assume the measurement and event processes are conditionally independent given type.

First find the entrant weights. The probability of surviving to entry in the source population is

0.5e^{-0.10(2)}+0.5e^{-0.30(2)} = 0.5e^{-0.2}+0.5e^{-0.6} \approx 0.6837712.

The corresponding entrant weights are \(0.5e^{-0.2}/0.6837712\approx0.5987\) and \(0.5e^{-0.6}/0.6837712\approx0.4013\). The normal measurement densities at \(Y=0.2\) are

f_Y(0.2\mid \mu=0) = \frac{1}{\sqrt{2\pi}}e^{-(0.2-0)^2/2} \approx0.3910, \qquad f_Y(0.2\mid \mu=1) = \frac{1}{\sqrt{2\pi}}e^{-(0.2-1)^2/2} \approx0.2897.

Because the subject is censored at year 4, the event-time factor for each type is the survival ratio from year 2 to year 4. The integrated conditional contribution is therefore

\begin{aligned} L^{\mathrm{entry}} &\approx 0.5987(0.3910)e^{-0.10(4-2)} + 0.4013(0.2897)e^{-0.30(4-2)}\\ &\approx 0.2555. \end{aligned}

This is a joint density for the measured value and the censoring information; its units are inverse units of \(Y\). It is not a probability that the subject belongs to either type. The calculation also shows why both the measurement density and the event contribution must be integrated using the entrant distribution.

Diagnostics, Alternatives, and Limitations

Several checks help detect errors in a delayed-entry likelihood. Confirm that every included subject satisfies \(\tilde T_i>L_i\), that entry and event times use the same origin and units, and that the event hazard is evaluated at the correct time scale. For a censoring record, the survival ratio must be at most 1. For an event, the conditional density should have units of inverse time. Numerically, the entrant weights should be nonnegative and sum to 1; if all random effects have the same survival probability to entry, the entrant distribution should equal the source distribution.

The main alternative is to omit the entry adjustment and use the right-censoring contribution from time zero. That is only suitable when subjects are observed from the time origin, or when the sampling design and estimand justify a different likelihood. With genuine left truncation, omission typically overrepresents subjects with high hazards. Resetting the clock at entry is also not generally equivalent: it changes the time argument of the hazard and is valid only under a suitable model for time since entry, not simply because observation begins then.

Approach	Random-effects averaging	What it represents
Delayed-entry conditional likelihood	Uses \(p(b_i\mid T_i>L_i)\), or the equivalent ratio of source-population integrals	Observed data conditional on surviving to entry
Unadjusted right-censoring contribution	Uses the source distribution without entry conditioning	Does not account for survival-based selection into observation
Entry as a new time origin	Depends on a model for time since entry	A different time-scale model, not a generic correction for delayed entry

This conditional likelihood has limitations. If entry depends on unobserved random effects for reasons beyond survival, the entrant distribution needs a model for that entry mechanism. If censoring is informative after conditioning on modeled variables, its process cannot simply be ignored. If longitudinal observations before entry are included, or observation times themselves carry information, the longitudinal and sampling contributions must match that design. Delayed-entry conditioning corrects survival-based selection; it does not automatically resolve other forms of selection or misspecification.

Common pitfall: It is wrong to integrate the post-entry conditional event contribution against the source density \(p(b_i)\) and assume that division by \(S_i(L_i\mid b_i)\) inside each random-effect-specific contribution has handled selection. The correct integral uses \(p(b_i\mid T_i>L_i)\). Equivalently, use the ratio of population integrals shown above. The denominator of that ratio is shared across random effects; it is not a reason to retain the source weights in an entrant-distribution average.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

A subject enters at year 3 and is censored at year 7. With a constant hazard of \(0.08\) per year, what is the conditional event-time contribution?
In a two-type population, one type has a higher event hazard than the other. Which type is generally less common among subjects who survive to a fixed positive entry time, and why?
For a subject who has an event at time \(t>L\), state the conditional event density given survival to entry.
When is the source-population random-effects distribution the same as the entrant distribution?

Solutions. (1) The contribution is \(S(7)/S(3)=\exp[-0.08(7-3)]=e^{-0.32}\approx0.7261\). (2) The higher-hazard type is generally less common, because its survival probability to entry is lower and Bayes’ rule weights types by that survival probability. (3) The density is \(\lambda(t\mid b)S(t\mid b)/S(L\mid b)\), conditional on the random effects. (4) They coincide when survival to entry does not vary with the random effects, so conditioning on entry does not reweight their source distribution.