Longitudinal Records Before Event or Censoring

Why the Record Cutoff Matters

A joint model links a subject’s repeated marker measurements to their time-to-event outcome through shared random effects. For a subject whose follow-up ends, the longitudinal part of the likelihood must use the measurements that were actually observed while the subject was under follow-up—not every value that might have appeared on a planned visit schedule. An event or censoring time therefore creates a cutoff for selecting the longitudinal records.

This selection is more than a data-cleaning detail. Including a measurement recorded after an event would use information that could not have been observed for that subject in the stated follow-up. Conversely, omitting a valid measurement taken before follow-up ended discards an observed-data contribution. The event or censoring outcome and the available marker records must be aligned on the same time scale and with a clearly stated boundary convention.

Definition: For subject \(i\), let \(T_i\) be the event time, \(C_i\) the censoring time, \(\tilde T_i=\min(T_i,C_i)\) the observed follow-up endpoint, and \(\delta_i=I(T_i\le C_i)\) the event indicator. Let \(y_{ij}\) be a marker measurement recorded at time \(t_{ij}\). The strictly pre-endpoint records are the observations for which \(t_{ij}<\tilde T_i\). Write their number as \(n_i^{\mathrm{pre}}\).

In this tutorial, “before” means strictly earlier than the endpoint. A marker value recorded exactly at the endpoint is not included in this strictly pre-endpoint set; its treatment requires an explicit convention and is not assumed here. A value with \(t_{ij}>\tilde T_i\) cannot be a measurement made during the subject’s observed follow-up under this setup, and should prompt a check of the time origin, data dates, or endpoint definition.

Where the Records Enter the Likelihood

Write \(b_i\) for the subject-specific random effects and \(m_i(t;b_i)\) for the latent marker trajectory conditional on those effects. A common longitudinal submodel has conditionally independent normal measurement errors:

y_{ij}\mid b_i \sim N\!\left(m_i(t_{ij};b_i),\sigma^2\right).

Under this assumption, the longitudinal density for the pre-endpoint records, conditional on \(b_i\), is the product of the measurement densities at their observed times. The product stops at the endpoint; it is not a product over unobserved planned visits.

L_{i,\mathrm{long}}^{\mathrm{pre}}(b_i) = \prod_{j:\,t_{ij}<\tilde T_i} f\!\left(y_{ij}\mid b_i,t_{ij}\right).

For the event submodel, let \(h_i(t\mid b_i)\) be the subject-specific hazard and \(S_i(t\mid b_i)\) the corresponding conditional survival function. The event-time contribution through the observed endpoint is \(h_i(\tilde T_i\mid b_i)S_i(\tilde T_i\mid b_i)\) when an event is observed, and \(S_i(\tilde T_i\mid b_i)\) when follow-up is right-censored. Equivalently, it is \(h_i(\tilde T_i\mid b_i)^{\delta_i}S_i(\tilde T_i\mid b_i)\).

L_i(b_i) = \left\{\prod_{j:\,t_{ij}<\tilde T_i} f\!\left(y_{ij}\mid b_i,t_{ij}\right)\right\} h_i(\tilde T_i\mid b_i)^{\delta_i} S_i(\tilde T_i\mid b_i).

This is a likelihood contribution conditional on the random effects and on the observed follow-up and measurement times. To obtain the subject’s marginal contribution, integrate over the random-effects distribution \(p(b_i)\):

L_i = \int \left\{\prod_{j:\,t_{ij}<\tilde T_i} f\!\left(y_{ij}\mid b_i,t_{ij}\right)\right\} h_i(\tilde T_i\mid b_i)^{\delta_i} S_i(\tilde T_i\mid b_i) p(b_i)\,db_i.

The integral matters because the same \(b_i\) links the marker trajectory and the hazard. In general, the longitudinal and event factors cannot be integrated separately and then multiplied. If the longitudinal model has residual correlation rather than conditional independence, replace the product by the joint density of the observed pre-endpoint marker vector. The rule for selecting records remains the same.

Worked Examples of Record Selection

Worked Example: An Observed Event With Earlier Marker Records

Suppose a hypothetical subject has an event at month 6 and is measured at months 0, 2, 4, and 7. The recorded marker values at months 0, 2, and 4 are 10, 12, and 9. The value at month 7 is listed in the file but occurs after the event.

The endpoint is \(\tilde T_i=6\) and \(\delta_i=1\). The strictly pre-event times are 0, 2, and 4, so \(n_i^{\mathrm{pre}}=3\). The month-7 value is not a longitudinal observation available before the event and does not enter this subject’s pre-endpoint longitudinal factor.

For an arithmetic illustration, suppose the conditional latent means at months 0, 2, and 4 are 10, 11, and 10, with measurement-error standard deviation \(\sigma=1\). The residuals are 0, 1, and \(-1\). The normal density is \(f(y\mid b_i,t)=(1/\sqrt{2\pi})\exp\{-(y-m_i(t;b_i))^2/2\}\). Thus the longitudinal factor is

L_{i,\mathrm{long}}^{\mathrm{pre}}(b_i) = \phi(0)\phi(1)\phi(-1) = 0.398942(0.241971)(0.241971) \approx 0.02336.

For a simple conditional event illustration, suppose the hazard is constant at \(0.08\) per month, so \(S_i(6\mid b_i)=e^{-0.08(6)}=e^{-0.48}\approx0.618783\). The event factor is \(h_i(6\mid b_i)S_i(6\mid b_i)=0.08(0.618783)\approx0.04950\). The conditional joint contribution is therefore approximately \(0.02336(0.04950)=0.001156\). This is conditional on the specified trajectory and hazard values; a model with random effects would still require integration to obtain the marginal contribution.

Worked Example: Right Censoring and a Boundary Record

Consider a hypothetical subject known to be event-free through month 10, when follow-up is censored. Marker observations are recorded at months 1, 4, 8, and 10. Under the strict pre-endpoint convention, the included measurements are at months 1, 4, and 8; the month-10 record is at the endpoint, not before it.

Suppose the conditional latent means at months 1, 4, and 8 are 5, 6, and 7, and the corresponding observed values are 5, 8, and 5. Let \(\sigma=2\). The residuals are 0, 2, and \(-2\), so the longitudinal density factor is

\begin{aligned} L_{i,\mathrm{long}}^{\mathrm{pre}}(b_i) &= \frac{1}{2}\phi(0)\, \frac{1}{2}\phi(1)\, \frac{1}{2}\phi(-1)\\ &= 0.199471(0.120985)(0.120985) \approx 0.002920. \end{aligned}

Here the factor \(1/2\) in each density converts the standard normal density to a normal density with standard deviation 2. If the conditional event hazard is constant at \(0.05\) per month, the censoring contribution is survival alone: \(S_i(10\mid b_i)=e^{-0.05(10)}=e^{-0.5}\approx0.606531\). There is no hazard multiplier because \(\delta_i=0\). The conditional subject contribution is approximately \(0.002920(0.606531)=0.001771\).

This example uses a stated strict cutoff, not a claim that endpoint-time measurements must always be discarded in every study design. If such a measurement is to be used, the data convention and its likelihood treatment must be defined consistently.

Worked Example: Different Endpoints, Different Record Sets

Suppose a hypothetical study schedules marker measurements at months 1, 3, 6, and 9. Subject A has an event at month 5; subject B is censored at month 8. Under the strict pre-endpoint rule, subject A contributes measurements at months 1 and 3, while subject B contributes measurements at months 1, 3, and 6. The month-6 and month-9 scheduled visits do not contribute for A, and the month-9 visit does not contribute for B.

The sets follow directly from the inequalities \(1<5\), \(3<5\), \(6\not<5\), \(9\not<5\), and \(1<8\), \(3<8\), \(6<8\), \(9\not<8\). Hence \(n_A^{\mathrm{pre}}=2\) and \(n_B^{\mathrm{pre}}=3\). Subject B’s extra measurement is available because B remained under observation longer; it is not a reason to assign A a hypothetical value at month 6 or to treat A’s missing post-event visit as an ordinary missing marker measurement.

Justification and Assumptions

The record cutoff follows from how the observed data are defined. Conditional on \(b_i\), the longitudinal observations made at the included times contribute their conditional density. The event information contributes a density at the endpoint if \(\delta_i=1\), or a survival probability through that endpoint if \(\delta_i=0\). Combining these factors gives the conditional likelihood above; marginalization then averages that joint contribution over the random-effects distribution.

This factorization relies on a specified joint model. In the basic version, the longitudinal errors are conditionally independent given the random effects and covariates, and the event process has its stated hazard conditional on those same random effects and covariates. The observed marker values and event outcome are modeled through these components. The likelihood written here conditions on the recorded measurement times; if the observation schedule itself is modeled or is informative under the analysis, its contribution must be handled as part of the broader model rather than silently treated as irrelevant.

Censoring also requires an appropriate model or assumption for the analysis. The expression above gives the event-process contribution when censoring is represented by the observed endpoint and event indicator; it does not by itself prove that censoring is conditionally independent. The likelihood treatment of censoring must be compatible with the study design and assumptions. Similarly, with delayed entry, the risk interval and conditioning at entry must follow the delayed-entry formulation rather than simply treating follow-up as if it began at the time origin.

Diagnostics, Alternatives, and Limitations

A practical audit is to sort each subject’s marker records by time, compare every time with \(\tilde T_i\), and verify that the likelihood’s measurement count equals the number satisfying the chosen cutoff. Also check that marker times and event/censoring times use the same origin and units. An apparently post-event marker often signals a date conversion error, an inconsistent definition of event time, or a measurement that was recorded in the database later than it was actually obtained.

A common mistake is to include every scheduled marker visit in the longitudinal product and treat unavailable post-event values as though they were ordinary missing values. That is wrong because the risk interval has ended: the subject cannot provide a marker measurement during follow-up after an observed event, and censoring ends the observed follow-up even though the subject’s later event status is unknown. The correction is to include the actual records strictly before the endpoint, and to specify separately how endpoint-time records are defined and handled.

The main alternative to conditioning on the observed measurement schedule is to model the visit process jointly with the marker and event processes. That may be important when visit timing itself carries information, but it adds modeling assumptions and parameters. A further alternative within the longitudinal component is to use a correlated-error density instead of a product of independent measurement densities. Neither alternative changes the basic chronology: only records in the modeled observation window can enter as observed marker values.

The pre-endpoint rule is a bookkeeping principle, not a guarantee that the complete model is correct. It cannot fix a misspecified trajectory, an inadequate hazard model, dependence in measurement errors that has been ignored, or an inappropriate censoring or observation-process assumption. Nor does the rule say that later unobserved marker values are zero or equal to the last observed value: they are simply not observed data for this subject’s likelihood contribution.

Key takeaway: First identify the observed endpoint and the measurement times. Include exactly those marker records satisfying \(t_{ij}<\tilde T_i\) under the strict convention, combine their density with the event or survival factor conditional on random effects, and integrate the resulting joint contribution over the random-effects distribution when forming the marginal likelihood.

Check Your Understanding

Answer each question using the strict pre-endpoint convention. Brief solutions follow the questions.

A subject has an event at month 7 and marker records at months 2, 7, and 9. Which records enter the strictly pre-endpoint factor?
A subject is censored at month 5 with records at months 1, 3, and 6. What is \(n_i^{\mathrm{pre}}\), and which event-process term applies?
Why is the conditional longitudinal product not generally a product over all planned visit times?
Suppose a longitudinal model has correlated errors conditional on \(b_i\). What changes in the likelihood expression, and what does not?

Solutions. (1) Only the month-2 record satisfies \(t_{ij}<7\); the month-7 value is at the boundary and month 9 is after the endpoint. (2) The month-1 and month-3 records enter, so \(n_i^{\mathrm{pre}}=2\); since the endpoint is censored, the event-process term is \(S_i(5\mid b_i)\), without a hazard multiplier. (3) Planned visits after the endpoint do not provide observed marker values during follow-up, so only actual pre-endpoint records contribute measurement densities. (4) Replace the product of individual measurement densities by the joint conditional density of the observed pre-endpoint marker vector. The selection rule \(t_{ij}<\tilde T_i\) and the event or survival factor remain as stated.