The Joint Contribution Under Right Censoring

Why a Subject-Level Contribution Needs Both Data Sources

In a joint model, one subject can contribute repeated biomarker measurements and information about when an event occurred. A subject who experiences the event contributes an event-time density; a subject whose follow-up ends first contributes information that the event had not happened by the censoring time. In either case, the event-process contribution must be combined with the subject’s longitudinal data contribution. The shared random effects link these pieces: conditional on those effects, the model specifies both the biomarker trajectory and the event hazard.

The task here is to assemble that contribution correctly. In particular, right censoring does not mean that the event time is known to equal the censoring time. It means only that the event time exceeds the observed follow-up time. That distinction determines which event-process term appears in the likelihood.

Definition: For subject \(i\), let \(y_i=(y_{i1},\ldots,y_{in_i})\) denote the repeated biomarker values observed at the subject’s measurement times. Let \(T_i^*\) be the event time, \(C_i\) the censoring time, \(\tilde T_i=\min(T_i^*,C_i)\) the observed follow-up time, and \(\delta_i=I(T_i^*\leq C_i)\) the event indicator. Thus \(\delta_i=1\) means the event was observed, while \(\delta_i=0\) means the subject was right-censored.

Write \(b_i\) for the subject’s shared random effects and \(x_i\) for the relevant observed covariates. Let \(f_Y(y_i\mid b_i,x_i)\) be the joint density of all the subject’s repeated measurements, conditional on the random effects and covariates. Let \(h_i(t\mid b_i,x_i)\) be the event hazard and let

H_i(t\mid b_i,x_i)=\int_0^t h_i(u\mid b_i,x_i)\,du,\qquad S_i(t\mid b_i,x_i)=\exp\{-H_i(t\mid b_i,x_i)\}.

The hazard is an instantaneous event rate, and \(H_i\) is the cumulative hazard. The survival function \(S_i(t\mid b_i,x_i)\) is the probability, conditional on the random effects and covariates, of remaining event-free through time \(t\).

Assumptions Behind the Factorization

The subject-level contribution below relies on specific assumptions. First, conditional on \(b_i\) and \(x_i\), the model treats the longitudinal measurement process and the event process as independent. Their marginal association is represented through their shared dependence on \(b_i\). Second, the specified joint density \(f_Y\) must account for dependence among repeated measurements. If the measurements have correlated residual errors, for example, \(f_Y\) is their multivariate density; it should not be replaced by a product of univariate densities unless conditional independence of those errors is part of the model.

Third, censoring is assumed independent of the event process and the longitudinal outcomes, conditional on the modeled covariates, with a censoring mechanism that is ignorable for the parameters of interest. Under this assumption, the censoring distribution contributes no parameter-dependent factor to the likelihood being used. If censoring is informative, or its distribution depends on unobserved random effects in a way that carries information about those effects, its contribution may need to be modeled rather than discarded.

Finally, the visit and measurement schedule is treated as fixed or ignorable under the model. If visit timing or dropout depends on unmodeled health status, the observed biomarker values alone may not have the stated density. These assumptions are substantive, not merely algebraic: the formula is justified only when they reasonably describe how the data were generated.

Theorem: Under the conditional independence and censoring assumptions above, the likelihood contribution for subject \(i\), conditional on \(b_i\), is

L_i(b_i) = f_Y(y_i\mid b_i,x_i) \left\{h_i(\tilde T_i\mid b_i,x_i)\right\}^{\delta_i} S_i(\tilde T_i\mid b_i,x_i).

When \(\delta_i=1\), this is \(f_Y(y_i\mid b_i,x_i)h_i(\tilde T_i\mid b_i,x_i)S_i(\tilde T_i\mid b_i,x_i)\): the longitudinal density multiplied by the event-time density at the observed event time. When \(\delta_i=0\), the hazard factor is raised to zero and the contribution is \(f_Y(y_i\mid b_i,x_i)S_i(\tilde T_i\mid b_i,x_i)\): the longitudinal density multiplied by the probability of surviving beyond the censoring time.

This is a likelihood contribution conditional on the random effects, not yet the marginal contribution based only on observed data. If \(b_i\) has density \(f_b(b_i)\), the marginal contribution is obtained by averaging over its distribution:

L_i = \int f_Y(y_i\mid b_i,x_i) \left\{h_i(\tilde T_i\mid b_i,x_i)\right\}^{\delta_i} S_i(\tilde T_i\mid b_i,x_i) f_b(b_i)\,db_i.

Why the Event and Censoring Terms Differ

Conditional on \(b_i,x_i\), the event-time density is \(f_T(t)=h_i(t)S_i(t)\). This follows from \(h_i(t)=f_T(t)/S_i(t)\), or equivalently from \(f_T(t)=-dS_i(t)/dt\) and \(dS_i(t)/dt=-h_i(t)S_i(t)\). Thus an observed event at \(\tilde T_i\) contributes \(h_i(\tilde T_i)S_i(\tilde T_i)\).

For a censored subject, the observation tells us \(T_i^*>\tilde T_i\), not \(T_i^*=\tilde T_i\). Its event-process likelihood is therefore \(P(T_i^*>\tilde T_i\mid b_i,x_i)=S_i(\tilde T_i\mid b_i,x_i)\), with no hazard factor. Combining each of these two event-process terms with \(f_Y\) gives the stated formula. This derivation also explains why replacing a censored subject’s survival contribution with an event density at the censoring time is incorrect.

Worked Example: An Observed Event

Suppose a hypothetical subject has two biomarker values, \(10.2\) and \(9.8\), with conditional means \(10+b_i\), independent normal measurement errors with standard deviation \(0.5\), and \(b_i=0.2\). The subject experiences the event at year 3. At this random-effect value, suppose the hazard is constant at \(0.08\) per year over follow-up. Calculate the conditional subject contribution.

The conditional means are both \(10.2\). The standardized residuals are \(0\) and \(-0.8\). The longitudinal density is therefore

f_Y = \left(\frac{1}{0.5\sqrt{2\pi}}e^{-0^2/2}\right) \left(\frac{1}{0.5\sqrt{2\pi}}e^{-(-0.8)^2/2}\right) = 0.7979\times0.5794 \approx0.4623.

For a constant hazard of \(0.08\) per year, \(H(3)=0.08(3)=0.24\), so \(S(3)=e^{-0.24}\approx0.7866\). Because the event is observed, \(\delta_i=1\), and the event-time factor is

h(3)S(3)=0.08e^{-0.24}\approx0.06293.

The conditional joint contribution is

L_i(b_i)=0.4623\times0.06293\approx0.02909.

This contribution includes both the repeated biomarker density and the event-time density. The event-time factor has units of inverse years; the longitudinal density has units determined by the biomarker measurement scale.

Worked Example: A Right-Censored Subject

Consider a different hypothetical subject with biomarker values \(4.1\) and \(3.9\). Conditional on the random effects, suppose both have mean \(4\), independent normal errors with standard deviation \(0.2\), and the event hazard is constant at \(0.12\) per month. The subject is event-free when observation ends at month 4. Calculate the conditional contribution.

Each standardized residual is either \(0.5\) or \(-0.5\). Each normal density is

\frac{1}{0.2\sqrt{2\pi}}e^{-0.5^2/2} \approx1.7603.

Thus \(f_Y\approx1.7603^2=3.0987\). The subject is censored, so \(\delta_i=0\). The event-process term is survival through month 4, not the hazard at month 4:

S(4)=e^{-0.12(4)}=e^{-0.48}\approx0.6188.

The joint conditional contribution is \(3.0987\times0.6188\approx1.917\). A density can exceed one when its measurement scale is sufficiently small; this is not a probability and does not need to be at most one.

Conditional Contributions and Marginal Contributions

In a shared-random-effects model, the two data sources are conditionally independent given \(b_i\), but they are generally associated after \(b_i\) is integrated out. For likelihood calculations based only on observed data, the contribution is the integral of the conditional contribution against the random-effects distribution. It is not generally correct to substitute an average random effect into the conditional formula. In general, the average of a function of \(b_i\) is not that function evaluated at the average of \(b_i\).

The following small example makes the distinction concrete. A two-point random-effects distribution is used only to make the integration arithmetic transparent; it is not a claim that this is the usual distributional choice in a joint model.

Worked Example: Averaging Over a Shared Random Effect

Suppose \(b_i=-1\) or \(b_i=1\), each with probability \(0.5\). The hypothetical repeated measurements are \(y_i=(0,1)\), conditionally independent normal values with variance \(1\) and mean \(b_i\). The subject is censored at year 2, and the conditional hazard is constant at \(0.1e^{b_i}\) per year. Find the marginal subject contribution.

At \(b_i=-1\), the two normal densities are \(\phi(1)\approx0.2420\) and \(\phi(2)\approx0.0540\), so \(f_Y\approx0.01306\). The hazard is \(0.1e^{-1}\approx0.03679\) per year, giving survival \(e^{-2(0.03679)}\approx0.9291\). The conditional joint contribution is approximately \(0.01306(0.9291)=0.01214\).

At \(b_i=1\), the normal densities are \(\phi(-1)\approx0.2420\) and \(\phi(0)\approx0.3989\), so \(f_Y\approx0.09653\). The hazard is \(0.1e\approx0.2718\) per year, giving survival \(e^{-2(0.2718)}\approx0.5806\). The conditional joint contribution is approximately \(0.09653(0.5806)=0.05605\).

Weighting each value by its probability gives the marginal contribution:

L_i = 0.5(0.01213)+0.5(0.05604) \approx0.03409.

The contributions are averaged after multiplying the longitudinal and survival terms at each possible random effect. Averaging the two components separately and then multiplying would generally give a different answer.

Assembly Checks, Alternatives, and Limitations

Before using a subject contribution, verify that its pieces match the data actually observed. Check that an event indicator of one selects the hazard-times-survival term, while an indicator of zero selects survival alone. Check that the cumulative hazard runs from the appropriate time origin to the observed follow-up time, and that the longitudinal density includes all measured values with their modeled dependence. Confirm that the hazard’s units are inverse time and that its time scale matches \(\tilde T_i\).

A separate longitudinal and event-time analysis is a simpler alternative, but it does not use the shared random-effects structure to represent their association. Multiplying separate marginal likelihoods as if the two outcomes were independent can therefore misrepresent the joint data-generating model. A joint model with an explicitly modeled censoring process is another alternative when censoring is informative; the simple contribution above is not adequate unless the relevant censoring mechanism is accounted for.

The contribution also depends on the chosen longitudinal distribution, hazard specification, and association structure. A correctly assembled likelihood does not protect against a misspecified biomarker model, an unsuitable hazard, an inadequate random-effects distribution, or informative visit timing. These are modeling limitations to assess with subject-matter reasoning and diagnostics, not errors that can be repaired by changing the censoring indicator.

Pitfall: treating censoring as an event. The incorrect reasoning is “follow-up ended at month 4, so the event-time density is evaluated at month 4.” The correction is that censoring establishes only \(T_i^*>4\); the contribution is \(S_i(4)\), with no hazard multiplier.
Pitfall: multiplying marginal pieces. The incorrect reasoning is to integrate the biomarker and event terms separately and multiply their results. When both depend on the same random effects, integrate their product against the shared random-effects distribution.
Pitfall: multiplying repeated measurement densities without checking. This is justified only when the conditional measurement model gives the required independence. Otherwise use the specified joint longitudinal density.

Key takeaway: Under the stated assumptions, the subject-level contribution is the longitudinal density multiplied by \(h_i(\tilde T_i)^{\delta_i}S_i(\tilde T_i)\), conditional on the shared random effects. For observed-data likelihood, average that entire product over the random-effects distribution.

Check Your Understanding

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

A subject is censored at time 5. What event-process term belongs in the contribution, and why is the hazard at time 5 absent?
For an observed event at time \(t\), write the event-process contribution using \(h(t)\) and \(H(t)\).
Why can the longitudinal and event contributions be multiplied conditional on the random effects, yet remain associated marginally?
A constant hazard is \(0.05\) per month and a subject is censored at month 6. Calculate the event-process contribution.

Solutions. (1) The term is \(S(5)=\exp\{-H(5)\}\), because censoring says the event time exceeds 5; it does not establish an event at 5. (2) The event-time density is \(h(t)S(t)=h(t)\exp\{-H(t)\}\). (3) Conditional independence given the shared random effects justifies multiplying the conditional densities. After integrating over random effects, both pieces depend on the same latent value, inducing marginal association. (4) Here \(H(6)=0.05(6)=0.30\), so the contribution is \(S(6)=e^{-0.30}\approx0.7408\).