Subject-Level Likelihood Contributions in Joint Models

Why Construct a Subject-Level Contribution?

A joint model connects repeated biomarker measurements with an event process, such as time to disease progression or death. The subject-level likelihood contribution makes that connection explicit: it records how well the model accounts for all the observed data from one subject, conditional on that subject’s shared random effects. This conditional contribution is a building block for inference, but it is not by itself the likelihood after accounting for the distribution of the random effects.

The key is to keep three pieces distinct: the longitudinal measurement density, the event-time or censoring contribution, and the assumptions that allow those pieces to be multiplied. The shared random effects link the two outcomes by entering both the biomarker trajectory and the event hazard.

Model and Assumptions

Write the longitudinal model’s conditional mean as \(m_i(t\mid b_i)\). For a simple Gaussian specification, suppose \(Y_{ij}=m_i(t_{ij}\mid b_i)+\epsilon_{ij}\), where the errors are conditionally independent and \(\epsilon_{ij}\sim N(0,\sigma^2)\). Conditional independence here means that, given \(b_i\), the covariates, and the model parameters, the measurement errors at the observed times are independent. Other longitudinal models can be used, but their appropriate joint density must replace the Gaussian product below.

Let \(h_i(t\mid b_i)\) be the event hazard conditional on the shared random effects and covariates. Define the cumulative hazard and conditional survival function by \(H_i(t\mid b_i)=\int_0^t h_i(u\mid b_i)\,du\) and \(S_i(t\mid b_i)=\exp\{-H_i(t\mid b_i)\}\). The hazard must be nonnegative and integrable over the time interval of interest.

The usual event-time contribution assumes that, conditional on the random effects and modeled covariates, the event process has the specified hazard. It also assumes that censoring is noninformative for the event-time likelihood: censoring does not add information about the event time after conditioning on the modeled variables. If the censoring process itself is modeled, its likelihood must be handled separately. The standard factorization further assumes that the longitudinal measurements and event process are conditionally independent given the shared random effects and covariates. Their marginal association is then induced through the shared effects.

The survival factor appears for both observed events and right-censored observations. For an observed event, the continuous event-time density is \(f_i(t\mid b_i)=h_i(t\mid b_i)S_i(t\mid b_i)\). For right censoring at time \(c\), the observation tells us only that the event time exceeds \(c\), which contributes \(P(T_i>c\mid b_i)=S_i(c\mid b_i)\). Combining these cases gives \(h_i(\tilde T_i\mid b_i)^{\delta_i}S_i(\tilde T_i\mid b_i)\).

For conditionally independent Gaussian measurement errors, the longitudinal part becomes \(\prod_j (2\pi\sigma^2)^{-1/2}\exp[-\{Y_{ij}-m_i(t_{ij}\mid b_i)\}^2/(2\sigma^2)]\). A subject with no recorded measurements has no such product contribution (the empty product is 1); this does not remove that subject’s event-time contribution. Conversely, a subject censored before any measurement still contributes survival information.

Worked Examples

Why the Product Is Justified—and When It Is Not

The factorization follows from the conditional-independence assumption. Let \(\mathbf Y_i\) denote all of subject \(i\)’s observed longitudinal measurements and \(D_i=(\tilde T_i,\delta_i)\) the observed event data. If \(\mathbf Y_i\) and \(D_i\) are conditionally independent given \(b_i\) and covariates, then \(f(\mathbf Y_i,D_i\mid b_i)=f(\mathbf Y_i\mid b_i)f(D_i\mid b_i)\). Substituting the longitudinal density and the event/censoring likelihood derived above yields \(L_i=L_{Yi}L_{Ti}\). This is a statement about the conditional data model, not a claim that the two outcomes are marginally independent.

The assumption is inappropriate if dependence remains after conditioning on the shared effects and modeled covariates. For example, measurement errors may have serial correlation not captured by the longitudinal model, or an unmodeled visit process may be related to health status. In such cases, multiplying independent Gaussian measurement densities and the usual event contribution does not represent the stated data-generating process; the relevant dependence or observation process must be modeled.

The event contribution also depends on the hazard definition. If the hazard contains a time-varying biomarker trajectory, such as \(h_i(t\mid b_i)=h_0(t)\exp\{\alpha m_i(t\mid b_i)\}\), then the current trajectory value enters the hazard at the event time, while the entire trajectory over follow-up affects the cumulative hazard integral. Replacing the integral by a single hazard value times follow-up time is generally wrong unless the hazard is constant over that interval.

Diagnostics, Alternatives, and Limitations

Check a constructed contribution before using it in computation. Confirm that each measurement density uses the observed value and its matching time; verify that the event indicator is coded as 1 for an observed event and 0 for right censoring; and check that the cumulative hazard is integrated from the time origin to the observed time. For a nonnegative hazard, \(H_i(t\mid b_i)\) must be nonnegative and nondecreasing, and the resulting survival probability must lie between 0 and 1. Numerical integration should be checked for stability when the hazard has no simple antiderivative.

A useful computational check is the log contribution:

This form avoids underflow when many densities or small event contributions are multiplied. It also makes a sign error in the cumulative-hazard term easier to detect: survival contributes \(-H_i\), not \(+H_i\). An event-time density has units of inverse time, whereas a censoring survival contribution is dimensionless; the different units are expected.

The main alternatives change the data model, not merely the arithmetic. One can specify correlated longitudinal errors and use their joint multivariate density instead of a product of independent densities. One can explicitly model informative censoring or the visit process when the assumptions for omitting those mechanisms do not hold. A model that omits shared random effects may treat longitudinal and event outcomes as unrelated conditional on covariates, but then it does not represent their association through subject-specific effects. Each alternative requires assumptions suited to the scientific and observation processes.

A frequent limitation is that shared random effects capture only the dependence represented by their chosen structure. A random intercept alone, for example, cannot represent every possible subject-specific trajectory difference. Further, the conditional contribution is not the integrated subject-level likelihood: it conditions on \(b_i\). The distribution of random effects and the treatment of unknown parameters are separate ingredients. Keeping this distinction clear avoids interpreting a conditional contribution as a complete marginal likelihood or as a posterior distribution.