Likelihood Construction For Joint Longitudinal-Survival Models

Why the Likelihood Must Integrate Over Shared Effects

A subject in a longitudinal-survival study may contribute several omics measurements and either an observed event time or a censoring time. A shared-random-effects model links these sources of information: the same latent subject-specific effects help describe the repeated measurements and the event hazard. The likelihood for that subject must therefore combine both sources conditional on the latent effects, then average over their distribution.

Multiplying a longitudinal likelihood by a survival likelihood evaluated at an estimated random effect is generally not the joint likelihood. It ignores uncertainty about that effect and can distort the contribution, especially when the measurements and event time both provide information about it.

Conditional Models and Their Contributions

Write the longitudinal trajectory as \(m_i(t;b_i)=x_i(t)^\mathsf{T}\beta+z_i(t)^\mathsf{T}b_i\), where \(x_i(t)\) and \(z_i(t)\) are known covariate vectors, \(\beta\) is a fixed-effect vector, and \(b_i\sim N(0,D)\). A common measurement model is

with conditionally independent measurement errors across visits. Thus the conditional longitudinal density is the product of the normal densities. Other measurement distributions can be used when appropriate; the likelihood construction is the same, with those densities substituted.

For illustration, take a proportional hazards model whose hazard depends on the current latent trajectory:

Here \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of event-model covariates, and \(H_i(t\mid b_i)\) is the cumulative hazard. Conditional on \(b_i\), the event-time contribution for the observed time and event indicator is \(h_i(T_i\mid b_i)^{\delta_i}\exp\{-H_i(T_i\mid b_i)\}\). The hazard factor appears only for an observed event; the survival factor appears for both an event and a censoring time.

Deriving the Subject-Level Joint Likelihood

Conditioning on \(b_i\) makes the factorization possible: the repeated-measurement density multiplies the event-time contribution. Since \(b_i\) is unobserved, the observed-data contribution averages that conditional density over the random-effects distribution.

This follows by the law of total probability: the joint density of observed data is the integral of its conditional density given \(b_i\), multiplied by the density of \(b_i\). The conditional-independence assumptions justify the product inside the integral. Crucially, the integration is over the product of the longitudinal and event contributions. In general, integrating each factor separately and multiplying the resulting marginal contributions would not give the same answer.

A useful equivalent form makes the role of the measurements explicit. Define \(L_{Yi}=\int [\prod_j f_Y(Y_{ij}\mid b_i)]f_b(b_i)\,db_i\), the marginal longitudinal density. The posterior density of \(b_i\) given the measurements is proportional to the longitudinal density times its prior density. Therefore

The expectation is over the distribution of \(b_i\) given \(Y_i\), not over the prior distribution alone. This identity is also a practical way to organize numerical calculations: first account for the longitudinal data, then average the conditional event contribution using the measurement-informed distribution of the random effects.

Numerical Integration and Censoring

Except in special cases, the integral over \(b_i\) has no closed form. Gaussian quadrature, adaptive quadrature, or Monte Carlo integration can approximate it. Whatever the numerical method, the target remains the same integral: each evaluation must include the conditional longitudinal density, the event contribution, and the random-effects density. The posterior-expectation form can be convenient when the longitudinal integral and its induced distribution for \(b_i\) are available.

Checking the Construction and Choosing a Calculation

Before fitting the model, check that each factor matches the data and assumptions. A useful construction sequence is:

The main computational alternatives differ in how they evaluate the integral. Gaussian quadrature is often efficient for a low-dimensional random-effect vector, while Monte Carlo methods can be more flexible as its dimension grows but may require many draws for stable estimates. A plug-in calculation evaluates the event contribution at an estimated \(b_i\); it is simpler but does not integrate over uncertainty and is not generally equivalent to the marginal likelihood. A conditional likelihood that treats \(b_i\) as known likewise answers a different calculation unless the effects truly are observed.

Diagnostics include comparing likelihood estimates across quadrature orders or integration tolerances, checking convergence from different initial values, and inspecting whether the fitted model reproduces the observed measurement patterns and event-time distribution. A rapidly changing hazard requires careful cumulative-hazard integration; using a constant-hazard shortcut when the model specifies a time-varying hazard is an error. Sensitivity to the random-effects distribution or to the assumed measurement-error structure can also signal that the likelihood is not capturing the data adequately.

• Common mistake: multiplying by the endpoint hazard for censored subjects. A censored subject has \(\delta_i=0\), so the event contribution is survival through the censoring time, not a hazard times survival.
• Common mistake: replacing \(H_i(T_i\mid b_i)\) with \(T_i h_i(T_i\mid b_i)\). That is valid only for a constant hazard over the interval. In general, \(H_i\) is the integral of the hazard over all times up to \(T_i\).
• Common mistake: integrating the two data sources separately and multiplying. Because both depend on the same latent \(b_i\), the correct operation integrates their product, or equivalently averages the event contribution under the measurement-informed distribution of \(b_i\).

The likelihood relies on the stated conditional-independence and censoring assumptions. If visit times or censoring are informative beyond the modeled quantities, or if measurements after an event are structurally unavailable in a way not addressed by the model, additional modeling may be needed. The likelihood construction alone does not establish that those assumptions are appropriate for a particular study.