Adding Survival to the Multimarker Likelihood

Why Add Survival to the Marker Likelihood?

A subject in a multimarker study contributes more than a set of repeated measurements. The subject may also experience an event during follow-up, or may leave follow-up without an observed event. A joint likelihood must account for both the marker records and what is known about the event time. The distinction matters: an observed event contributes an event-time density, whereas a right-censored record contributes an event-time survival probability.

The multivariate longitudinal likelihood describes the marker observations conditional on the subject’s latent random effects. To add survival, we multiply that marker density by the appropriate event-observation contribution, and then integrate over the latent effects. If censoring depends on those latent effects, its contribution belongs inside that same integral. It cannot generally be discarded just because event and censoring times are conditionally independent given the latent effects.

Conditional Marker, Event, and Censoring Models

Condition throughout on observed covariates \(X_i\). A standard multivariate linear mixed model specifies a multivariate normal marker density conditional on \(b_i\), for example \(Y_i\mid b_i,X_i\sim N\{\mu_i(b_i),\Sigma_i\}\). Here \(\mu_i(b_i)\) is the stacked conditional mean and \(\Sigma_i\) is the covariance matrix for the observed marker vector. It may include covariance across markers within a visit and across observations, as specified by the marker model. The distribution of the random effects is denoted \(f_b(b_i\mid X_i)\); a common specification is \(b_i\sim N(0,D)\).

Write the conditional event hazard as \(\lambda_{T,i}(t\mid b_i,X_i)\). Its corresponding conditional survival function and density are

The event-time contribution for the observed pair \((U_i,\delta_i)\) is therefore

If \(\delta_i=1\), this is the event density at \(U_i\). If \(\delta_i=0\), it is the probability that the event time exceeds the censoring time \(U_i\). A censored observation does not say that the event occurred at \(U_i\).

For an explicit censoring model, let \(f_{C,i}\) and \(S_{C,i}\) be the conditional censoring-time density and survival function. If \(T_i\) and \(C_i\) are conditionally independent given \(b_i,X_i\), the censoring contribution is

This reverses the density/survival pattern in \(L_{T,i}\): an observed event tells us that censoring lasted at least until \(U_i\), whereas a censored record tells us that censoring occurred at \(U_i\). These expressions assume continuous event and censoring times; ties require a convention or a model that allows them.

Deriving the Subject-Level Joint Likelihood

Suppose that, conditional on \(b_i,X_i\), the marker data are independent of the event and censoring times, and that \(T_i\) and \(C_i\) are conditionally independent of each other. The conditional density contribution for all the observed data on subject \(i\) is then the product of the three components. Because \(b_i\) is not observed, integrate that product over its distribution:

This is the subject-level observed-data likelihood contribution. The sample likelihood is \(\prod_i L_i\), assuming subjects are independent conditional on the model parameters and covariates. The integral is essential: it averages the conditional likelihood over the random-effects distribution rather than treating a subject’s latent effects as known.

The factorization follows directly from the conditional density. For example, if \(\delta_i=1\), the observed time information is \(T_i=U_i\) and \(C_i\ge U_i\). Under conditional independence, its density contribution is \(f_{T,i}(U_i\mid b_i,X_i)S_{C,i}(U_i\mid b_i,X_i)\). If \(\delta_i=0\), it is \(T_i>U_i\) and \(C_i=U_i\), giving \(S_{T,i}(U_i\mid b_i,X_i)f_{C,i}(U_i\mid b_i,X_i)\). Multiplying by the conditional marker density and the random-effects density, then integrating, gives the displayed likelihood.

A censoring contribution may be factored out only under a suitable ignorability condition. For instance, suppose \(C_i\) is independent of \((T_i,b_i,Y_i)\) conditional on \(X_i\), and the censoring parameters are distinct from the event and marker parameters. Then \(L_{C,i}\) is a function of observed data and censoring parameters, not \(b_i\), and

For inference on the event and marker parameters, the factored censoring term can then be omitted from the likelihood if the censoring parameters are nuisance parameters. Fixed administrative follow-up may likewise be conditioned on as part of the study design. By contrast, conditional independence of \(T_i\) and \(C_i\) given \(b_i,X_i\) is not enough: if censoring depends on \(b_i\), the observed censoring record can change the latent-effects weighting inside the integral.

Assumptions, Diagnostics, and Alternatives

The likelihood derivation relies on assumptions that should be checked against the study design and data-generating process. The conditional marker density must represent the joint distribution of the observed marker vector, including relevant covariance across markers and visits. The event model must provide a valid nonnegative hazard and the implied survival function. The product form also requires the stated conditional independence of marker data from the event and censoring times given the latent effects and covariates, plus conditional independence between event and censoring times if their contributions are modeled separately.

Useful diagnostics include checking whether censoring patterns vary across observed covariate groups or marker histories, comparing the assumed censoring model with observed follow-up patterns, and examining whether event-model and marker-model residual or fit diagnostics reveal systematic lack of fit. Such checks cannot prove independence from unobserved random effects. If censoring plausibly depends on latent health status represented by \(b_i\), the censoring process should be modeled conditionally on those effects, with its likelihood contribution retained inside the integral. If event and censoring times remain dependent even after conditioning on the modeled latent quantities and covariates, the simple product of separate event and censoring terms is not justified; a joint model for their dependence is needed.

A common alternative for event-parameter inference is to use the marker-and-event integral alone. This is appropriate only when censoring is ignorable for the parameters of interest under assumptions such as independence of censoring from latent effects and event time conditional on observed covariates, with a distinct censoring mechanism. It is not licensed merely by assigning separate censoring parameters. Including a censoring likelihood is more explicit and can accommodate latent-dependent censoring, but requires a credible censoring model and can add substantial estimation difficulty. Neither approach repairs misspecification of the marker covariance or event model.

A frequent mistake is to reason, “Event and censoring times are independent given the random effects, so the censoring likelihood can be omitted.” The correction is that the random effects are integrated out, not observed. If censoring depends on them, \(L_{C,i}(b_i)\) changes the weights assigned to different \(b_i\) values inside the integral. The censoring term may be omitted from event-parameter inference only under an ignorability condition that allows it to factor out, or under an explicitly adopted assumption that it is irrelevant to the target likelihood.