Constructing a Conditional Subject Likelihood

Why Construct a Conditional Subject Likelihood?

In a joint model, one subject contributes repeated measurements and an event-time record. Both types of data can depend on the same latent subject-specific random effects. Holding those effects fixed lets us write the subject’s longitudinal and event-time contributions on a common footing: each is a density or likelihood contribution conditional on that same set of effects.

The result is a building block, not yet a likelihood with the random effects removed. In particular, a conditional subject likelihood is a function of the model parameters and the fixed value of the subject’s random effects. It does not average over possible values of those effects.

Model and Assumptions

Suppose subject \(i\) has measurements \(y_{ij}\) at times \(t_{ij}\), for \(j=1,\ldots,n_i\). Write a linear mixed model as \(y_{ij}=x_{ij}^{\mathsf T}\beta+z_{ij}^{\mathsf T}b_i+\varepsilon_{ij}\), where \(\beta\) is a vector of fixed effects, \(b_i\) is a \(q\)-dimensional random-effects vector, and \(x_{ij}\) and \(z_{ij}\) are known design vectors. For the Gaussian example used here, conditional on \(b_i\), the errors are independent with \(\varepsilon_{ij}\sim N(0,\sigma^2)\). Thus the conditional mean is \(m_{ij}(b_i)=x_{ij}^{\mathsf T}\beta+z_{ij}^{\mathsf T}b_i\).

Let \(T_i\) be the event time, \(C_i\) the censoring time, \(Y_i=\min(T_i,C_i)\) the observed time, and \(\delta_i=I(T_i\le C_i)\) the event indicator. Conditional on \(b_i\) and covariates, specify the event hazard \(h_i(t\mid b_i)\), with cumulative hazard \(H_i(t\mid b_i)=\int_0^t h_i(u\mid b_i)\,du\). For example, a current-value association model can use \(h_i(t\mid b_i)=h_0(t)\exp\{w_i^{\mathsf T}\gamma+\alpha m_i(t;b_i)\}\), where \(h_0(t)\) is a baseline hazard, \(w_i\) is a vector of event-model covariates, and \(m_i(t;b_i)\) is the longitudinal model’s latent mean trajectory at time \(t\). The hazard form is part of the specified model; it is not inferred from the observed measurements alone.

The product construction relies on assumptions. The longitudinal errors are conditionally independent across measurement occasions under the stated Gaussian model. Conditional on \(b_i\) and covariates, the longitudinal measurement errors and event-time process are independent, while the event hazard may depend on the latent trajectory through \(b_i\). Censoring is noninformative for the event-time likelihood conditional on the variables being conditioned on, so its distribution contributes no parameter-dependent factor to the likelihood for the event process. The observation schedule is treated as fixed or noninformative under the model. If these assumptions do not hold, the corresponding dependence or observation process may need its own model.

Why the Event-Time Factor Has This Form

For a continuous event time conditional on \(b_i\), the event-time density is the hazard multiplied by the event-free survival probability: \(f_T(t\mid b_i)=h_i(t\mid b_i)S_i(t\mid b_i)\). Since \(S_i(t\mid b_i)=\exp\{-H_i(t\mid b_i)\}\), an observed event at \(Y_i\) contributes \(h_i(Y_i\mid b_i)\exp\{-H_i(Y_i\mid b_i)\}\). A right-censored observation at \(Y_i\) contributes the probability of surviving at least to that time, \(\exp\{-H_i(Y_i\mid b_i)\}\). Both cases are represented by the indicator exponent \(\delta_i\).

The conditional-independence assumption then gives the product of the longitudinal density and this event-time contribution. This is a product of likelihood contributions, not a claim that the observed measurements and event time are unconditionally independent. They can be associated marginally because they share \(b_i\).

Worked Construction

What Is Included, and What Is Not?

The conditional likelihood includes the longitudinal density and the event-time factor evaluated at \(b_i\). It does not include the random-effects density. If the random effects are modeled as \(b_i\sim N(0,D)\), then their density is a separate factor in a likelihood that treats the random effects as latent quantities. The distinction matters: adding that density changes the object from a likelihood conditional on \(b_i\) to a joint contribution involving \(b_i\).

The conditional likelihood also differs from the marginal likelihood for the observed data. The latter accounts for uncertainty about the unobserved random effects rather than treating their values as fixed. Its construction requires an additional operation beyond the conditional product; the conditional contribution above is the integrand-level building block, not the result of that operation.

The subject likelihood may be written on the log scale for computation. Taking logarithms converts the product over measurements and the event-time factor into a sum:

This form is useful for checking calculations: the cumulative hazard enters with a minus sign for both events and censored observations, while the log hazard appears only when an event is observed. The expression presumes a positive hazard at an observed event time.

Diagnostics, Alternatives, and Limitations

Several checks can reveal a construction error. Confirm that every measurement mean uses the same \(b_i\) held fixed for the event contribution. Check that \(H_i(Y_i\mid b_i)\) integrates the hazard over the full interval from the time origin to the observed time, and that its value is nonnegative. For an event, the factor must include the hazard at \(Y_i\); for right censoring, it must not. Units provide another check: a hazard has units of inverse time, whereas a cumulative hazard is dimensionless. The event contribution for an exact event time is a density and therefore has units of inverse time; a censoring contribution is a probability and is dimensionless.

A common mistake is to multiply by the event hazard for every subject, including those who were censored. That treats a censoring time as if an event occurred there. The correction is to use \(h_i(Y_i\mid b_i)^{\delta_i}\): when \(\delta_i=0\), the hazard factor is \(1\), and the contribution is survival through \(Y_i\).

Another mistake is to multiply the longitudinal and event-time contributions while silently assuming they are unconditionally independent. The factorization is conditional on \(b_i\) and the modeled covariates. Because both components depend on the same random effects, they need not be independent after those effects are left unconditioned. A third error is to include the random-effects density in a quantity described as conditional on \(b_i\); keep that factor separate.

The main alternative at the next level of analysis is a marginal likelihood that no longer conditions on the random effects. That alternative targets the observed data without treating each \(b_i\) as a fixed known value, but it is not the conditional subject likelihood constructed here. The conditional expression also depends on the chosen longitudinal distribution, hazard model, association structure, and censoring assumptions. For example, correlated measurement errors, informative visit times, or informative censoring can invalidate the simple product unless those processes are represented appropriately.