Why factor the complete-data likelihood?
A joint model connects a subject’s repeated measurements to their event time, often through subject-specific random effects. Writing down its likelihood as one large expression can hide which part of the model each observation informs. Factoring the likelihood makes those parts visible and provides a clear starting point for likelihood-based estimation.
The key qualification is complete data: in addition to the observed measurements and event information, the data representation includes the subject-specific random effects. Those effects are generally latent in practice. Here they are treated as given arguments of the complete-data density, not integrated out.
The event indicator is \(\delta_i=1\) when the event is observed at \(t_i\), and \(\delta_i=0\) when follow-up ends at \(t_i\) without an observed event. Write \(\theta_Y\), \(\theta_T\), and \(\theta_b\) for parameters governing the longitudinal model, event-time model, and random-effects distribution, respectively. These parameter blocks may share parameters; the labels identify model components, not a requirement that every parameter be distinct.
Assumptions behind the factorization
A standard factorization uses two conditional-independence assumptions. First, given \(b_i\) and covariates, the longitudinal measurements have a specified joint density \(f_Y(Y_i\mid b_i;\theta_Y)\). Second, conditional on \(b_i\) and covariates, assume that the longitudinal measurement process, event time, and censoring time are mutually independent. Their association can still arise through their common dependence on \(b_i\). The random effects have density \(f_b(b_i;\theta_b)\).
For a simple version, assume subjects are independent, the random effects are independent across subjects, and the event time \(T_i\) and censoring time \(C_i\) are conditionally independent given the modeled covariates and random effects. Let \(\theta_C\) denote censoring parameters distinct from the event parameters \(\theta_T\). The censoring mechanism contributes a factor to the exact observed-data density. With \(b_i\) held fixed, this factor can be omitted from the complete-data likelihood kernel for \(\theta_T\) when \(\theta_C\) is distinct; in the observed-data likelihood, retain it when integrating over \(b_i\), unless additional conditions make censoring ignorable. If censoring is informative or shares parameters with the event model, it must be modeled in the likelihood.
Under these assumptions, the subject-level complete-data density includes a censoring contribution as well as the longitudinal, event-process, and random-effects contributions:
Here \(x_i\) and \(z_i\) denote covariates used in the longitudinal and event models. The factor \(f_{T,\Delta}\) is the event-process likelihood contribution, and \(f_{C,\Delta}(t_i,\delta_i\mid b_i,z_i;\theta_C)=S_C(t_i\mid b_i,z_i;\theta_C)^{\delta_i}f_C(t_i\mid b_i,z_i;\theta_C)^{1-\delta_i}\) is the censoring contribution, where \(f_C\) and \(S_C\) are the conditional density and survival function of the censoring time. The exact complete-data likelihood for \(n\) independent subjects is therefore
where \(\theta\) collects the model parameters, including \(\theta_C\). This is a product over subjects and, within each subject, a product of four model contributions. With \(b_i\) held fixed, the censoring factor can be omitted from the complete-data likelihood kernel for \(\theta_T\) because it involves only the distinct censoring parameters. In the observed-data likelihood, retain this factor when integrating over \(b_i\), unless additional conditions make censoring ignorable. The factorization does not assert that the observed longitudinal data and event time are independent without conditioning on \(b_i\). Indeed, integrating over a shared random effect can induce dependence between them.
Why the product follows
The chain rule writes the joint density as a product of conditional densities. One useful ordering is to first specify the random effects, then the longitudinal data and event information conditional on them:
The second equality uses the conditional independence of the longitudinal process, event time, and censoring time given the random effects and covariates. Without those assumptions, the first conditional density need not split into the displayed factors. This is the essential justification: the product is a consequence of the model’s conditional structure, not an algebraic rearrangement that is valid for every joint distribution.
Worked Example: Evaluating Two Complete-Data Contributions
Consider a hypothetical model with one measurement per subject, \(Y_i\mid b_i\sim N(b_i,0.25)\), and \(b_i\sim N(0,0.25)\). Suppose the event hazard is constant over time conditional on \(b_i\), with \(\lambda_i=0.1\exp(0.4b_i)\). For this model, the event-process contribution is \(\lambda_i\exp(-\lambda_i t_i)\) for an observed event and \(\exp(-\lambda_i t_i)\) for censoring at \(t_i\). The example below omits the censoring factor, so the products are event-parameter likelihood kernels rather than exact complete-data likelihoods. Assume the stated conditional-independence assumptions hold.
For subject 1, take \(Y_1=0.5\), \(b_1=0.25\), and an observed event at \(t_1=2\). The longitudinal density is
The random-effects density at \(b_1=0.25\) is
The hazard is \(\lambda_1=0.1\exp(0.4\times0.25)=0.1\exp(0.1)\approx0.1105\). Thus the event-time contribution is
Multiplying the three displayed factors gives the subject’s event-parameter likelihood kernel:
For subject 2, take \(Y_2=0\), \(b_2=0\), and censoring at \(t_2=2\). The longitudinal and random-effects densities are both \(1/(0.5\sqrt{2\pi})\approx0.7979\). The hazard is \(\lambda_2=0.1\), so the event-process contribution for censoring is \(\exp(-0.1\times2)\approx0.8187\). Therefore,
Under subject independence, the product of the two event-parameter likelihood kernels at these specified values is \(K_c=K_{c,1}K_{c,2}\approx(0.04393)(0.5212)=0.02290\). These are likelihood-kernel contributions with the censoring factors omitted, not probabilities assigned to the exact continuous values.
What belongs inside each factor?
The longitudinal factor is the joint density of all measurements for a subject conditional on that subject’s random effects. It is not automatically a product of separate visit densities. If measurement errors are conditionally independent across visits, then it can be written as \(\prod_{j=1}^{n_i} f_Y(Y_{ij}\mid b_i,x_i;\theta_Y)\). If the model specifies serially correlated residual errors, use the corresponding joint density instead.
The event-process factor uses the event/censoring information actually observed. For example, if \(h_i(t\mid b_i)\) is the hazard and \(H_i(t\mid b_i)=\int_0^t h_i(u\mid b_i)\,du\) is the cumulative hazard, the event-process likelihood contribution can be written \(h_i(t_i\mid b_i)^{\delta_i}\exp\{-H_i(t_i\mid b_i)\}\). This is a density contribution when \(\delta_i=1\) and a survival contribution when \(\delta_i=0\), but the exact observed-data density also includes the censoring factor.
The random-effects factor is the density of \(b_i\) under the assumed population distribution, often a multivariate normal distribution. Its presence matters: treating the random effects as part of the complete data does not mean they are fixed covariates with no probability model. Their density is one of the factors in the joint likelihood.
Worked Example: A Correlated Longitudinal Vector
Suppose a hypothetical subject has two measurements and, conditional on \(b_i\), their measurement-error vector has a bivariate normal distribution with covariance matrix
The conditional correlation is \(0.6\), so the two measurements are not conditionally independent. The longitudinal contribution must be the bivariate density \(f_Y((Y_{i1},Y_{i2})^\mathsf{T}\mid b_i;\theta_Y)\), not the product of two univariate normal densities. The full subject contribution is still
Thus the three-part factorization can hold while the longitudinal factor itself remains a joint density across visits. Multiplying marginal visit densities would impose a different covariance structure and generally produce the wrong likelihood.
Common mistakes, diagnostics, and alternatives
- Dropping the random-effects density. The complete-data likelihood includes \(f_b(b_i;\theta_b)\). Omitting it changes the specified joint model.
- Calling the factors marginally independent. Conditional factorization given \(b_i\) does not imply independence after \(b_i\) is unobserved or averaged over.
- Using an event density for a censored observation. A censored subject contributes survival through \(t_i\), not an event at \(t_i\).
- Multiplying visit densities despite residual correlation. Check the longitudinal model’s conditional covariance structure before splitting its joint density by visit.
- Assuming any factorization is harmless. The factorization depends on the specified conditional-independence and censoring assumptions. If residual dependence links the event process to the measurements beyond \(b_i\), or censoring is informative, the displayed factors may be incomplete.
One alternative is to write a single joint conditional density for all observed subject data given \(b_i\), then multiply by the random-effects density. That can be useful when the component assumptions are not naturally separable, but it hides which observations inform which model component. The factorized form is clearer when its assumptions hold. It is also distinct from a likelihood that treats only observed data as data: the latter does not include \(b_i\) as an observed coordinate. Handling the latent effects by integrating them out is a separate construction.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- What does “complete data” add to the observed subject record in a shared-random-effects model?
- Why can longitudinal measurements and event times be associated even when their processes are conditionally independent given \(b_i\)?
- With a constant conditional hazard of \(0.2\) per year, what event-time contribution does a subject censored at 3 years make?
- If two measurements have correlated residual errors conditional on \(b_i\), may their marginal visit densities simply be multiplied to form the longitudinal contribution?
Solutions. (1) It includes the latent random-effects vector \(b_i\) as part of the complete-data representation. (2) Both processes depend on the same random effects, so averaging over those effects can induce association. (3) The survival contribution is \(\exp(-0.2\times3)=\exp(-0.6)\approx0.5488\). (4) No. Use the specified joint longitudinal density; multiplying visit densities would incorrectly impose conditional independence.