Placing Covariates in Both Submodels

Why Covariate Placement Matters

In a joint model, a covariate can be associated with the longitudinal marker, the event hazard, or both. These placements answer different questions. A covariate in the longitudinal submodel changes the modeled marker trajectory. A covariate in the event submodel changes the hazard directly, conditional on the marker value and shared random effects. When the same covariate appears in both, the two effects have separate coefficients and distinct interpretations.

For example, a hypothetical treatment could be associated with a lower marker level over time and also with a lower event hazard among subjects with the same marker value. A model that includes treatment only in the longitudinal submodel does not estimate that additional conditional hazard association. Conversely, including treatment only in the event submodel does not allow treatment to shift the modeled marker trajectory.

Writing Both Submodels

Let \(y_{ij}\) be the longitudinal measurement for subject \(i\) at time \(t_{ij}\). Write the latent marker trajectory as \(m_i(t)\), the subject-specific marker value at time \(t\) before measurement error. A standard linear mixed model for that trajectory is

Here \(x_i(t)\) contains covariates and time terms for the fixed part of the trajectory, \(\beta\) contains their longitudinal coefficients, \(z_i(t)\) describes the random-effects structure, and \(b_i\) is the subject-specific random-effects vector. In the usual basic model, \(b_i\sim N(0,D)\) and the measurement errors are independent with \(\epsilon_{ij}\sim N(0,\sigma^2)\), independently of \(b_i\). The trajectory \(m_i(t)\) is latent; an observed measurement also includes \(\epsilon_{ij}\).

For a current-value association, a corresponding event submodel is

Here \(h_0(t)\) is the baseline hazard, \(w_i(t)\) contains covariates with direct event-submodel effects, and \(\gamma\) contains their coefficients. The parameter \(\alpha\) links the latent current marker value to the hazard. The hazard is conditional on the shared random effects through \(m_i(t)\), as well as on the event covariates. If a covariate is in both submodels, it appears in \(x_i(t)\) and \(w_i(t)\), with a coefficient in \(\beta\) and a separate coefficient in \(\gamma\).

The standard assumptions include an adequately specified mean trajectory and hazard structure, an appropriate random-effects distribution, and conditional independence of the longitudinal measurement errors and event process given the shared random effects and covariates. The event submodel also imposes proportional hazards conditional on those quantities: a covariate’s log-hazard contribution is additive in the exponent. These assumptions make the conditional model interpretable; they do not guarantee that a chosen covariate set or functional form is appropriate for a particular application.

How the Two Effects Combine

Suppose \(A_i\) is a time-invariant covariate that appears in both submodels. To see its contributions clearly, consider a model in which it shifts the trajectory by \(\beta_A A_i\), and its direct event effect is \(\gamma_A A_i\). Compare \(A_i=1\) with \(A_i=0\) at the same time, holding other covariates and the random effects fixed. The trajectory contrast is \(\beta_A\). Because the hazard also depends on the current marker, the log-hazard contrast includes both contributions.

This contrast assumes the covariate enters the trajectory as the stated additive shift and that the other model terms are held fixed. The direct event-submodel hazard ratio, comparing subjects at the same marker value, is \(\exp(\gamma_A)\). The contrast allowing the marker trajectory to differ is \(\exp(\gamma_A+\alpha\beta_A)\). It is a conditional model contrast, not automatically a population-average hazard ratio or a causal effect.

Covariates in Only One Submodel

A covariate need not appear in both submodels. If it appears only in the longitudinal model, it can still be associated with the event hazard through the latent marker link \(\alpha m_i(t)\). If it appears only in the event model, it has a conditional event association but does not shift the modeled trajectory. The placement should reflect the scientific model being specified, not a rule that every available covariate must occur twice.

Likelihood, Diagnostics, and Limitations

The covariate placements are reflected in different parts of the subject’s conditional likelihood. For a subject with event or censoring time \(T_i\), event indicator \(\delta_i\), longitudinal measurements \(y_{ij}\), and random effects \(b_i\), a standard conditional contribution has the form

where \(f\) is the measurement density and \(S_i\) is conditional event-free survival. The longitudinal covariate effects enter through the measurement model and \(m_i(t)\); direct event covariate effects enter through \(h_i(t\mid b_i)\) and the associated survival. The shared latent trajectory also affects the event contribution through the current-value link. The marginal subject likelihood averages this conditional contribution over the random-effects distribution. This explains why coefficients from the two submodels are not interchangeable: they affect distinct components, even when they refer to the same covariate.

Useful checks begin with the specification. Confirm that a covariate intended to have both effects is actually present in both equations, with separate coefficients and appropriate units. Check that categorical reference groups and continuous-variable scaling are stated consistently. Examine whether the marker trajectory form captures its observed time pattern and whether the assumed hazard relationship is plausible. Large standard errors, unstable estimates, or strong changes under modest changes in covariate specification can indicate weak information or collinearity—for example, a covariate closely aligned with the marker trajectory.

A common mistake is to interpret \(\gamma_A\) as the whole hazard contrast when \(A_i\) also shifts \(m_i(t)\). That coefficient describes the direct event-submodel contrast at a fixed marker value. When the marker differs because of the longitudinal effect, the conditional log-hazard contrast in the additive model also contains \(\alpha\beta_A\). The reverse mistake is to call \(\exp(\gamma_A+\alpha\beta_A)\) the direct effect: it includes the marker pathway. Both interpretations depend on the specified model and the stated conditioning.

Finally, these model coefficients are not automatically causal effects. A treatment coefficient can reflect confounding if treatment assignment is not adequately addressed, and an association between a covariate and the marker or event does not establish causation. The model also relies on its trajectory, hazard, random-effects, and conditional-independence assumptions. Incorrect functional forms, omitted relevant variables, or a poorly represented random-effects distribution can make the resulting contrasts misleading. Joint modeling accounts for the shared latent marker structure; it does not remove the need to assess whether the scientific and statistical assumptions are defensible.